Shuhei Inoue

Takeshi Nakajima

Kazuya Nomura

Yoshihiro Kikuchi

Growth Mechanism and Selective Synthesis of SWNTs

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INTRODUCTION

Single-walled carbon nanotubes (SWNTs) [START_REF] Iijima | Single-shell carbon nanotubes of 1-nm diameter[END_REF] are considered as a special material; however, their growth mechanism remains unclear. This incomplete knowledge prevents their application to any field because of the difficulty in controlling their growth. In particular, controlling the diameter and chiral indices of SWNTs are major factors in any application. In order to control their diameter and chirality, knowledge of the growth mechanism of SWNTs is essential. To date, some promising growth mechanisms have been proposed [START_REF] Thess | Crystalline ropes of metallic carbon nanotubes[END_REF][START_REF] Yudasaka | Mechanism of the effect of NiCo, Ni and Co catalysts on the yield of single-wall carbon nanotubes formed by pulsed Nd : YAG laser ablation[END_REF][START_REF] Kataura | Diameter control of single-walled carbon nanotubes[END_REF][START_REF] Dai | Single-wall nanotubes produced by metalcatalyzed disproportionation of carbon monoxide[END_REF], furthermore, recently Shibuta & Maruyama [START_REF] Shibuta | Molecular dynamics simulation of formation process of single-walled carbon nanotubes by CCVD method[END_REF] and Ding et al. [START_REF] Ding | Molecular dynamics study of the catalyst particle size dependence on carbon nanotube growth[END_REF] have demonstrated the nucleation of SWNTs by molecular dynamics simulation, and also Fan et al. [START_REF] Fan | Nucleation of single-walled carbon nanotubes[END_REF] have shown the nucleation pathway for SWNTs on a metal surface by DFT calculation. In their papers carbon atoms are directly provided to the catalysts or metals. On the other hand Yudasaka et al. [START_REF] Yudasaka | Causes of different catalytic activities of metals in formation of single-wall carbon nanotubes[END_REF] have tried to explain the synthetic model from the experimental aspect, they have explained the nucleation of SWNTs in terms of graphitization and solubility of catalytic metals in graphite.

However the growth mechanism of SWNTs has not been clear. We also proposed a simple growth model [START_REF] Inoue | Diameter Control and Growth Mechanism of Single-Walled Carbon Nanotubes[END_REF] that predicted the possibility of low-temperature synthesis of SWNTs using the Rh and/or Pd catalysts. Some reports [START_REF] Kataura | Diameter control of single-walled carbon nanotubes[END_REF][START_REF] Kataura | Formation of thin single-wall carbon nanotubes by laser vaporization of Rh/Pd-graphite composite rod[END_REF] have described the use of the Rh/Pd catalyst in a laser furnace technique; however, the synthesis SWNTs using this technique requires a reaction field in high temperature. On the other hand, the macroscopic production of SWNTs was almost realized [START_REF] Maruyama | Low-temperature synthesis of high-purity single-walled carbon nanotubes from alcohol[END_REF]. They overcome this difficult issue of macroscopic production using the water-assisted CVD technique, so-called "Supergrowth". In this paper, we have presented the selective synthesis of SWNTs with different diameters using the combined catalysts.

EXPERIMENTAL

A schematic view of the experimental apparatus, which is similar to that of Maruyama group [START_REF] Maruyama | Low-temperature synthesis of high-purity single-walled carbon nanotubes from alcohol[END_REF], is shown in Fig. 1. Ethanol vapor is made to flow in a quartz tube (I.D. 27 mm, O.D. 30 mm and 1200 mm) with a consumption rate of 0.9 cm3/min in liquid. The metal catalysts supported with zeolite [START_REF] Mukhopadhyay | A simple and novel way to synthesize aligned nanotube bundles at low temperature[END_REF] are prepared in the quartz boat, which is set at the center of an electric furnace. In the apparatus, the electric furnace consists of two parts, which can be controlled well (± 0.5%), individually. A typical procedure is as follows:

Setting the quartz boat at the center of the electric Raising the temperature of the electric furnace up to the preset value, and maintaining it for 150 minutes.

Opening the ethanol gate and flowing gaseous-alcohol for 10 minutes at 400 Pa. The alcohol vessel is set in the water reservoir (10 °C) to keep the alcohol temperature constant. The temperature of the reactor, which is directly measured, rises gradually but not as high as the electric furnace temperature.

