Side-branch growth in two-dimensional dendrites . I . Experiments

Laboratoire de Physique Statistique, Ecole Normale Supérieuere, 24 rue Lhomond, 75231 Paris Cedex 05, France Departament de Fisica Aplicada, Universitat Politècnica de Catalunya, Av. del Canal Olímpic s/n E-08860 Castelldefels (Barcelona), Spain Departament ECM, Facultat de Fisica, Universitat de Barcelona, Diagonal 647, E-08028 Barcelona, Spain sReceived 9 September 2004; published 17 March 2005 d


I. INTRODUCTION
The growth of monocrystals in an undercooled solution gives rise to the characteristic formation of dendrites.These structures have a parabolic tip in two dimensions ͑a paraboloid in three dimensions͒ directed along one of the main crystallographic directions where the growth is fastest.Experimental, numerical, and theoretical efforts have resulted in a complete understanding of the selection mechanism ͓1-6͔ of the radius of the parabola in two dimensions or paraboloid in three dimensions.This type of structure is in fact an archetype of anisotropic growth in a diffusion field ͓7͔ and is also observed in other phenomena.Dendritic structures can also be obtained in Saffman-Taylor viscous fingering ͓7,8͔ when a singular disturbance of the tip gives a preferred direction of growth to the viscous finger.Similar dendrites also grow in limited aggregation ͑DLA͒ in the presence of anisotropy ͓9,10͔.
While the tip of the dendrite is stable, lateral protrusions are observed to form and grow.In the frame of reference moving with the dendrite tip, they form at some distance from the extremity, then grow in amplitude as they are advected away from the tip region.This is the side-branching phenomenon, which has already been widely investigated ͓11-23͔, both in three dimensions and two dimensions.As already proposed some time ago ͓13-16͔, side branching results from the selective amplification along the parabolic profile of noisy fluctuations near the tip.The initial growth of these disturbances was investigated by Dougherty and Gollub ͓17,18͔.These authors showed that the very early development of the protrusions is exponential.In their experiment, the typical initial spacing of the protrusions is of the order of 5 ͑ being the parabola radius of curvature͒.Both the initial amplitude and the spacing of these protrusions tend to be irregular and there is no correlation between the side branch-ing observed on each side of the dendrite.These results supported the hypothesis of the selective amplification of noise.
Correlatively, working on the dendritic fingers generated in anomalous Saffman Taylor fingering, Rabaud et al. ͓8͔ were able to apply a localized disturbance of the parabolic tip and to observe the growth and spreading of the resulting wave packet.They observed directly, in the frame of reference of the dendrite's tip, that the waves which disturb the parabolic sides are advected away from the tip with a velocity which is larger than their own.This is the characteristic feature of a class of instabilities called convective instabilities in the context of fluid dynamics where they are common in, e.g., shear flow instabilities ͑see Ref. ͓25͔ for a review͒.Their common feature is that they give rise to irregular structures because the unstable medium serves as a selective amplifier of incident noise.In such instabilities, it is possible to replace the natural noise by a periodic forcing and thus to generate a periodic structure.This was done in both dendritic viscous fingers ͓8͔ and dendrites ͓26,27͔.
The period of exponential growth of the protrusions is short, and followed by their transformation into proper side branches as they become sensitive to the crystallographic anisotropy.In the case of a cubic lattice where the main crystallographic directions of growth are ͑100͒, the branches are perpendicular to the main dendrite.As they grow the branches acquire a stable parabolic tip.The neighboring branches interact because of the nonlocal nature of the diffusion field in which they grow.Figures 1͑a͒ and 1͑b͒ shows two sketches of the isoconcentration lines, for slow and fast growing dendrites, respectively.The growth and interaction of the branches are expected to be different when their lengths are small compared to the local diffusion length ͓Fig.1͑a͔͒ or when both length scales are of the same order of magnitude ͓Fig.1͑b͔͒.
A theoretical incentive for the study of the growth of slow dendrites on a large scale comes from results obtained on anisotropic DLA, which showed the formation of a selfaffine fractal structure ͓9,10͔.In the context of crystal growth a fractal regime is only observed in the limit of very longrange interaction and thus very large diffusion length.This was investigated in the quasi-two-dimensional situation by Couder et al. ͓19͔, in experiments performed in the crystallization of ammonium bromide diluted in water and contained in a quasi-two-dimensional cell.Ahead of a crystallization front growing at a velocity U the impurity diffusive field is characterized by its length scale l D =2D / U where D is the diffusion constant.At weak under-cooling, the growth is slow and l D is so large that the distances between all branches are small in comparison.In these conditions an approximation of a Laplacian growth is obtained and the side branches are not free: they remain prisoners of the lateral front of the main dendrite.Their velocity is not constant and they keep interacting, screening off the slow branches, so that the structure continues coarsening.This leads to the buildup of a self-affine fractal structure.The fractal dimension of such a dendrite was measured ͓19͔, far from the tip, by a box counting method and found to be d f = 1.58± 0.03 in agreement with the dimension ͑d f = 1.57͒ found ͓10͔ for fourfold anisotropic DLA.
In the same limit, the area occupied by the dendrite in two dimensions was also investigated ͓19͔.In spite of the fluc-tuations due to the side-branching instability, this area is the same ͑on the average͒ as that of a smooth parabolic shape.The parameter of this average parabola is the same as that of the stable dendrite.This means that the mean fractal object occupies the same area that the dendrite would have covered if it had remained a stable parabola with tip radius .This parabolic dependence of the unstable dendrite could be expected from Ivantsov's ͓1͔ qualitative argument: the tip of the dendrite moves at constant velocity so that its position is proportional to time t; the growth of the lateral sides results from a diffusive process and their position moves as t 0.5 .
In three dimensions the situation is slightly more complicated ͓20-22͔.The dendrite has the shape of a paraboloid only near the tip but the axisymmetry around the dendrite axis breaks down even before the lateral destabilization.First there is formation of four ridges which then destabilize to form side branches.These grow in the main crystallographic directions perpendicular to the dendrite's direction.The cross section of the dendrite thus develops in a four armed petal shaped structure.If xЈ is their distance to the tip, the theoretical prediction is that the mean length of these arms ͓21͔ grows as ͉xЈ͉ 3/5 ͑i.e., as t 3/5 ͒ while their width in the orthoradial direction grows as ͉xЈ͉ 1/5 .The growth in a paraboloid mean shape is only recovered if an averaging is done over all the radial directions around the axis of the dendrite.
In the present paper we will perform experiments aimed at characterizing the late evolution of the side branches in an approximately two-dimensional situation.
We will first study the regime of growth at very small Peclet numbers.The work will be focused on the characterization of the coarsening phenomenon responsible for the buildup of ever larger branches.This will shed light on how the growth of the average parabolic front can be sustained by an ever-decreasing number of branches.Some preliminary results about it were published in Ref. ͓24͔.
We will then turn to the specific effects observed for larger velocities and smaller l D .The same initial coarsening leads to a greater spacing of the growing side branches.But when this spacing becomes of the order of the local thickness of the mean diffusion front, the side branches have the possibility of escaping and form independent dendrites.
Finally this first paper ͑Part I͒ is exclusively experimental.Part II is devoted to a phase field numerical computation of this growth of the dendrites lateral branches.

