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Onset and amplitude of sidebranches in three dimensional growth of xenon dendrites
Materials Science and Engineering A 413–414 (2005) 447–450
Onset and amplitude of sidebranches in three dimensional growth of xenon dendrites O. Wittwer∗ , J.H. Bilgram Laboratorium f¨ur Festk¨orperphysik, Eidgen¨ossische Technische Hochschule, CH-8093 Z¨urich, Switzerland Received in revised form 1 July 2005
Abstract The growth of sidebranches of xenon dendrites has been studied in a 3D experiment. Sidebranches grow at the ridges of the four fins of the nonaxisymmetric crystal. They can be initiated by selective amplification of thermal noise (type N) or by a macroscopic perturbation (type P). Both types of sidebranches have been identified in our experiments. Type N sidebranches start to grow 3–7 tip radii behind the tip. The sidebranches growing at the four fins are not correlated and the amplitudes measured in our experiments are in quantitative agreement with predictions of Brener and Temkin. Type P sidebranches start to grow at the tip. They are highly correlated at the four fins. © 2005 Elsevier B.V. All rights reserved. PACS: 68.70.+w; 64.70.Dv; 81.10.Fq Keywords: Structure formation; Sidebranching; Solidification; Dendritic growth
1. Introduction Applications in automotive industry call for aluminum castings to be mass produced without casting defects. Predictive models are necessary to prevent casting defects. During the growth of a solid from its undercooled parent melt microstructures are formed. The best known case of such structures are dendrites. Microstructural changes can be observed as undercooling is increased, namely the typical length scales like the tip radius decrease with increasing undercooling [1]. These are quantitative changes. In addition to that, for some materials it has been observed that they also undergo an abrupt decrease in the micro scale. This effect is known as spontaneous grain refinement. Evidence has been found that spontaneous grain refinement is initiated by changes in the morphology of the microscopic structures. We perform 2D/3D-numerical simulations and experiments which allow in situ investigations of three-dimensional growth of xenon crystals into undercooled pure melt. Dendrites, seaweed, doublons, triplons etc. can be produced in experiments and numerical studies depending on initial conditions. In an interplay of numerical studies and experiments we use simulations to interpret and to plan experiments. We use in our experiments ∗
Corresponding author. Tel.: +41 44 633 22 97; fax: +41 44 633 12 31. E-mail address: [email protected] (O. Wittwer).
0921-5093/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2005.08.189
xenon as a transparent model substance for metals. Experimental results are used as a basis of model calculations and the development of models of complex shapes. Three-dimensional growth shapes of crystals are reconstructed using sophisticated image processing combined with experimentally determined shape parameters. In this paper we focus at the origin of sidebranches. We have observed two types of sidebranches: “type N” which are initiated by the selective amplification of thermal noise and “type P” which are initiated by macroscopic perturbations of the dendrite tip. In the following, all lengths are scaled with tip radius (Eq. (2)). The initiation of type N sidebranches has been studied in some detail, for a review see [2]. An important step forward has been made by Brener and Temkin [3,4], who have determined the shape of the tip and the sidebranch amplitude of the nonaxisymmetric needle crystal analytically. The sidebranch amplitude has been found to be 2(5/3)9/10 2/5 √ AB.–T. (z) = ξ12 (z)1/2 ∼ S¯ exp (1) z 3 3σ ∗ where S¯ 2 = 2kB T 2 cl Dth /(L2 vtip R4tip ) is the dimensionless noise or fluctuation strength and σ ∗ = 2Dth d0 /(vtip R2tip ) the so called stability constant with the capillary length d0 = γsl cl Tm /L2 . Where Dth , vtip , Rtip , γsl , cl , Tm , L are the thermal diffusivity,
O. Wittwer, J.H. Bilgram / Materials Science and Engineering A 413–414 (2005) 447–450
448
the tip velocity, the tip radius, the free energy of the solid–liquid interface, the specific heat, the equilibrium melting temperature and the latent heat, respectively. The material properties used to calculate the fluctuation strength S¯ and the stability constant σ ∗ can be found in [5]. Tip radius and tip velocity have been determined by H¨urlimann et al. [5] Rtip = (5.2 ± 0.4) × 10−3 T −0.83±0.03 , vtip = (1.888 ± 0.086) × 10−1 T 1.745±0.017
(2)
where Rtip is given in m, vtip in m/s, and T in K. Bisang and Bilgram [6] compared the tip shape and the position of the onset of sidebranches at the fins of experimentally grown three-dimensional xenon dendrites with the predictions made in [4]. In order to test the predicted sidebranch amplitude they used the mean distance z¯ SB between the tip and the position where the sidebranches have a root-mean-square amplitude Aexp (z) = (x(z, t) − x(z, t))2 (3) of about 1Rtip . According to Brener and Temkin [4] this position behind the tip is z¯ SB (1Rtip ) ≈
(27σ ∗ )5/4 ¯ 5/2 . | ln S| 25/2 (5/3)9/4
(4)
z¯ SB was found not to depend on undercooling and the mean value was z¯ SB = 17.5 ± 3. Type P sidebranches have been observed in quasi-2D experiments. In [7], a collision of a dendrite tip with a finite disturbance led to two symmetrical branches growing significantly faster than other branches. In [8], localized heat pulses were applied to succinonitrile dendrite tips undergoing directional solidification. In these experiments the perturbations led to sidebranches growing closer to the tip than the spontaneous noise-generated sidebranches. The resulting sidebranches were fitted with an analytical description of the response of the dendrite tail to perturbations applied to the tip region [9,10]. Also convective flow can significantly influence the amplitude of sidebranches [11].