RESULTS AND DISSCUSSION

Figure 2 shows the typical Raman spectra of the raw soot, which was synthesized at 950 °C using the (a) Rh/Co, (b) Pd/Co, (c) Ni/Co, and (d) Fe/Co catalysts. The peaks in the image of each spectrum are quite similar-strong peaks around 1590 cm -1 , broad peaks around 1550 cm -1 , and a very weak peak at approximately 1350 cm -1 called the D band. The group theory predicted that SWNTs have six Raman-active modes within their tangential G-band region [START_REF] Jorio | Polarized Raman Study of Single-Wall Semiconducting Carbon Nanotubes[END_REF][START_REF] Rao | Diameter-Selective Raman Scattering from Vibrational Modes in Carbon Nanotubes[END_REF][START_REF] Saito | Raman intensity of single-wall carbon nanotubes[END_REF][START_REF] Rahmani | Nonresonant Raman spectrum in infinite and finite single-wall carbon nanotubes[END_REF], which extends from approximately 1550 cm -1 to 1600 cm -1 . The G-band feature can be used for distinguishing the metallic and semi conducting SWNTs in a sample. Four symmetric Lorentzian features at 1553, 1569, 1592, and 1607 cm -1 are assigned as peaks from semi conducting SWNTs, and they correspond to the E 2 (E 2g ), A(A 1g ) + E 1 (E 1g ), A(A 1g ) + E 1 (E 1g ), and E 2 (E 2g ) modes, respectively. On the other hand, metallic SWNTs have two strong peaks [START_REF] Brown | Origin of the Breit-Wigner-Fano lineshape of the tangential G-band feature of metallic carbon nanotubes[END_REF]: one is located at approximately 1540 cm -1 and exhibits an asymmetric Breit-Wigner-Fano feature, which is expressed as following equation.

( ) [ ] ( ) [ ] 2 0 2 0 0 / 1 / 1 ) ( Γ - + Γ - + = ω ω ω ω ω q I I
Generally, these two modes of G and D are used for simply discussing the purity of SWNTs. Figure 3 shows this G/D ratios of SWNTs obtained from the different types of catalysts, and indicates that the rhodium and palladium catalyst (Rh/Pd) exhibits poor catalysis when they are applied individually or together even in the higher temperature region (below 1000 °C it was impossible to measure this ratio). However, effective catalysts can be obtained by blending them with cobalt. In particular, the Rh/Co (▼) and Fe/Co (•) catalysts exhibit good G/D ratios. The latter is usually considered as one of the best catalysts in ACCVD. On the other hand, the Pd/Co catalyst does not show such a good G/D ratio; however, it exhibits a remarkable increase as compared to the Rh/Pd catalyst. These results indicate that in the ACCVD technique, blending cobalt can enhance the catalyst ability. Moreover, the Ni/Co catalyst (►) also shows a good G/D ratio. That is why combining cobalt with other kinds of metal might be a good catalyst for ACCVD.

Figure 4 and Fig. 5 are the low-frequency Raman spectra of SWNTs (RBM mode) where the catalyst density is different. It is speculated that the catalyst particle grows in size with an increase in its density. RBM mode is dependent on the diameter of the SWNTs, so that in this paper, we use the relationship d (nm) = 248 / ω (cm -1 ), where ω indicates the RBM frequency. As shown in Fig. 4, three main peaks are observed, the first peak is approximately at 216 cm -1 , second is at 254 cm -1 , and the last is at 280 cm -1 . Referring to this figure, as catalyst density becomes larger (i.e. a particle size becomes larger) a diameter distribution becomes larger.

Figure 5 shows also RBM mode with different catalyst density. Basically observed phenomenon shows a same tendency with Fig. 4, and this tendency is much clear. Almost most of detected SWNTs in Fig. 5(b) are 1.3 nm in diameter, on the other hand in Fig. 5(a) there are four or five dominant SWNTs are detected from 0.9 nm to 1.3 nm. Furthermore in Fig. 5(b) two peaks are shown around 350 cm -1 and 370 cm -1 indicated by arrows, however since these peaks are not consistent with Kataura Plot, further experiment is essential.

In any case of these experiments shown in Fig. 4 and Fig. 5, assuming that every surface area of zeolite can support the catalyst particles, the number of particles must be equal and particle size is proportional to the density. In case of this assumption a correlation between the particle size and catalyst density is expressed as follows:

2 1 3 1 1 1       ≤ ≤       ρ ρ N r N ,
where N is the number of supportable areas of zeolite, ρ is catalyst density, and r is particle size. According to this assumption in case of Fig. 4 a difference of catalyst density is 10 times and in case of Fig. 5 this is 2.5 times, so that the difference of diameter should be 2.15 -3.16 times and 1.36 -1.58 times respectively. On the other hand 1.2 nm / 0.9 nm = 1.33 times in Fig. 4 and 1.3 nm /0.9 nm = 1.44 times in Fig. 5 are experimentally shown. In case of Fig. 4 there is a big gap between ideal value and experimental value; however, in case of Fig. 5 there shows a very good agreement. It might be a reason for this disagreement in Fig. 4 that lower wave number regions (ω ≤ 200 cm -1 ) could not be observed owing to some *Owing to the filter problem ω < 200 cm -1 could not be observed unlikely Fig. 5. experimental problems. Referring to these results, changing the particle size can be an effective technique to control the diameter of SWNTs. Figure 6(A) shows the RBM spectra of the different compositions of catalytic metal species. Since the cobalt and the iron atoms have almost same atomic masses, the size of the catalyst particles supported on zeolite may be almost equivalent. Hence, there is no influence of the size of the catalyst particle under these conditions, although the peak distribution is quite different. The RBM spectra of SWNTs synthesized on the iron-rich catalyst exhibit three peaks, which are almost equivalent. On the contrary, the RBM spectra of SWNTs from the cobalt-rich catalyst show one strong peak (216 cm -1 ) and two weak peaks (254 cm -1 and 280 cm -1 ). A similar phenomenon is observed in Fig. 6(B). In this experiment, the gross weights of catalytic metals are less than those shown in Fig. 6(A), but the amount of catalytic metal is almost equivalent in both (Fe/Co 2.5 wt.% and Co 5.0 wt.%). The differences in diameter distributions in Fig. 6(B) are more distinct than those shown in Fig. 6(A). In case of the heterogeneous catalyst (Fe/Co 2.5 wt.%), the peak at 280 cm -1 is dominant and the peak at 216 cm -1 is very weak; however, in the case of the homogeneous catalyst (Co 5.0wt.%), the peak at 216 cm -1 is apparently dominant. On the other hand, SWNTs are not synthesized on the iron (only) catalyst (Fe 5.0 wt.%).

100
These experimental results clarify the role of catalytic metals; cobalt atoms can catalyze the pyrolysis of ethanol; however, iron atoms cannot decompose ethanol. Hence, the iron (only) catalyst (Fe 5.0 wt.%) could not synthesize SWNTs. Both iron and cobalt atoms promote the growth of SWNTs; they support the core of SWNTs, as shown in Fig. 7(A), which is a schematic diagram of the growth model of SWNTs. Yudasaka et al. [START_REF] Yudasaka | Causes of different catalytic activities of metals in formation of single-wall carbon nanotubes[END_REF] mentioned the graphitization but we consider, typically, the carbon atoms tend to take the sp 2 structure and not the sp 3 structure under low pressure, especially in this synthetic technique the time scale is much longer than that of laser ablation or arc discharge that is why carbon atoms can be annealed fully. Since the electric orbits of the sp 2 structure spread along the x-y plane, it is difficult for carbon atoms to achieve the original cylindrical form. In order to maintain a cylindrical form against the distortion energy (Fig. 7(B)), support materials are a prerequisite. Particularly, to generate narrow SWNTs, carbon atoms must be strongly supported by more associative materials. Since the iron atoms are much associative and exhibit stronger interactions between carbon atoms than those of cobalt atoms, SWNTs synthesized on the iron-rich catalyst have narrow diameters. The relationship between synthetic temperature and diameter, which has been elucidated in some previous reports, can be easily explained through this model. At a high temperature, vibration energy is also high. Therefore, the total repulsion energy, which is the sum of the vibration and distortion energies and originated from the distortion of sp 2 orbits, is too large to maintain a narrow diameter. Hence, wider diameters are obtained at higher temperatures. Based on these assumptions, the simple growth mechanism of SWNTs is as follows: First, the "core" is nucleated on the catalyst particles and supported against the distorted energy of sp 2 orbits. The diameter of the "core" is determined by balancing of energy in terms of size of catalyst particle, temperature, and the interaction between the catalyst and carbon atoms. The carbon source (ethanol is used in this work) is decomposed by the cobalt atoms and supplied to the "core." The SWNTs are synthesized after some time. This growth model can briefly explain the other synthetic technique. For example, cobalt atoms are not always required for the laser-furnace technique and arcdischarge techniques. Since carbon atoms are directly provided in these techniques, the role of catalytic metals is limited to supporting the "core." Some reports stating that using Rh/Pd catalyst may generate very narrow SWNTs are available; however, a higher temperature is required for the decomposition of ethanol and the interaction between carbon atoms. Since rhodium or palladium atoms cannot decompose ethanol at a low temperature, both or either has much stronger interactions between carbon atoms than that of the iron and cobalt atoms. Therefore, very narrow SWNTs are generated at higher temperatures. If Co/Rh or a Co/Pd catalyst is employed, narrower SWNTs can be synthesized at low temperatures. This growth model can also be extended to the growth mechanism of multi-walled carbon nanotubes (MWNTs). In the absence of a metal catalyst in the laser-furnace or the arc-discharge techniques, only MWNTs are synthesized and the synthesis of SWNTs is nearly impossible. Further, in the absence of catalyst particles, the "core" must maintain its structure by itself against the total repulsion energy, and the diameter must be larger to relax their distortion energy. Moreover, since a single layer is too weak to maintain a cylindrical structure, a pillar is needed in the center of the wide tube to enable a multi layer growth processes. MWNTs are generated following these processes with the support of their π bonds, which are along the direction of a radius, against the distortion energy.