II. EXPERIMENTAL SETUP
We grow ammonium bromide crystals from supersaturated aqueous solution in conditions similar to those previously described ͓19,28͔.The solution, of concentration c o , is initially enclosed in a thin cell formed of two glass plates of radius 50 mm.The distance between the plates is fixed by a mylar spacer of thickness e =20 m.The cell is entirely immersed in a circulation of water thermally regulated with an accuracy of 20 mK.A few runs are first performed at large undercooling so as to obtain several crystals in the cell.The cell is then warmed up so as to melt these crystals almost completely, leaving only a few germs.The desired undercooling is then imposed and the growth of the dendrites can be observed.The experiment is only recorded when one isolated, well oriented germ ͑with two ͓100͔ axis in the plane of the cell͒ has been obtained.We then wait until a long dendrite is obtained, far away from the boundaries and from the other main dendrites grown from the same germ.
As the thermal diffusivity is three orders of magnitude larger than the mass diffusivity, the system is limited by mass diffusion for which D =2ϫ 10 −9 m 2 / s.Several different solutions were used for these experiments with concentrations corresponding to several equilibrium temperature T E .Solutions with T E = 35 °C were chosen for the cases where a small undercooling was to be applied, leading to slowly growing dendrites.Fast dendrites were grown in solutions with an equilibrium temperature.T E = 50 °C where undercoolings as large as 35 °C could be chosen.Altogether, dendrites growing in a very large range of velocities 3 -90 m/s were thus obtained.
The dendrites were observed with an inverted microscope which allowed magnifications up to 480.The growth was recorded with a video camera and digitized with a resolution 768 by 512 pixels.We used four different magnifications corresponding respectively to resolutions 0.36, 1.19, 2.97, and 4.76 / pixel.Since we wanted to record the growth of individual branches until they reached large amplitudes, we generally used the magnification corresponding to 2.97 m / pixel so that the early growth of the side branching was poorly resolved.For this reason, in the case of the dendrites of low velocities, before recording the branch growth, we briefly recorded the growth of the tip with a large magnification so as to measure its shape.The results, concerning the relation between the selected radius of curvature and the velocity, were in good agreement with the values previously reported ͓28͔.
Finally we can note that in spite of its small thickness ͑e =20 m͒, the cell does not impose a strict two dimensionality to the growth.The tip, having a radius smaller than d, remains a three-dimensional paraboloid.However, the limited thickness prevents the growth of branches in the direction perpendicular to the cell's plane.Furthermore, for the slow dendrites, the cell thickness is always small compared to the distance of the competing branches.The diffusion field in which these branches grow can thus be considered as two dimensional.The fact that most of our results are recovered in strictly two-dimensional simulations ͑see Part II͒ is confirmation of the validity of the two-dimensional ͑2D͒ aproximation for the investigation of the side branches' growth.