Fig. 1. Type N sidebranch initialization by noise. (a) Half-width x(z, t) of the dendrite. (b) Two-dimensional color plot of the sidebranch activity x˜ z (t) (normalized amplitude of x(z, t) around its mean value x(z, t)t calculated separately for each z) of a typical dendrite: red corresponds to −1, white to 1. The dashed lines drawn at z = 3Rtip show the border where the first indications for sidebranches can be observed.
coordinate system. For each contour the half-widths x(z, t)left and x(z, t)right (horizontal distance between the vertical axis (x = 0) and the contour as a function of the vertical distance z from tip and time t) of both sides of the dendrite contour have been determined (Fig. 1a).
2. Experiments
3. Experimental results
The base of our data were image sequences of xenon dendrites growing into their undercooled melt. The undercooling was in the range of 30 mK< T < 230 mK corresponding to about 10−3 ≤ ≤ 10−2 in dimensionless units, where the dimensionless undercooling is defined as = T/(L/cl ) = (Tm − T∞ )/(L/cl ). A description of the experimental setup can be found in [6]. We used a digital CCD-camera with 1280 × 1024 pixels. The resolution of the optical system was about 1 m. Timesteps between the images ranged from 0.2 to 5 s. In order to observe longer sequences of undisturbed growing crystals a tip-tracking system has been developed allowing us to keep the crystal tip in the optimal region of the optical image system ten times longer as without this system. The contours of the dendrites were extracted from the images by an image processing technique described in [12]. The contours have been transformed back to their original proportions and the tips have been shifted to the origin of the
3.1. Sidebranches induced by selective amplification of thermal noise We analysed about 30 sequences of dendrites growing in a steady-state. As there are two modes to initiate sidebranches we had to carefully select the sequences where the tip shapes can be described by a 5/3 power law deduced in [3] and where the first sidebranches become visible at about 3–7 tip radii Rtip behind the tip and reach an amplitude of 1Rtip at zSB 17.5. The sidebranch activity x˜ z (t) of such a dendrite sequence is illustrated in Fig. 1b, where x˜ z (t) is the normalized amplitude around the mean value x(z, t)t calculated for each z. x˜ z (t) is plotted in colors (red = −1, white = +1) versus t (horizontal axis) and z (vertical axis). The upper part of the plot corresponds to the left and the lower part to the right side of the dendrite. Fig. 2a shows a plot of the sidebranch amplitude Aexp (z) versus distance from tip z of a typical dendrite sequence. In
O. Wittwer, J.H. Bilgram / Materials Science and Engineering A 413–414 (2005) 447–450
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All tested sequences were within the uncertainty corresponding to the errors of Eq. (2). The mean value and the scatter of the points following a horizontal line might pretend a relatively high noise level. In fact it is below 0.1Rtip (< 2 m) in all analysed sequences. The noise level is an artefact due to the pixel-based contour extraction of the digital images and does not depend on z. Fig. 2c shows the root-mean-square amplitude of a sequence of synthetic contours with no sidebranches and the same tip radius as the radius of the sequence of Fig. 2a and b. The contours were rendered at the same pixel-resolution and the same calculations were applied to the contours in order to get Aexp (z). 3.2. Sidebranches induced by macroscopic perturbations By repeated shifting the dendrite in the melt we could induce the growth of sidebranches. Fig. 3a shows the shift ztip (t) versus time and the half-widths xl/r (z, t) at z = 200 m. The t-axis of the x versus t plot has been shifted to the left hand side by t = 25 s in order to match the x and the ztip curves. The curves of ztip (t) and xl/r (z, t) are similar and the two sides of the dendrite are almost perfectly matching (symmetric sidebranches). The delay of 25 s corresponds to about z = vtip t 125 m. The effect of the perturbation can be compared to a local perturbation applied at z 75 m behind the tip. In difference to thermally induced sidebranches, the sidebranches induced by macroscopic disturbances started to grow