CONCLUSION

In this paper, the most simple growth model of SWNTs is proposed. The balance of the total repulsion energy and the association energy, which are interacted with the "core" and support catalysts, determines the diameter of the SWNTs. The explanation to the question why the dual atom catalyst is ideal for the ACCVD technique is because that one specie functions as a decomposer and the other supports the "core." According to this idea, changing the supporter can easily control the diameter of SWNTs; further, the interaction between carbon atoms is also modified.

Fig. 2

 2 Fig. 2 Raman spectra of as-grown SWNTs grown using (a) Rh/Co, (b) Pd/Co, (c) Ni/Co, and (d) Fe/Co catalysts of 2.5 wt.% each at a furnace temperature of 950°C. The excitation wavelength is 632.8 nm.

Fig. 4

 4 Fig. 4 The RBM spectra of SWNTs. The density of catalytic metals is different. The excitation wavelength is 632.8 nm. *Owing to the filter problem ω < 200 cm -1 could not be observed unlikely Fig. 5.

Fig. 7 A

 7 Fig. 7 A schematic diagram of the growth mechanism of SWNTs. (A) The "core" is generated and supported by the catalytic particles. (B) The repulsing energy and the stabilizing energy.

©TIMA Editions/ENS 2006-page-ISBN:

ACKNOWLEDGEMENT

The Raman spectroscopy measurements were obtained using JRS-SYS2000 at Shimane Institute for Industrial Technology and NRI-1866M at the Natural Science Center for Basic Research and Development (N-BARD), Hiroshima University. This study was partly supported by a research grant from the Mazda Foundation.


R Rao

J E Bradby

J S Williams

NANOINDENTATION-INDUCED PHASE TRANSFORMATION IN SILICON

Nanoindentation-induced phase transformation in silicon has been studied. A series of nanoindentations were made with the sharp diamond Berkovich tip. During nanoindentations, maximum load ranged from 2000 µN to 5000 µN, with a 1000 µN/sec loading rate. Slow unloading rate at 100µN/sec was chosen to favor the formation of the crystalline end phases, high pressure phase (Si-III and Si-XII). Fast unloading rate at 1000µN/sec was used to obtain amorphous phase. The phase transformation was examined by Raman spectroscopy and plan-view transmission electron microscopy (TEM). HPP have been identified even if no "pop-in" and "pop-out" observed in load-depth characteristics curves. HPP appeared in c-Si when the maximum load up to 3000 µN. TEM images have been revealed that the optimization HPP transformation in c-Si at the nanoscale occurred when the maximum load applied at 5000 µN.

INTRODUCTION

It has been known that the cubic diamond structure of silicon (Si-I) can transform to a metallic phase (Si-II) under indentation load. Upon pressure release Si-II undergoes further transformation. The structure of the final phase can be either a-Si or poly-Si (Si-III/Si-XII). Fast unload rates result in the amorphous phase and slower unload rates in Si-III/Si-XII. Many works have reported the phase transformation in silicon by microindentation [START_REF] Zarudi | The difference of phase distributions in silicon after indentation with Berkovich and spherical indenters[END_REF][START_REF] Bradby | Mechanical Deformation in silicon by microindentation[END_REF][START_REF] Bradby | Transmission electron microscopy observation of deformation microstructure under spherical indentation in silicon[END_REF] . Recently, although a few works have examined the phase transformation by nanoindentation [6][7] , the mechanisms of the phase transformations and the corresponding microstructural changes in silicon by nanoindentation have never been explored in detail.

Obviously, it is significant to study characterization of such phase transformation in Si at the nanoscale.

In this work, we studied characterization of such phase transformation in crystalline silicon (c-Si) at the nanoscale. We focused on the behavior of c-Si under sharp Berkovich indentation during loading and unloading. We analyzed the microstructure changes by Raman spectroscopy and examined the deformation regions using plan-view TEM.

EXPERIMENT

In the experiments, p-type c-Si wafers with (100) crystallographic orientation and resistivity of 8-12 Ωcm were used. A series of nanoindentations were performed with the sharp diamond Berkovich tip using TriboIndenter (Hysitron Inc., USA). During nanoindentations, maximum load ranged from 2000 µN to 5000 µN, with a 1000 µN/sec loading rate. Slow unloading rate at 100µN/sec was chosen to favor the formation of the crystalline end phases, Si-III and Si-XII. Fast unloading rate at 1000µN/sec was used to obtain amorphous phase.

After indentations, Raman spectra were recorded with Renishaw 2000 Raman Imaging Microscope using the 647.1nm excitation lines of a heliumneon laser focused to a spot of ~ 1 µm radius. According to the atomic force microscopy (AFM) image, the width of the nanoindents is about 500 nm. Therefore, nanoindents closely spaced with 0.5 µm in an effort to obtain enough Raman signal to examine the phase changed. Furthermore, the deformed regions were investigated by plan-view TEM.