Global structure
We will first investigate the regime of slow growth at weak undercoolings.A well oriented initial seed in a flat cell gives rise to four arms.The diffusion length being large, these arms interact with each other so that each of them grows in a petal shape ͓29͔.Only the front part of each petal, can be considered parabolic.For this reason we always waited long enough for the dendrite to have grown far away from the germ.In the following, the axes of the laboratory frame of reference are chosen so that the dendrite grows along the O x axis and its branches along O y .Taking the origin OЈ at the dendrite's tip OЈ x Ј , OЈ y Ј will be the frame of reference moving with it.
The results we will first report in detail were obtained on a dendrite which grew with a velocity V = 3.77 m / s.Using a strong magnification we measured by image processing ͓28͔ the shape of the tip parabola and deduced a tip radius = 1.4± 0.1 m.This value corresponds to a value 2 V = k = 7.5 m 3 / s which is in good agreement with the value found for crystals of NH 4 Br grown in a solution with T E = 35 °C, as reported by Maurer et al. ͓28͔.
We then turn our interest to the large scale structure of the dendrite.The system is limited by mass diffusion for which the diffusion constant is D =2 10 −9 m 2 / s, so that for this dendrite the Peclet number ͑Pe= V /2D͒ is Peϳ 10 −3 .At such a low velocity of growth the diffusion length is large and a self-affine regime of growth is observed in a large range of length scales.As in our previous work ͑Couder et al. ͓19͔͒ the half area covered by the structure ͑on the yЈ Ͼ 0 side͒ grows ͑on the average͒ with the distance to the tip as where xЈ is the distance to the tip.This means that the selfaffine dendrite has a mean lateral front.The same front that would have existed if the dendrite had not undergone a lateral destabilization and had remained parabolic.It is thus located at