Fig. 2. Root-mean-square amplitude A(z) of steady-state sidebranches. The solid drawn lines are AB.–T. (z) calculated with the central values of Rtip and vtip from Eq. (2) and the dashed lines correspond to the uncertainties given in (2). (a) Dimensionless coordinates in a linear scale. (b) A(z) plotted in a logarithmic scale vs. z2/5 . (c) A(z) of a synthetic contour of the form z = a|x|5/3 with no sidebranches and the same tip radius as in (a) and (b). The amplitude is an artefact due to the image processing methods used to obtain the contours originating from the pixel grid of the digital images.
order to illustrate the dependencies we rewrite Eq. (1): 2(5/3)9/10 2/5 ln A(z)B.–T. ∼ ln S¯ + √ z . 3 3σ ∗
(5)
When plotting Aexp (z) in a logarithmic scale versus z2/5 (Fig. 2b), a straight line denotes a linear dependence of ln A(z) on z2/5 according √ to Eq. (5). The proportionality is given by 2(5/3)9/10 /(3 3σ ∗ ). A horizontal line indicates independence of z. The solid line in these plots is A(z)textrmB.−T. derived from the central values Rtip and vtip corresponding to Eqs. (2) and the dashed lines correspond to the borders of their uncertainties.
Fig. 3. Type P sidebranch initialization by macroscopic perturbations. (a) The perturbation shift ztip (t) applied to the dendrite and the half-widths x(z, t)l/r at z = 200 m are plotted vs. time. The upper time scale is shifted against the lower one by 25 s corresponding to the time the crystal takes to grow 125 m. The perturbations can be compared to a local perturbation applied at z 75 m behind the tip. (b) 2D color plot of the sidebranch activity x(z, t) of the same sequence.
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O. Wittwer, J.H. Bilgram / Materials Science and Engineering A 413–414 (2005) 447–450
Fig. 4. Transient from type N to type P. (a) After a perturbation of a dendrite with a smooth tip, the dendrite begins to produce sidebranches at the tip on all four fins. (b) Sidebranch activity of a dendrite with a smooth tip that begins to produce sidebranches at the tip at t = 100 s and again becomes a dendrite with a smooth tip at about t = 250 s.
very close to the tip (Fig. 3b). Fig. 4a shows three images of a dendrite where sidebranches began to grow close to the tip. Fig. 4b shows the sidebranch activity of a similar sequence where sidebranches started to grow at the tip at t = 100 s. After about 150 s the dendrites returned to a steady-state mode where sidebranches became again visible at about 5Rtip . 4. Discussion We have performed experiments using xenon as a transparent model substance for metals. For an undisturbed dendrite the contour is a smooth line and according to Brener [3] and Bisang and Bilgram [6] it can be described by a 5/3 power law (see Fig. 1). Our experiments provided data which allowed the characterization of sidebranches according to three criteria: (i) the distance from the dendrite tip where sidebranches start to grow, (ii) the amplitude of sidebranches as a function of the distance from tip and (iii) the symmetry of sidebranches growing at the fins. If only thermal fluctuations are active at the tip of the dendrite then Brener and Temkin [4] have predicted the onset of
sidebranches as a function of undercooling. This prediction has been verified by Bisang [6]. We have shown that sidebranches get visible at about 3–7 tip radii behind the tip (Fig. 1). The amplitude of the sidebranches as a function of the distance from the dendrite tip has been predicted by Brener and Temkin [4]. This prediction has been verified in our experiments (Fig. 2). There are two quantitative predictions for type N dendrites we have verified quantitatively without any fitting parameters. For type P dendrites, sidebranches have been initiated by macroscopic perturbations at the tip. In this case, the growth of sidebranches has been initiated at the same time at all four fins. The sidebranches started at the tip