RESULTS AND DISCUSSION

Figure 1 shows a load versus penetration plot for a range of maximum loads. It is well known that c-Si exhibits a discontinuity on loading and unloading during micro-indentation. The discontinuity on loading is called "pop-in" and the discontinuity on unloading is called "pop-out". Such discontinuities during micro-indentation have been reported to occur for c-Si as a result of pressure-induced phase transformations [2] .

However, under the nanoindentation conditions shown in Fig. 1, no discontinuities on either the loading or unloading sections of the curve are observed and it is interesting to examine if phase transformation have occurred in these cases. We use Raman spectroscopy (shown in Fig. 2) to examine if phase transformation occurred during these nanoindentations. The multiple narrow bands at 161, 350 382, 394 and 430 cm -1 suggest the presence of the SI-III and Si-XII phases. The bands at 350 and 394 cm -1 have been assigned to Si-XII, and the bands at 166, 382, 430 cm -1 , to Si-III. We can see that the Si-III/Si-XII phases start to appear when the maximum load applied reaches about 3000 µN. The Si-III/Si-XII peaks become sharper and stronger with increasing maximum load. Therefore, it is clear that phase transformation can occur in c-Si during nanoindentation even if no discontinuities on either the loading or unloading curve are observed.

Even if we observed HPP peaks appeared at maximum load applied up to 3000 µN in Raman spectra, TEM images (Fig. 4) taken from this deformation region show the nanoindent is amorphous state. However, the SAD pattern from the indent made by 5000 µN shown in Fig. 4(b) is in agreement with the Raman result. Compared the two bright field images shown in Fig. 3(a) and Fig. 4(a), it is obviously that the plump shape of the indents can be obtained at 5000 µN load value. According to the plan-view TEM results, we think that the optimization high pressure phase transformation in c-Si at the nanoscale occurred when the maximum load applied at 5000 µN.

CONCLUSION

In conclusion we have observed nanoindentationinduced phase transformation in c-Si resulting in the end phases Si-III and Si-XII, even though no "pop-in" and "pop-out" are observed in nanoindentation curves. HPP began to appear in c-Si when the maximum load up to 3000 µN. TEM images have been revealed that the optimization HPP transformation in c-Si at the nanoscale occurred when the maximum load applied at 5000 µN. The results indicate that further investigation is needed to understand the nanoindentation behavior of Si.

Fig. 1

 1 Fig.1 Load-depth curves of nanoindentation in c-Si

Fig. 2

 2 Fig.2 Raman spectra of nanoindentations in c-Si


Agustín E González

CLUSTER ANISOTROPIES IN COLLOIDAL AGGREGATION IN THE PRESENCE OF GRAVITY

The colloidal aggregation problem coupled with the sedimentation experienced preferentially by the large clusters is stratified, the structural and dynamical quantities describing the aggregates depending on the depth at which they are measured. In this work we will use a new computer algorithm for colloidal aggregation coupled with sedimentation, that is capable of proportioning the average of these quantities inside a layer at a fixed depth and of arbitrary thickness in the aggregation prism. The dynamics, using this algorithm, will be shown to be in accordance with the experimental results. We have also found that (i) in some cases of sedimentation strengths and layer depths, the mean width (perpendicular to the gravitational field direction) and the mean height of the large settling clusters scale with the size as a power law, with the same scaling power, in some range of cluster sizes. This leads to self-similar clusters with an appreciably higher fractal dimension (d f ) than the d f of aggregating clusters driven purely by diffusion (DLCA). However, (ii) there are some other cases for which the parallel and perpendicular scaling powers differ. (iii) There are still cases for which only the mean width or the mean height scale as a power law. Finally, (iv) there are further cases for which neither the mean width nor mean height scale as a power law with the size. In the last (ii), (iii) and (iv) cases the settling clusters are anisotropic and a fractal dimension cannot be defined by them.

INTRODUCTION

Particle aggregation is of great interest not only due to the variety of roles it plays in biological systems, medical diagnostics, paints and coatings, and numerous foods, but also as a model system for growth under non-equilibrium conditions. In the past two decades, work has focused on the formation of aggregates, including their geometry and growth kinetics, in the absence of sedimentation. After the proposal of a single-aggregate model by Witten and Sander [1], and more adequate for our purpose of colloidal aggregation, after the development of a diffusive model for cluster aggregation, simultaneously made by Meakin [2] and by Kolb et al [3], the number of works dealing with the aggregation of particles has increased considerably. However, many real aggregation phenomena rarely take place under quiescent conditions but instead occur in the presence of gravitational fields or macroscopic flows. In addition, external fields, including electrical, magnetic, and optical, have proven useful for assembling and moving colloidal particles, techniques that should be useful when dealing with particles of nanometers in size, since they are barely affected by gravity. Among the many examples for which the sedimentation effect is important we can cite the clearing or clarifying of liquids, the settling of bacteria clusters in quiet water, the aggregation and deposition of asphaltenes in crude oil, and a number of precipitation techniques employed by the chemical industry.