͑2͒
In the laboratory frame of reference, at a large distance xЈ from the tip, the velocity normal to this front is thus on the average Taking into account the selection of the tip radius by which 2 V is a constant k and the normal velocity becomes The decrease of the normal velocity along the mean parabolic profile induces an increase of the diffusion length.In the tip region it is l D T =2D / V.But along the sides the diffusion length, l D S grows with the distance to the tip as

͑5͒
in a given region of the laboratory frame of reference, the diffusion length grows as the square root of time,

͑6͒
In the case of the dendrite growing at V = 3.77 m/s, the diffusion length at the tip is l D T = 0.53 mm.In Fig. 2 going from ͑b͒ to ͑d͒ the normal velocity of the mean front decreases from ϳ0.1 to ϳ 0.06 m / s.Correlatively, the diffu-sion length grows from l D S ϳ 2 mm to ϳ3 cm.These values are always very large as compared to the size of the branches and to their spacing.The cell thickness is small as compared to l D S , so that the diffusion field can be considered as two dimensional.

The growth of the dominant branches
The dendrite is observed with a microscope and a video camera which records a fixed field that the dendrite crosses during its growth.Figure 2 shows five images of such a recording concerning 2.2 mm of the length of the dendrite.Note that this region was more than 1 cm away from the initial seed so that the influence of the other dendrites grown from the same seed was negligible.In this case the growth was observed during a period of an hour and a half.Figure 2͑a͒ is typical of the first frames where the tip region of the dendrite is observed.The distribution of branches resulting from the primary instability form an irregular array with branches having an initial average spacing of approximately 5 and various amplitudes.The four others show the typical evolution of the side branches.The process of coarsening is clearly visible: fewer and fewer branches continue growing as time elapses.The branches which stop growing remain as fossil structures.Much later, however, it can be noted that the tip of these branches becomes facetted, before eventually melting.The sequence shown in Fig. 2 exhibits another obvious feature.The largest branches were already locally dominant at their formation.In other terms, the dominance of a branch appears to be entirely determined from the start.The following measurements set this observation on a quantitative basis.
On an image such as Fig. 2͑c͒ we use a home-made image processing which detects the profile of the dendrite, then its protrusions in the O y direction.Each of these protrusions is ascribed a number n.It is easy to follow a branch on the successive images because its abscissa x n is practically constant.The origin of the time t n used for a given branch n is the time when the tip of the main dendrite has passed at the abscissa x n .We measure the position y n ͑t n ͒ of the tip of each branch as a function of its own time.
Figure 3͑a͒ shows the plot y n ͑t n ͒ of the growing length of five of the branches of the dendrite shown in Fig. 2. They all have a velocity which decreases as a function of time.This means that none of the lateral branches becomes a free dendrite.All branches remain prisoner of the diffusion front of the main dendrite.In a loose sense, the lateral branches can thus be considered as dendrites resulting from the instability of a directional growth in the mean diffusion field imposed by the main dendrite.As time elapses the front slows down and this diffusive field thickens: correlatively there is coarsening and the wavelength separating active branches keeps increasing.
A remarkable result is that the hierarchy of size of the branches is fixed initially at the formation of the lateral branches.A better characterization is obtained in a log-log plot ͓Fig.3͑b͔͒ where each branch is observed to be fitted by a power law growth of the type The velocities thus all decrease with time as V n ϰ t ␣ n −1 .The exponent ␣ n is constant for a given branch but depends on how dominant the branch is at the start, i.e., how strong the initial disturbance was.The values of the ␣ n for the branches of Fig. 2 are given in Table I.For this dendrite, they range from 0.51 to 0.64.We can note that this power law behavior is only observed in a finite range of values of t.At large times the velocity of all observed branches decreases due to their screening off by faster branches.Even branch 6, which is the fastest, is observed to slow down at times larger than 2000 s as it begins to be screened off by a larger branch located outside the observed region.This screening effect is analyzed in Sec.III A 3. These results suggest that the stronger the initial disturbance, the larger the growth exponents.For this reason it is interesting to observe situations in which a dendrite is submitted accidentally to a strong localized disturbance.Figure 4 shows such an event where a dendrite of velocity 7.7 m/s collided with a speck of dust present in the cell.The violent disturbance of the tip leads to a symmetrical destabilization of the front which already has a finite amplitude very close to the tip ͓Fig.4͑a͔͒.As a result, two symmetrical branches form, which grow faster than usual.Their initial growth velocity is close to being V / 2, half of the velocity of the main dendrite, ͑see below Sec.III B 2͒.The logarithmic plot ͑Fig.5͒ shows that these branches also have a power law growth of the type given by Eq. ͑7͒ but with an unusually large value ␣ = 0.80.Such branches are naturally dominant over their neighbors.In this range of Peclet numbers and in observed regions as large as 1 cm they are observed to slow down ͑␣ Ͻ 1͒ and thus they still remain prisoner of the mean lateral front.