of the dendrite and a symmetric arrangement of sidebranches at the four fins was observed (Fig. 3). The results show that the initial conditions determine the growth of sidebranches (see also [13]). Dendrites with type N branches seem to exhibit deterministic chaos with sensitivity to initial conditions. The branches on the four fins develop individually. For dendrites with type P branches macroscopic perturbations determine the initial conditions and all four fins develop in the same way. Acknowledgements We thank Prof. Dr. H.R. Ott for his support of our experiments. This work was supported by the Swiss National Science Foundation. References [1] J.P. Gollub, J.S. Langer, Rev. Mod. Phys. 71 (1999) 396–403. [2] J.S. Langer, J. Souletie, J. Vannimenus, R. Stora (Eds.), Chance and Matter, Elsevier Science Publishers B.V, Amsterdam, Netherlands, 1987, pp. 629– 711. [3] E. Brener, Phys. Rev. Lett. 71 (1993) 3653–3656. [4] E. Brener, D. Temkin, Phys. Rev. E 51 (1995) 351–359. [5] E. H¨urlimann, R. Trittibach, U. Bisang, J.H. Bilgram, Phys. Rev. A 46 (1992) 6579–6595. [6] U. Bisang, J.H. Bilgram, Phys. Rev. E 54 (1996) 5309–5326. [7] Y. Couder, J. Maurer, R. Gonz´alez-Cinca, A. Hern´andez-Machado, Phys. Rev. E 71 (2005) 031602–1-12. [8] X.W. Qian, H.Z. Cummins, Phys. Rev. Lett. 64 (1990) 3038–3041. [9] R. Pieters, J.S. Langer, Phys. Rev. Lett. 56 (1986) 1948–1951. [10] R. Pieters, Phys. Rev. A 37 (1988) 3126–3143. [11] X. Tong, C. Beckermann, A. Karma, Q. Li, Phys. Rev. E 63 (2001) 0616011-16. [12] H.M. Singer, J.H. Bilgram, J. Cryst. Growth 261 (2004) 122–134. [13] M. Fell, H.M. Singer, J.H. Bilgram, Mater. Sci. Eng. A, in press.
Onset and amplitude of sidebranches in three dimensional growth of xenon dendrites O. Wittwer∗ , J.H. Bilgram Laboratorium f¨ur Festk¨orperphysik, Eidgen¨ossische Technische Hochschule, CH-8093 Z¨urich, Switzerland Received in revised form 1 July 2005
Abstract The growth of sidebranches of xenon dendrites has been studied in a 3D experiment. Sidebranches grow at the ridges of the four fins of the nonaxisymmetric crystal. They can be initiated by selective amplification of thermal noise (type N) or by a macroscopic perturbation (type P). Both types of sidebranches have been identified in our experiments. Type N sidebranches start to grow 3–7 tip radii behind the tip. The sidebranches growing at the four fins are not correlated and the amplitudes measured in our experiments are in quantitative agreement with predictions of Brener and Temkin. Type P sidebranches start to grow at the tip. They are highly correlated at the four fins. © 2005 Elsevier B.V. All rights reserved. PACS: 68.70.+w; 64.70.Dv; 81.10.Fq Keywords: Structure formation; Sidebranching; Solidification; Dendritic growth
1. Introduction Applications in automotive industry call for aluminum castings to be mass produced without casting defects. Predictive models are necessary to prevent casting defects. During the growth of a solid from its undercooled parent melt microstructures are formed. The best known case of such structures are dendrites. Microstructural changes can be observed as undercooling is increased, namely the typical length scales like the tip radius decrease with increasing undercooling [1]. These are quantitative changes. In addition to that, for some materials it has been observed that they also undergo an abrupt decrease in the micro scale. This effect is known as spontaneous grain refinement. Evidence has been found that spontaneous grain refinement is initiated by changes in the morphology of the microscopic structures. We perform 2D/3D-numerical simulations and experiments which allow in situ investigations of three-dimensional growth of xenon crystals into undercooled pure melt. Dendrites, seaweed, doublons, triplons etc. can be produced in experiments and numerical studies depending on initial conditions. In an interplay of numerical studies and experiments we use simulations to interpret and to plan experiments. We use in our experiments ∗
Corresponding author. Tel.: +41 44 633 22 97; fax: +41 44 633 12 31. E-mail address: [email protected] (O. Wittwer).