By adding salts or flocculants, which screen the electrostatic repulsion between the particles or which establish bridges between them, it is induced the aggregation of the particles into clusters, which themselves collide with other clusters, stick together and become larger. Initially, the aggregates are small and essentially move by diffusion. As the aggregation proceeds, the settling velocity of the large aggregates becomes significant, and a new mechanism for the movement of these large clusters appears. As the sedimentation velocity of the small aggregates is lower than that for the larger ones, these large clusters catch up with the smaller ones, actually sweeping them and becoming in the process larger and larger. As a result, the hitting rate between aggregates increases, which leads to an acceleration of the aggregation kinetics. However, as the colloidal matter in the bulk becomes more and more scarce, the aggregation kinetics eventually slows down. In the final state and for the non-gelling cases (low concentrations), the system is composed of a tenuous sediment on the bottom and a clear fluid above.

Experimentally, this new problem of aggregation coupled with sedimentation has been studied more extensively by Allain and collaborators [4,5,6], who found an increase in the fractal dimension (d f ) of the large, settling aggregates (reaching values as high as 2.2) [5,6]. They attributed this increase in the d f of the large clusters to a restructuring mechanism, due to the hydrodynamic stresses felt by their branches when they drift downwards. They also found the large settling aggregates to be non-rotating. This behavior was interpreted in Refs. [7,8] in terms of a non-homogeneity of these clusters, being denser in their lower part, which is the one that is sweeping the smaller clusters that attach to them. The large clusters therefore act as badminton shuttlecocks that, when left rotating at a certain altitude, at the end they fall at constant velocity without rotation.

On the simulational side and in our first works [7,8,9] that consider both sedimentation and deposition (through a rarefaction of colloidal matter in the bulk), due to computational limitations, the structural (d f ) and dynamical quantities (basically the weight-average cluster size S w (t)) were obtained from averages over the clusters inside the aggregating prism, independent of their vertical position. This assumption was necessary in order to have good statistics for the evaluation of the averages of highly fluctuating quantities. However, the colloidal aggregation problem coupled with sedimentation is not homogeneous on the vertical direction. In Fig. 1 we are reproducing figure 1 of Chapter III of Ref. [4] (the horizontal straight lines were added by the present author). In their system with a colloidal volume fraction of φ = 0.001, the Calcium Carbonate spherical colloidal particles were of a diameter of 0.07 µm; therefore, they cannot be optically seen unaggregated, and even the smallest points shown in the confinement prism are actually clusters of many particles. The prism was illuminated from behind in such a way that what we actually see are the shadows of the clusters. We can clearly see in this figure that, for a sufficiently long time, the large clusters lie preferentially deeper than the smaller ones, which leads to a stratification of the system, as mentioned. Moreover, in each of the seven sublayers shown, we can also see that the average cluster size increases with time and peaks at a certain time, diminishing afterwards. The deeper the layer is, the longer the time it takes to peak and the higher the value at which it peaks.

More recently, in Ref. [10], the authors considered a two-dimensional lattice model for aggregation coupled with sedimentation; that is, they considered the z coordinate on the direction of the field and only one horizontal coordinate. They found that, for some cases of high sedimentation strengths and within some range of sizes, the mean width of the large settling clusters scaled as a power law with the cluster size while the mean height did not, leading to anisotropic clusters for which a fractal dimension could not be defined. To proceed further in this problem, what we actually need is an algorithm capable of proportioning the average aggregation quantities at a fixed depth in the prism, and consider a good number of cases of sedimentation strengths and layer depths. In Ref. [11] such an algorithm was presented in the form of a letter in which, by the nature of its size, it was not possible to show all the different cases of sedimentation strengths and depths, that lead to different behaviors in this problem. In this paper we consider a number of interesting cases of sedimentation strengths and depths, with such algorithm, presenting in more detail two of the cases.

To simulate such a system of diffusing, sedimenting and aggregating particles and clusters, one needs to know by which amount a cluster of N particles undergoes sedimentation as compared to the distance diffused, during a certain Monte Carlo time. Let us consider the sedimentation velocity v s experienced by a cluster of N spherical particles of radius a and mass m o :

v s = m o (1 -ρ/ρ o )gN f = m o (1 -ρ/ρ o )g k B T DN , (1)
where ρ o is the particles density, ρ is that of the suspension fluid, f is the cluster's friction coefficient, D (= kB T 6πηRg ) is its diffusion coefficient, R g is its radius of gyration, η is the solvent viscosity and T is the temperature. Let t o be the time for which the cluster diffuses a particle diameter (d = 2a), that is, t o = 2a 2 /3D. During the same Peclet number gives the sedimentation strength felt by the particles, being a number much smaller than one for most colloidal systems with a size generally smaller than 1µm. It is in fact not difficult to show that if the colloidal particles are 1 µm in diameter, 1 -ρ/ρ o is less than but of the order of unity, and T is room temperature, P e is of the order of unity. However, if the diameter is 0.1 µm such quantity is of the order of 10 -4 , while if the diameter is 10 µm, P e goes as high as 10 4 . Therefore, 1 µm marks the transition between diffusive and drifting behavior for individual particles, with density different from that of the medium.