The slower branches' screening off
The second process is screening off.As time passes, the slower branches stop growing altogether.This is observed in Fig. 3 where the branches ͑e.g., 2 and 4͒ reach asymptotic lengths, y 2 Max and y 4

Max
, respectively.These branches have stopped growing because of the screening effect due to faster growing branches.This effect is dominated by the geometrical positions of neighboring branches as can be observed in Fig. 2. Neglecting the curvature of the parabola near the tip, the relative position of these branches can be sketched ͓Fig.6͑a͔͒ as uneven branches growing from a linear base.On this sketch we draw two straight lines from the extremity of each branch and tangent to the most protruding neighboring structures.Whenever these neighbors are larger, they have larger ␣ n 's so that the difference in size will increase with time.The velocity of the small branch then decreases until it finally stops completely.This process repeats itself, leading to a hierarchy of screening-off processes proceeding from the nearest to the furthest.On Fig. 6͑a͒ branch n will be screened FIG. 3. ͑a͒ The lengths y n ͑t n ͒ of the lateral branches ͑1͒, ͑2͒, ͑3͒, ͑4͒, ͑6͒, as a function of time ͑in seconds͒.Branches 2 and 4 reach limit lengths y 2 Max and y 4

Max
, respectively.The dotted line shows the linear motion x T ͑t͒ of the tip of the dendrite on the same scale.͑b͒ Logarithmic plot for branches ͑1͒, ͑2͒, ͑3͒, and ͑6͒.The best fits for the power law regions correspond to ␣ 1 = 0.58, ␣ 2 = 0.51, ␣ 3 = 0.54, ␣ 6 = 0.64, respectively.off by p and q, then branch p will be screened off by q and r and later q will stop because of the influence of r and s.The process repeats itself on larger and larger scales.
If y n Max is the maximum length reached by branch n we define the nondimensional quantity Y n as The geometrical situation of the branch n between p and q is characterized by the screening angles n p and n q as sketched in Fig. 6͑b͒.It turns out that the dynamically relevant parameter is with tan n p = ͑y p − y n ͒ / ͑x p − x n ͒ and tan n q = ͑y q − y n ͒ / ͑x q − x n ͒.The time evolution of S n can be computed from the y n ͑t n ͒, y p ͑t p ͒, y q ͑t q ͒. Figure 7͑a͒  The two parameters Y 0 and S 0 are obtained from the best fits of the linear part of the semilog plots ͓Fig.7͑a͔͒.The values found for various branches are approximately constant and given in Table I.It can be noted that they depend neither on the scale of the process nor on the symmetry or asymmetry of the positions of branches p and q with regard to branch n.The mean value of S 0 is 0.44.The values of Y 0 are more scattered.When the screening-off process of branch n begins early we find values of Y 0 ϳ 1.This value is found when branch n is mainly screened by a previous branch p.In the opposite situation where the main screening comes from a branch q, grown after n, we obtain Y 0 ϳ 0.75.From these values we can deduce that in a symmetrical situation, branch n will reach an amplitude 0.99y n Max for n q = n q = 45.4°͑Y0 ϳ 1͒ or for n q = n q = 43.5°͑Y0 ϳ 0.75͒.We have thus defined the screening-off process in purely geometrical terms.Can this lead to a temporal description of the growth of a given branch?We find empirically that the time evolution of the screening factor is linear ͓see Fig. 7͑b͔͒ so that S can be written as a result combining relations ͑8͒ and ͑9͒ the time evolution of the branch length can be written with = Y 0 exp͑− a/S 0 ͒ and = b/S 0 .
The growth of an infinite array of parallel needles in a Laplacian field was investigated by M. Adda Bedia ͓30͔.In his lattice shorter and larger needles alternate.The smaller needles are then observed to undergo an exponential decay.There are differences between this theoretical situation and the experimental one.For instance the lattice of parallel needles is infinite, and there is no central base from which the needles grow.However, the screening effect is of the same nature as observed experimentally.