0921-5093/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2005.08.189
xenon as a transparent model substance for metals. Experimental results are used as a basis of model calculations and the development of models of complex shapes. Three-dimensional growth shapes of crystals are reconstructed using sophisticated image processing combined with experimentally determined shape parameters. In this paper we focus at the origin of sidebranches. We have observed two types of sidebranches: “type N” which are initiated by the selective amplification of thermal noise and “type P” which are initiated by macroscopic perturbations of the dendrite tip. In the following, all lengths are scaled with tip radius (Eq. (2)). The initiation of type N sidebranches has been studied in some detail, for a review see [2]. An important step forward has been made by Brener and Temkin [3,4], who have determined the shape of the tip and the sidebranch amplitude of the nonaxisymmetric needle crystal analytically. The sidebranch amplitude has been found to be 2(5/3)9/10 2/5 √ AB.–T. (z) = ξ12 (z)1/2 ∼ S¯ exp (1) z 3 3σ ∗ where S¯ 2 = 2kB T 2 cl Dth /(L2 vtip R4tip ) is the dimensionless noise or fluctuation strength and σ ∗ = 2Dth d0 /(vtip R2tip ) the so called stability constant with the capillary length d0 = γsl cl Tm /L2 . Where Dth , vtip , Rtip , γsl , cl , Tm , L are the thermal diffusivity,
O. Wittwer, J.H. Bilgram / Materials Science and Engineering A 413–414 (2005) 447–450
448
the tip velocity, the tip radius, the free energy of the solid–liquid interface, the specific heat, the equilibrium melting temperature and the latent heat, respectively. The material properties used to calculate the fluctuation strength S¯ and the stability constant σ ∗ can be found in [5]. Tip radius and tip velocity have been determined by H¨urlimann et al. [5] Rtip = (5.2 ± 0.4) × 10−3 T −0.83±0.03 , vtip = (1.888 ± 0.086) × 10−1 T 1.745±0.017
(2)
where Rtip is given in m, vtip in m/s, and T in K. Bisang and Bilgram [6] compared the tip shape and the position of the onset of sidebranches at the fins of experimentally grown three-dimensional xenon dendrites with the predictions made in [4]. In order to test the predicted sidebranch amplitude they used the mean distance z¯ SB between the tip and the position where the sidebranches have a root-mean-square amplitude Aexp (z) = (x(z, t) − x(z, t))2 (3) of about 1Rtip . According to Brener and Temkin [4] this position behind the tip is z¯ SB (1Rtip ) ≈
(27σ ∗ )5/4 ¯ 5/2 . | ln S| 25/2 (5/3)9/4
(4)
z¯ SB was found not to depend on undercooling and the mean value was z¯ SB = 17.5 ± 3. Type P sidebranches have been observed in quasi-2D experiments. In [7], a collision of a dendrite tip with a finite disturbance led to two symmetrical branches growing significantly faster than other branches. In [8], localized heat pulses were applied to succinonitrile dendrite tips undergoing directional solidification. In these experiments the perturbations led to sidebranches growing closer to the tip than the spontaneous noise-generated sidebranches. The resulting sidebranches were fitted with an analytical description of the response of the dendrite tail to perturbations applied to the tip region [9,10]. Also convective flow can significantly influence the amplitude of sidebranches [11].
Fig. 1. Type N sidebranch initialization by noise. (a) Half-width x(z, t) of the dendrite. (b) Two-dimensional color plot of the sidebranch activity x˜ z (t) (normalized amplitude of x(z, t) around its mean value x(z, t)t calculated separately for each z) of a typical dendrite: red corresponds to −1, white to 1. The dashed lines drawn at z = 3Rtip show the border where the first indications for sidebranches can be observed.
coordinate system. For each contour the half-widths x(z, t)left and x(z, t)right (horizontal distance between the vertical axis (x = 0) and the contour as a function of the vertical distance z from tip and time t) of both sides of the dendrite contour have been determined (Fig. 1a).