The details of the algorithm are too long to be reproduced here. The interested reader is referred to Ref. [11] for those details. The volume fraction was fixed at the value φ = 0.001. We considered 4 depths: Z = 500, 5000, 50000 and 500000, measured in terms of the diameter. For each of those depths, 5 Peclet numbers were considered: P e = 0.0001, 0.001, 0.01, 0.1 and 1.0 . In turn, for each Peclet number, a series of 10 simulations of 274625 particles were made in order to have enough statistics to evaluate the structural and dynamical quantities.

DYNAMICAL RESULTS

In Fig. 2 are shown the weight-average cluster sizes as a function of time, S w (t), for the Peclet numbers (a) 1.0, (b) 0.1, (c) 0.01 and (d) 0.001. As we can see in the figures, S w increrases with time, peaks at a certain time and diminishes afterwards, a behavior that is in accordance to what we saw qualitatively in Fig. 1. We also note in all four figures that the deeper the layer we are considering, the longer the time it takes to peak and the higher the value at which it peaks, again in correspondance with Fig. 1.

STRUCTURE

The addition of a gravitational field on the vertical z direction in the colloidal aggregation problem, felt preferentially by the big clusters, introduces some degree of anisotropy in the structure of these clusters, as we will see below. It is therefore necessary to define and study some anisotropy measures of the clusters before trying to do a radius of gyration (R g ) vs size (N ), log-log analysis, in order to see if it is possible to extract a fractal dimension from such plots. Let us define the anisotropy measures A xz ≡ < R gx /R g > / < R gz /R g > and A yz ≡ < R gy /R g > / < R gz /R g >, where R gx , R gy and R gz are the diagonal components of the radius of gyration tensor, such that R 2 g = R 2 gx + R 2 gy + R 2 gz , and where the average values are calculated over all clusters inside segments of constant magnitude in the logarithmic size scale. Note that for isotropic clusters, the measures A xz and A yz should be equal to one, up to the statistical uncertainties. After studying the A xz and A yz quantities, we will make plots of not only < R g > vs N , but also of < R gx > vs N , < R gy > vs N and < R gz > vs N , to find any regions for which we have a power law scaling of those quantities. Here again the averages of the different radii are made over the same segments of constant magnitude in the logarithmic size scale. We have found a whole variety of behaviors for the structure of the clusters in this problem, depending on the sedimentation strength (P e), the layer depth (Z) and the region of sizes considered. Generally speaking, the four radii (R gx , R gy , R gz and R g ) scale as a power law with N , with the same scaling power, for the small, non-settling clusters, except for a number of cases with a high sedimentation strength: P e = 1.0 for all depths and P e = 0.1 for Z = 500000. In those cases of cluster isotropy it is therefore possible to define a cluster fractal dimension, This behavior shall be called the quasi-DLCA regime, where the "quasi" means that, as there is still some sweeping of even smaller clusters, the fractal dimension is a little bit higher than the usual DLCA d f . For the large, settling clusters, we have found cases for which (i) the four radii again scale as a power law with N , with the same scaling power, making it possible to define a settling-clusters fractal dimension, a behavior that shall be called the sweeping scaling regime. There are however cases for which (ii) the scaling powers for the horizontal and vertical directions differ, obtaining therefore self-affine settling clusters. There are still some cases for which (iii) only the mean width or the mean height scale as a power law with N , and even further cases for which (iv) no scaling as a power law of the four radii is obtained.

The sweeping scaling regime

This regime is obtained at deep layers and intermediate Peclet numbers; the deeper the layer considered, the lower the P e that can be used to attain the regime. In our studies we have found this regime for Z = 500000 with P e = 0.01, 0.001 and 0.0001, and for Z = 50000 with P e = 0.001. In Fig. 3 we are showing the case for Z = 500000 and P e = 0.001. The anisotropy measure A yz is first shown in Fig. 3a. As we can see, A yz stays very close to one for all sizes up to about 10000, which is therefore the upper bound in this case for trying to obtain a fractal dimension from the log-log plots of the four radii vs N . After this upper bound, the clusters become definitely oblate. It should be mentioned that for the anisotropy measure A xz we obtain a quite similar plot, with the same upper bound. In Fig. 3b is shown the log-log plot of the average radius R gx vs N . After an initial curvature, correspondig to the corrections to scaling zone for small sizes, we obtain not one but two straight lines, before the upper bound is reached. This means that there are two zones for which we have scaling as a power law of the R gx with N , one defined by the arrows labeled "a" and the other by the arrows labeled "b". The first one corresponds to the quasi-DLCA regime, with an inverse of the scaling power equal to d f ax = 1.842 ± 0.008, while the second one has an inverse of the scaling power of d f bx = 2.086 ± 0.111. After the upper bound, the R gx does not scale with N as a power law. For the average radius R gy vs N a very similar plot is obtained, but this time with d f ay = 1.851 ± 0.010 and d f by = 2.003 ± 0.055. The log-log plot of the average R gz vs N is shown in Fig. 3c, where we can extract the following inverses of the scaling powers: d f az = 1.847 ± 0.006 and d f bz = 2.044 ± 0.106. Up to the statistical uncertainties, the scaling powers for the "a" zone coincide as well as those for the "b" zone. This means that it is therefore possible to obtain a fractal dimension for each zone. This is done in Fig. 3d, where we are plotting now the average R g vs N . The inverses of the scaling powers provide us now with the following fractal dimensions: d f a = 1.852 ± 0.005, which corresponds to the quasi-DLCA regime, and d f b = 2.060 ± 0.049, corresponding to what we have called the "sweeping scaling regime" [8,11].

