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Self-Assembly of Colloidal Pyramids in Magnetic Fields L. E. Helseth School of Physical and Mathematical Sciences, Division of Physics and Applied Physics, Nanyang Technological University, Singapore Received April 28, 2005. In Final Form: May 30, 2005
We study routes toward the construction of 2D colloidal pyramids. We find that magnetic beads may self-assemble into pyramids near a nonmagnetic 1D boundary as long as the number of beads in the pyramid does not exceed 10. We have also found that a strong magnetic field gradient could act as a boundary, thus assisting the self-assembly of magnetic colloids in water, and have observed the formation of stable microscopic pyramids within a certain magnetic field range. Our results indicate that colloidal pyramids can be formed in a number of ways by utilizing external fields.
1. Introduction During the last 4500 years, the pyramids of Egypt and Mexico have drawn considerable admiration and interest, ranging from grave robbery to religious worship. The largest pyramid, built for Pharaoh Khufu around 2530 B.C., was until the early twentieth century the largest building on our planet. The question of how these constructions were built has not yet been answered properly and therefore continues to fascinate mankind. Our fascination with pyramids also extends to the microscopic scale as our building skills improve. The assembly of colloidal structure in one, two, and three dimensions has been a major task in materials science the last few decades.1-15 A variety of colloidal structures can now be routinely reproduced using various selfassembly or lithographic techniques. Colloidal suspensions show particular promise because it has been demonstrated that such suspensions may self-assemble into nanostructures that sometimes mimic biological morphologies.2-10 The self-assembled structures exhibit size-dependent phase separation2 and show novel responses in external fluid flow and magnetic or electric fields.1-10 Recently, researchers have been interested in fabricating composites using colloidal particles as building blocks. To this end, pyramidal structures have attracted attention (1) Pieranski, P. Contemp. Phys. 1983, 24, 25. (2) Sommer, A. P.; Ben-Moshe, M.; Magdassi, S. J. Phys. Chem. B 2004, 108, 8. (3) Sommer, A. P.; Franke, R. P. Nano Lett. 2003, 3, 573. (4) Su, G.; Guo, Q.; Palmer, R. E. Langmuir 2003, 19, 9669. (5) Masuda, Y.; Itoh, T.; Itoh, M.; Koumoto, K. Langmuir 2004, 20, 5588. (6) Masuda, Y.; Itoh, M.; Yonezawa, T.; Koumoto, K. Langmuir 2002, 18, 4155. (7) Cong, H.; Cao, W. Langmuir 2003, 19, 8177. (8) Fudouzi, H.; Xia, Y. Langmuir 2003 19, 9653. (9) Abe, M.; Orita, M.; Yamazaki, H.; Tsukamoto, S.; Teshima, Y.; Sakai, T.; Ohkubo, T.; Momozawa, N.; Sakai, H. Langmuir 2004, 20, 5046. (10) Guo, Q.; Arnoux, C.; Palmer, R. E. Langmuir 2001, 17, 7150. (11) Helseth, L. E.; Wen, H. Z.; Hansen, R. W.; Johansen, T. H.; Heinig, P.; Fischer, T. M. Langmuir 2004, 20, 7323. (12) Wen, H. Z.; Helseth, L. E.; Fischer, T. M. J. Phys. Chem. B 2004, 108, 16261. (13) Shevchenko, E. V.; Talapin, D. V.; Rogach, A. L.; Kornowski, A.; Haase, M.; Weller, H. J. Am. Chem. Soc. 2002, 124, 11480. (14) Yin, Y.; Lu, Y.; Gates, B.; Xia, Y. J. Am. Chem. Soc. 2001, 123, 8718. (15) Yin, Y.; Li, Z. Y.; Xia, Y. Langmuir 2003, 19, 622.
because of their important role in technologies relying on sharp structures (e.g., force microscopy). The controlled construction of colloidal microscopic pyramids has until now relied on prepatterned substrates or the self-assembly of nanocrystals.13-15 Such techniques have proven very useful. Still, we would like to gain more knowledge about colloidal self-assembly in external fields and possibly be able to provide templates were the colloidal structures can be controlled in situ. Our previous studies showed how strong magnetic field gradients can be used to grow colloidal crystals or to change the conformation of dipolar chains.11,12 Here we explore theoretically how magnetic fields could assist the assembly of pyramidal shapes and offer experimental demonstrations as proof of this principle. 2. Colloidal Pyramids in a Homogeneous Field A 2D colloidal pyramid cannot be in a stable equilibrium configuration in a homogeneous magnetic field without any boundaries. To this end, consider N identical dipoles assembled into a 2D pyramid, with their magnetic moments aligned in the x direction. The colloids considered here are modeled as dipoles of radius a with dipole moment mi, and the dipolar interaction energy between two such dipoles is given by
Eij )
[
]
µ0 3(mi‚ri)(mj‚ri) mi‚mj 4π ri5 ri3
(1)
where µ0 is the permeability of the surrounding medium and ri is the distance between the colloids. To obtain the total energy, one must sum over all such pair potentials. We will assume that the magnetic moment of the beads is influenced only by the external magnetic field and that the field from the other particles does not play a role. That is, mi ) (4π/3)χa3H, where χ is the susceptibility of the beads and H ) Hxex is the external magnetic field in the x direction. This is a good approximation because the dipole moment of the paramagnetic beads considered here is rather small and the external magnetic field is relatively weak. The magnetic field required to saturate the magnetic moment of the beads is on the order of 100 kA/m, and we operate here with fields that are orders of magnitude smaller than this.