2. Experiments
3. Experimental results
The base of our data were image sequences of xenon dendrites growing into their undercooled melt. The undercooling was in the range of 30 mK< T < 230 mK corresponding to about 10−3 ≤ ≤ 10−2 in dimensionless units, where the dimensionless undercooling is defined as = T/(L/cl ) = (Tm − T∞ )/(L/cl ). A description of the experimental setup can be found in [6]. We used a digital CCD-camera with 1280 × 1024 pixels. The resolution of the optical system was about 1 m. Timesteps between the images ranged from 0.2 to 5 s. In order to observe longer sequences of undisturbed growing crystals a tip-tracking system has been developed allowing us to keep the crystal tip in the optimal region of the optical image system ten times longer as without this system. The contours of the dendrites were extracted from the images by an image processing technique described in [12]. The contours have been transformed back to their original proportions and the tips have been shifted to the origin of the
3.1. Sidebranches induced by selective amplification of thermal noise We analysed about 30 sequences of dendrites growing in a steady-state. As there are two modes to initiate sidebranches we had to carefully select the sequences where the tip shapes can be described by a 5/3 power law deduced in [3] and where the first sidebranches become visible at about 3–7 tip radii Rtip behind the tip and reach an amplitude of 1Rtip at zSB 17.5. The sidebranch activity x˜ z (t) of such a dendrite sequence is illustrated in Fig. 1b, where x˜ z (t) is the normalized amplitude around the mean value x(z, t)t calculated for each z. x˜ z (t) is plotted in colors (red = −1, white = +1) versus t (horizontal axis) and z (vertical axis). The upper part of the plot corresponds to the left and the lower part to the right side of the dendrite. Fig. 2a shows a plot of the sidebranch amplitude Aexp (z) versus distance from tip z of a typical dendrite sequence. In
O. Wittwer, J.H. Bilgram / Materials Science and Engineering A 413–414 (2005) 447–450
449
All tested sequences were within the uncertainty corresponding to the errors of Eq. (2). The mean value and the scatter of the points following a horizontal line might pretend a relatively high noise level. In fact it is below 0.1Rtip (< 2 m) in all analysed sequences. The noise level is an artefact due to the pixel-based contour extraction of the digital images and does not depend on z. Fig. 2c shows the root-mean-square amplitude of a sequence of synthetic contours with no sidebranches and the same tip radius as the radius of the sequence of Fig. 2a and b. The contours were rendered at the same pixel-resolution and the same calculations were applied to the contours in order to get Aexp (z). 3.2. Sidebranches induced by macroscopic perturbations By repeated shifting the dendrite in the melt we could induce the growth of sidebranches. Fig. 3a shows the shift ztip (t) versus time and the half-widths xl/r (z, t) at z = 200 m. The t-axis of the x versus t plot has been shifted to the left hand side by t = 25 s in order to match the x and the ztip curves. The curves of ztip (t) and xl/r (z, t) are similar and the two sides of the dendrite are almost perfectly matching (symmetric sidebranches). The delay of 25 s corresponds to about z = vtip t 125 m. The effect of the perturbation can be compared to a local perturbation applied at z 75 m behind the tip. In difference to thermally induced sidebranches, the sidebranches induced by macroscopic disturbances started to grow
Fig. 2. Root-mean-square amplitude A(z) of steady-state sidebranches. The solid drawn lines are AB.–T. (z) calculated with the central values of Rtip and vtip from Eq. (2) and the dashed lines correspond to the uncertainties given in (2). (a) Dimensionless coordinates in a linear scale. (b) A(z) plotted in a logarithmic scale vs. z2/5 . (c) A(z) of a synthetic contour of the form z = a|x|5/3 with no sidebranches and the same tip radius as in (a) and (b). The amplitude is an artefact due to the image processing methods used to obtain the contours originating from the pixel grid of the digital images.
order to illustrate the dependencies we rewrite Eq. (1): 2(5/3)9/10 2/5 ln A(z)B.–T. ∼ ln S¯ + √ z . 3 3σ ∗
(5)
When plotting Aexp (z) in a logarithmic scale versus z2/5 (Fig. 2b), a straight line denotes a linear dependence of ln A(z) on z2/5 according √ to Eq. (5). The proportionality is given by 2(5/3)9/10 /(3 3σ ∗ ). A horizontal line indicates independence of z. The solid line in these plots is A(z)textrmB.−T. derived from the central values Rtip and vtip corresponding to Eqs. (2) and the dashed lines correspond to the borders of their uncertainties.