Cases for scaling of the mean height or the mean width only

We have found cases for which the mean height scales as a power law with N while the mean width does not and viceversa. The cases of scaling of the mean height were for P e = 0.1 with Z = 50000 and 5000, while those of scaling of the mean width were for P e = 0.01 with Z = 50000, and for P e = 0.1 with Z = 500. In Fig. 4 we are showing the case P e = 0.1 with Z = 50000. In Fig. 4a we can see that the anisotropy measure A xz decreases very soon from one, reaching values around 0.8, which indicates somewhat elongated clusters. Afterwards it starts increasing very rapidly, crossing one and reaching values close to 3.0, which indicates this time very oblate clusters. A very similar plot was found for A yz . In these cases of high P e ′ s, the quasi-DLCA regime disappears, hidden in the curvature of the zone of the corrections to scaling. However, for the settling clusters we can see that there is no scaling as a power law of the average R gy vs size, as shown in Fig. 4b. The graph is all curved, with no possibility to define a straight line. A quite similar plot is obtained for the average R gx vs size. Notwithstanding this, in Fig. 4c we can clearly see a zone of the settling clusters with a well defined straight line, which indicates scaling as a power law of the average R gz vs N . The inverse of the scaling power was found as d f bz = 1.927 ± 0.018, above the DLCA fractal dimension. For the average of the whole R g vs N we see again a curved graph, with no possibility to define a fractal dimension, as shown in Fig. 4d.

CONCLUSIONS

As it was seen, it is possible to devise an algorithm to study the colloidal aggregation problem coupled with sedimentation, that can provide us with the average aggregation quantities at a fixed depth in the aggregating prism. The higher value of the d f for the sweeping scaling regime comes, in our case, from the sweeping of the small clusters by the large settling ones, which in turn occlude the holes and cavities of these large aggregates, increasing in this way their compacticity. Note that we cannot invoke a restructuring mechanism since it is not built-in in our model. Notice however that we have found a lower fractal dimension (∼ 2.05) for the settling clusters than the one found by the experimentalists (∼ 2.2) [6]. It is conceivable that also a restructuring of the large clusters would make them still more compact, helping to push its d f up to about 2.2 . However, some other mechanisms could be pre-sent, like a sticking probabilitry lower than one (which would increase the d f ), the breakage of the clusters above a certain size (which may also increase their d f ), or the inclusion of the hydrodynamic interactions (whose effect on the d f is unclear). As we have also seen, the present problem is richer than the colloidal aggregation problem driven purely by diffusion due to the whole variety of anisotropic cases obtained.

Figure 1 :

 1 Figure 1: The states at different times of a Calcium Carbonate aggregating suspension of spherical particles, for the following times: (a) 5 min 20 s, (b) 6 min 20 s, (c) 6 min 40 s, (d) 7 min, (e) 7 min 20 s, (f) 7 min 40 s, (g) 8 min (from Ref. [4]).

Figure 2 :

 2 Figure 2: A log-log plot of the time evolution of the weight-average cluster size, S w , for the Peclet numbers: (a) P e = 1, (b) P e = 0.1, (c) P e = 0.01 and (d) P e = 0.001. In all four figures the labels of each of the curves: a, b, c and d, correspond to the following depths: 500, 5000, 50000 and 500000, respectively.

Figure 3 :

 3 Figure 3: All four figures correspond to the layer at Z = 500000 and P e = 0.001. (a) A plot of the anisotropy measure (see the text) A yz vs the size. (b) A plot of R gx vs the size. (c) A plot of R gz vs the size. (d) A plot of R g vs the size. In Figs. (b), (c) and (d) the plotted radii are averages over all clusters inside segments of constant magnitude in the logarithmic size scale N .

Figure 4 :

 4 Figure 4: All four figures correspond to the layer at Z = 50000 and P e = 0.1. (a) A plot of the anisotropy measure A xz vs the size. (b) A plot of R gy vs the size. (c) A plot of R gz vs the size. (d) A plot of R g vs the size.

Acknowledgments

This work was supported in part by DGAPA-UNAM (PAPIIT Project IN118705-2).