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Figure 2. Schematic drawing of the experimental setup where the contact line of a water droplet is used to assemble the pyramids. Figure 1. Two possible configurations of paramagnetic spheres in an external magnetic field: (a) pyramid and (b) linear chain. Both configurations have a base consisting of NB beads.
In summing eq 1, it can be shown that the total dipolar energy associated with the 2D pyramid seen in Figure 1a is given by
3µ0m2 Ep ) (N - NB) 64πa3
(2)
Here NB is the number of beads in the base of the pyramid: NB
N)
i) ∑ i)1
NB(NB + 1)
(3)
2
However, if the beads assemble into a pearl chain then the total energy is given by
El ) -
µ0m2
(N - 1) 16πa3
(4)
It is seen that El < Ep under all circumstances, which means that pyramid formation is not favorable if the elements interact only via dipolar forces. One must therefore conclude that pyramids are prohibited from selfassembling in the absence of boundaries when all of the magnetic dipole moments are identical. To overcome this barrier, let us now assume that the base (x ) 0) of the pyramid or chain consisting of NB dipoles is fixed and cannot move (Figure 1a). The rest of the beads (N - NB) may then assemble into a pyramid or a pearl chain aligned along the y axis. The energy of the pyramid of Figure 1a is still given by eq 2, whereas the dipolar energy of the structure of Figure 1b is found to be
El ) -
µ0m2
3 N- N ( 2 16πa 3
B
+
1 2
)
(5)
It can therefore be seen that pyramid formation is favorable as long as NB < 4. When NB ) 4 (N ) 10), we find that Ep ) El, thus suggesting that ensembles fulfilling this requirement may be either chain-like or pyramidallike. For ensembles with NB > 4 (N > 10), we expect the linear chains to form as long as the dipole energy is larger than the thermal energy.
Figure 3. Formation of 2D colloidal pyramids in a drop of water. The gray lines on the left in a and b are the contact lines of the corresponding droplets. Formation of various shapes in a magnetic field H ) 800 A/m. See the text for details.
To test this theory experimentally, we investigated the behavior of magnetic beads in a drop of water. The paramagnetic beads used here had a radius of a ) 1.4 µm and a susceptibility of χ ) 0.17 and were manufactured by Dynal (Dynabeads M270 coated with a carboxylic acid group). The beads were immersed in deionized, ultrapure water at a density of 107 beads/mL and deposited in droplets of 10 µL volume. We visualized the system with a Leica polarization microscope equipped with a halogen light source and a Hamamatsu CCD camera. The temperature during the experiments was about T ) 300 K. A schematic drawing of the experimental setup is shown in Figure 2. After some initial random motion during drop settlement, the beads eventually drifted to the contact line because of weak evaporative fluid flow; see also refs 2 and 3. Thus, in our system the contact line functions as a 1D boundary to which the beads adhere. After a certain number of beads have reached the contact line, we turn on a magnetic field of 800 A/m (10 G) along the x axis, thus aligning the magnetic moments. The results of two such experiments are shown in Figure 3a and b. In Figure 3a, we can see a single standing pyramid as well as a pyramid with a “tail” (i.e., a linear chain attached to its apex). A physical explanation for the tail has not yet been revealed, although it is likely that it is related to the way the colloids were assembled. In the center of the droplet, only isolated linear chains were observed (no pyramids), and these were driven to the contact line by a radial flow gradient. The way that these chains conform when they enter the contact line may influence pyramid formation but is outside the scope of the current study. Experimentally, the method suggested in this section is not optimal for assembling single pyramids because of the fact that only limited sizes can be assembled. In fact,
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throughout numerous experiments we never found larger pyramids than that of Figure 3b, in agreement with our simple theory. It is therefore necessary to look for new solutions, and this is the topic of the next section. 3. Colloidal Pyramids in an Inhomogeneous Field Above we considered a system where all of the beads had identical magnetic moments. Then we could use the contact line of a droplet as a 1D boundary to assemble colloidal pyramids. However, if a spatial distribution of magnetic moments can be generated, then one may hope to generate pyramids in a more efficient manner. Previously we showed that 1D domain walls can assemble a variety of colloidal structures.8,9 Here we suggest that such a 1D structure may indeed function as a virtual boundary in addition to providing a magnetic field gradient. The magnetic field from a domain wall acting on a paramagnetic sphere resting on the magnetic film a distance r ) xex + aez from the domain wall can be approximated by