Fig. 3. Type P sidebranch initialization by macroscopic perturbations. (a) The perturbation shift ztip (t) applied to the dendrite and the half-widths x(z, t)l/r at z = 200 m are plotted vs. time. The upper time scale is shifted against the lower one by 25 s corresponding to the time the crystal takes to grow 125 m. The perturbations can be compared to a local perturbation applied at z 75 m behind the tip. (b) 2D color plot of the sidebranch activity x(z, t) of the same sequence.
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O. Wittwer, J.H. Bilgram / Materials Science and Engineering A 413–414 (2005) 447–450
Fig. 4. Transient from type N to type P. (a) After a perturbation of a dendrite with a smooth tip, the dendrite begins to produce sidebranches at the tip on all four fins. (b) Sidebranch activity of a dendrite with a smooth tip that begins to produce sidebranches at the tip at t = 100 s and again becomes a dendrite with a smooth tip at about t = 250 s.
very close to the tip (Fig. 3b). Fig. 4a shows three images of a dendrite where sidebranches began to grow close to the tip. Fig. 4b shows the sidebranch activity of a similar sequence where sidebranches started to grow at the tip at t = 100 s. After about 150 s the dendrites returned to a steady-state mode where sidebranches became again visible at about 5Rtip . 4. Discussion We have performed experiments using xenon as a transparent model substance for metals. For an undisturbed dendrite the contour is a smooth line and according to Brener [3] and Bisang and Bilgram [6] it can be described by a 5/3 power law (see Fig. 1). Our experiments provided data which allowed the characterization of sidebranches according to three criteria: (i) the distance from the dendrite tip where sidebranches start to grow, (ii) the amplitude of sidebranches as a function of the distance from tip and (iii) the symmetry of sidebranches growing at the fins. If only thermal fluctuations are active at the tip of the dendrite then Brener and Temkin [4] have predicted the onset of
sidebranches as a function of undercooling. This prediction has been verified by Bisang [6]. We have shown that sidebranches get visible at about 3–7 tip radii behind the tip (Fig. 1). The amplitude of the sidebranches as a function of the distance from the dendrite tip has been predicted by Brener and Temkin [4]. This prediction has been verified in our experiments (Fig. 2). There are two quantitative predictions for type N dendrites we have verified quantitatively without any fitting parameters. For type P dendrites, sidebranches have been initiated by macroscopic perturbations at the tip. In this case, the growth of sidebranches has been initiated at the same time at all four fins. The sidebranches started at the tip of the dendrite and a symmetric arrangement of sidebranches at the four fins was observed (Fig. 3). The results show that the initial conditions determine the growth of sidebranches (see also [13]). Dendrites with type N branches seem to exhibit deterministic chaos with sensitivity to initial conditions. The branches on the four fins develop individually. For dendrites with type P branches macroscopic perturbations determine the initial conditions and all four fins develop in the same way. Acknowledgements We thank Prof. Dr. H.R. Ott for his support of our experiments. This work was supported by the Swiss National Science Foundation. References [1] J.P. Gollub, J.S. Langer, Rev. Mod. Phys. 71 (1999) 396–403. [2] J.S. Langer, J. Souletie, J. Vannimenus, R. Stora (Eds.), Chance and Matter, Elsevier Science Publishers B.V, Amsterdam, Netherlands, 1987, pp. 629– 711. [3] E. Brener, Phys. Rev. Lett. 71 (1993) 3653–3656. [4] E. Brener, D. Temkin, Phys. Rev. E 51 (1995) 351–359. [5] E. H¨urlimann, R. Trittibach, U. Bisang, J.H. Bilgram, Phys. Rev. A 46 (1992) 6579–6595. [6] U. Bisang, J.H. Bilgram, Phys. Rev. E 54 (1996) 5309–5326. [7] Y. Couder, J. Maurer, R. Gonz´alez-Cinca, A. Hern´andez-Machado, Phys. Rev. E 71 (2005) 031602–1-12. [8] X.W. Qian, H.Z. Cummins, Phys. Rev. Lett. 64 (1990) 3038–3041. [9] R. Pieters, J.S. Langer, Phys. Rev. Lett. 56 (1986) 1948–1951. [10] R. Pieters, Phys. Rev. A 37 (1988) 3126–3143. [11] X. Tong, C. Beckermann, A. Karma, Q. Li, Phys. Rev. E 63 (2001) 0616011-16. [12] H.M. Singer, J.H. Bilgram, J. Cryst. Growth 261 (2004) 122–134. [13] M. Fell, H.M. Singer, J.H. Bilgram, Mater. Sci. Eng. A, in press.