Global structure
The global structures of slow and fast 2D dendrites are very different.As discussed above, the slow dendrites are self-affine fractals which retain a global parabolic mean shape.The velocities of their branches are always small compared to that of the main dendritic tip.In contrast, the fast dendrites cover a finite fraction of the 2D space and have a wedge shaped envelope as a result of the velocity of the branches which equals that of the main tip.The investigation of the growth of individual branches sheds some light on the origin of the crossover between these two regimes of growth.Two main features are observed which will be described successively.Near the tip there is a quantitative change in the values of the growth exponents.Far from the tip, there is a qualitative change as some branches escape from the diffusion field of the main dendrite.

The initial behavior of the fast branches
In the range of velocities 20Ͻ V Ͻ 50 m / s, as the diffusion length diminishes, different processes appear, both near the tip and away from it.In all this range of velocities the dendrite is still observed to have a classical parabolic extremity and the same mode of initial destabilization.The above-described processes of competition between branches are still observed.However, compared to the previous experiments the rate of growth of the lateral instability is now larger and the destabilization of the parabolic front occurs nearer to the tip.Correlatively there is a shift of the values of the growth exponents ␣ n resulting from the normal fluctuations.They are found to range from 0.6 to 0.9 for a dendrite of velocity 20 m / s and from 0.8 to 1 for a dendrite of velocity 50 m / s.As previously, the value of an ␣ n depends on the amplitude of the initial disturbance of the parabolic front.For V ജ 50 m / s large amplitude disturbances occur more frequently and generate branches which grow with ␣ = 1 and thus a constant velocity.A remarkable result is that in those cases the branch velocity is initially half that of the dendrite ͑as will be seen below in Fig. 10͒.This is in agreement with a recent finding by Hakim ͓31͔ in a model system.Working on linear needles growing in a Laplacian field, he considered a situation where, on a semi-infinite needle, two lateral branches grow at a right angle.He finds analytically a specific solution in the shape of a cross where the velocity of the two branches is half that of the main needle.
In the range of velocities 50Ͻ V Ͻ 100 m / s the tip itself appears strongly disturbed by instabilities which generate frequent and large lateral disturbances.In this instability the tip suddenly slows down and splits asymetrically, generating two secondary tips of unequal length.Unlike what is observed at larger velocities, this tip splitting fails to produce two independent structures.The larger of the two tips accelerates while the other stops.This leads to the formation of a new single tip, which in turn will become unstable by a new tip splitting.The whole structure thus undergoes relaxation oscillations.It is not known if this is an intrinsic destabilization mode, or if this instability is generated by an interaction with the cell's walls.A quantitative investigation of the tip structure in this range of velocities would have required working with a fast camera and was not the aim of the present paper.It is worth mentioning, however, because of its implication on the behavior of lateral branches on a long time range.Each tip splitting is a strong disturbance where two symmetrical side branches of large amplitude are generated ͑Fig.8͒.Finally, at very large velocities the tip of the dendrite can also undergo a complete tip splitting.As shown in Fig. 9, this gives rise to two dendrites which will continue growing alongside each other.

The escape of branches
We can now turn to the evolution of branches, as observed on a long time scale, using a weaker magnification of the microscope.
For dendrites with 20Ͻ V Ͻ 50 m / s, the branches usually have the same evolution as described in Sec.III A: there is coarsening and fewer and fewer branches continue growing as time elapses.In the cases where the dendrite has undergone a finite disturbance something different occurs.At some distance away from the tip the resulting branch is observed to accelerate ͑Fig.10, inset͒, before again reaching a constant velocity.Three successive regimes of growth are clearly identified on the plot of the length of the dominant branch as a function of time ͑Fig.10͒.In the initial region ͑I͒ the branch is prisoner of the main dendrite and grows with a power law of time ͑on the example shown here ␣ =1͒.Then it accelerates ͑region II͒.Ultimately, in region III, the velocity becomes constant and equal to the velocity V of the main dendrite.The branch has then escaped the main dendrite's diffusion front, to become an independent dendrite.For dendrites with V Ͻ 40 m / s, this escape was only observed to occur for those fast branches which resulted from an anomalous strong disturbance of the tip.If a dendrite is grown in the same conditions as those of Fig. 10, but without undergoing any finite disturbance, no branch escapes in the limited duration of our experiment.
Four successive images of a faster dendrite ͑V =51 m/s͒ are shown in Fig. 11.In the tip region the structure is similar to that discussed above, and the branches are still disturbances of a dendrite which remains parabolic on the average.At a certain distance from the tip some branches are repeatedly observed to accelerate until they reach the velocity V of the main dendrite.The most obvious feature of the side of the global structure of a fast dendrite is that it is formed of independent parallel branches, all growing at the same velocity ͑Figs.11 and 12͒.On a given dendrite, the branches which escape reach the same velocity after approximately the same delay , so that the global structure acquires a wedge shape.This organization is obvious on a photograph ͑Fig.12͒ of a very fast dendrite ͑V =87 m/s͒ at a weak magnification.Two straight lines at 45°from the direction of the initial dendrite form the envelope of the branches.The structure, though complex, is no longer a self-affine fractal; inside the wedge the pattern covers a finite fraction of the 2D space.The tip of the main dendrite protrudes slightly at the vertex of the wedge.This is related to the delay necessary for the branches to free themselves.At large velocities, branches growing with ␣ ϳ 1 become frequent as the loss of stability of the tip creates large initial disturbances, and the diffusive field becomes thinner.As a result the initial region of growth ͑region I in Fig. 10͒ becomes shorter and the fit by a power law becomes less and less accurate.Some branches are observed to have a velocity increasing with time very shortly after their formation.It is not clear if they have a growth exponent ␣ Ͼ 1 or if this is the effect of the crossover regime of region II.
Two typical lengths can be measured.The first is l E the distance from the tip at which the branches become free.This length is l E = V where is the typical escape time of the branches.This time is measured with a rather good accuracy on plots of the type of Fig. 10, where it is the limit between regions II and III.On various dendrites obtained at different undercoolings the evolution of this time with the velocity was measured.As shown in Fig. 13͑a͒, diverges at low velocities.The second typical length is the distance between free branches.Since they are not evenly spaced, we measured the distances of as many free branches as possible and deduced an average value ¯.We then repeated the measurements on dendrites grown at various velocities.In the low range of velocities the free branches are scarce so that the average values are not precise.For a given dendrite the two lengths l E = V and ¯have similar values.If both l E = V and ¯obtained for various dendrites are plotted as a function of the diffusion length l D T at the tip, no simple relation is found.In particular for slow dendrites l E = V and ¯become very large as compared to l D T .This may appear surprising because the independent growth of different dendrites would be expected to occur whenever the dendrites are separated by a distance larger than a few diffusion lengths.But the question of the independent growth of two side branches differs from that of the independent growth of parallel dendrites.In order to become free, the branches must first escape out of the diffusion field of the lateral side of the primary dendrite.For very fast dendrites this occurs rather close to the tip in a region where the lateral front is still growing fast and the diffusion length is of the order of l D T .In contrast, in slower dendrites, the escape occurs much further away from the tip in a region where the mean motion is slow and where the local diffusion length has thus become large.Only branches much further away from each other can then become independent.
A reasonable assumption is that a branch can only become free when it has become larger than some fraction of the local mean diffusion length of the lateral front.This length at a given abscissa x grows as the square root of time ͓relation ͑6͔͒.Since the growth exponents ␣ n are larger than 0.5, the branches can, in principle, approach the diffusion front.
For these reasons the relevant comparison is that of both l E = V and ¯with the local lateral diffusion length l D S in the region where this escape occurs.This length l D S can be computed fixing t = in relation ͑6͒ and using the values of D and grows faster than its neighbors by a simple point effect.It does so at the expense of the growth of its neighbors.Since the mean growth is always maintained, the material that would have crystallized on the weaker branches is redistributed and contributes to the growth of faster branches with larger ␣ n .As a dominant branch grows, however, it becomes sensitive to the presence of other dominant branches located at larger distances along the lateral front.We have shown that this screening is a function of the sum of tangents of the angles q r and q s .It will only have a noticeable effect when the lengths ͑or rather the length differences͒ of these branches will become comparable to their spacing.In practice, with branches resulting from the normal noise, the screening effect remains weak between branches separated by a distance larger than their own length.
For faster dendrites, the range of values of the exponents ␣ n corresponding to the normal random fluctuations of the tip, shifts to larger values.This can be understood if we consider the same disturbance of the interface in dendrites of various velocities.The smaller the diffusion length, the larger the number of isoconcentration lines disturbed by a protrusion of a given amplitude.For this intuitive reason the same disturbance will grow with larger ␣'s in faster dendrites.This is confirmed by the results of the simulations using the phase field model presented in Part II.
For fast dendrites there is a crossover; at a certain distance from the tip, the dominant branches are observed to accelerate before reaching the constant velocity of a free dendrite.The transition to this regime of growth is subcritical: in the intermediate range, the branches resulting from the normal noise at the tip remain prisoner of the thickening lateral diffusion field of the dendrite.Only branches due to a large disturbance of the tip can reach the escape velocity.At very large velocities this escape occurs near the tip so that a wedge shaped array is formed, which occupies a finite fraction of the plane.

FIG. 1 .
FIG.1.͑a͒, ͑b͒ Sketches of two dendrites growing at small and large velocity respectively and of the isoconcentration lines in their vicinity.

FIG. 2 .
FIG. 2. Five photographs taken respectively at a time t o and t o + 300 s, t o + 660 s, t o + 1260 s, t o + 3180 s, of one side of a dendrite growing at low velocity V = 3.77 m / s.The length of the region shown here is of the order of L = 2.2 mm.The labels ͑1͒¯͑10͒ show the branches which have been investigated.
is a semilogarithmic plot of Y n ͑t͒ as a function of S n ͑t͒ for branch 4.An exponential decay is clearly observed showing that Y n and S n satisfy Y n = Y 0 exp͑− S n /S 0 ͒.͑10͒

FIG. 5 .
FIG.5.Logarithmic plot of the position of the tip of the disturbed dendrite of Fig.4and of the length of its main lateral branch as a function of time ͑in seconds͒.This branch grows as t ␣ p with ␣ p = 0.80.

FIG. 8 .
FIG. 8. Photograph of the tip of a fast dendrite in the regime of tip oscillations.

FIG. 9 .
FIG. 9. Photograph of a dendrite of velocity V =51 m / s after its tip has split.

FIG. 11 .
FIG. 11.Four photographs taken respectively at a time t o and t o + 9.2 s, t o +13 s, t o + 20 s, of one side of a dendrite growing at velocity V =64 m / s.The length of the region shown here is of the order of L = 2.2 mm.

TABLE I .
Growth exponent ␣ n and screening parameters Y o and S o for branches 1-8.
FIG. 4. Four photographs taken, respectively, 1, 3, 5.4, and 15.1 s, after the collision of a dendrite growing at velocity V = 7.7 m / s with a finite disturbance.