HDW )
M sw r 2π r2
(6)
where Ms is the magnetization of the magnetic film containing the wall and w is the width of the domain wall. The force from the domain wall on a single magnetic bead is found by noting that the interaction can be associated with an energy E ) -µ0m‚HDW, where µ0 is the permeability of water and m is the magnetic moment of the bead. Moreover, we assume that only the external magnetic field H ) Hx aligns the magnetic moments of the beads, such that m ) (4π/3)a3χHx. Again there are two possible generic structures that can form near the domain wall. We first assume that the beads are attracted to the domain wall forming a pearl chain, and the interaction energy between the beads and the domain wall is given by
E⊥1 ) -
2µ0χa2MsHw
N
∑ i)0
3
2i + 1
1 + (2i + 1)2
(7)
where the closest bead is assumed to be located a distance x ) a from the center of the domain wall. The corresponding dipolar attraction due to interaction between the beads themselves is associated with the energy
µ0χ2πa3H2 (N - 1) E⊥2 ) 9
(8)
The total energy associated with the chain structure is then given by E⊥ ) E⊥1 + E⊥2. However, the energy of the beads attracted to the domain wall forming a pyramidal shape is given by
E∆1 ) -
2µ0χa2MsHw NB
∑ i)0
3
1 + ix3
1 + (1 + ix3)
(NB - i)
(9)
2
where the closest beads are assumed to be located a distance x ) a from the domain wall. Here NB is the number of beads in the base of the pyramid: NB
N)
i) ∑ i)1
NB(NB + 1) 2
(10)
Figure 4. Dipolar energy of pearl chains (blue and red lines) and pyramids (magenta and green lines) for two different values of N. See the text for details.
The dipolar energy associated with dipolar interactions in the pyramid is given by
E∆2 ) -
µ0χ2πa3H2 Nb(Nb - 1) 24
(11)
The total energy associated with the pyramid seen in Figure 1b can therefore be expressed as E∆ ) E∆1 + E∆2. It is seen that the E⊥ decreases more rapidly than E∆ as the magnetic field is increased. Moreover, there is a critical field Hc where E∆ e E⊥, thus suggesting that pyramids are more stable than chain configurations below this field as long as the magnetic field is sufficiently strong such that the dipolar interactions overcome the Brownian motion (H ≈ (9kT/πχ2µ0a3)1/2 ≈ 300 A/m). Figure 4 shows E⊥ (blue line) and E∆ (magenta line) as a function of the magnetic field Hx for N ) 6 (NB ) 3) assuming that Ms ≈ 105 A/m and w ) 100 nm. Note that the critical field is Hc ) 28 kA/m. However, when N ) 21 (NB ) 6), we see that E⊥ (red line) becomes smaller than E∆ (green line) at Hc ) 43 kA/m. To test our theoretical predictions experimentally, we investigated the self-assembly of colloidal pyramids near 1D magnetic domain walls. The domain walls were formed in a bismuth-substituted ferrite garnet film of thickness 4 µm and magnetization Ms ≈ 105 A/m. A glass ring with a diameter of about 2 cm was placed on top of the garnet film, and beads immersed in pure water at a density of ∼105 beads/mL were confined within this ring. The particles used here are paramagnetic beads as described above, manufactured by Dynal (Dynabeads M270) and coated with a carboxylic acid (COOH-) group. In the absence of colloids, the domain wall is easily recognizable in a polarization microscope. However, the presence of light scattering from the colloids makes it more difficult to visualize the domain wall, but the behavior of the colloids clearly demonstrates its presence. A schematic drawing of the experimental setup is shown in Figure 5. The domain wall (lying along the y axis) acts as a 1D nanomagnet where the beads adsorb. An external magnetic field parallel to the x axis aligns the magnetic moments of the beads perpendicular to the wall. In weak magnetic fields between 400 and 1000 A/m, we observe the formation of pyramids, whereas in stronger fields >1000 A/m colloidal chains are dominant. Typically, the pyramids occurred in formation as seen in Figure 6. The domain wall is located at the base of the pyramids, and
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Figure 8. Mondisperse system of colloidal pyramids with N ) 3.
Figure 5. Schematic drawing of the experimental setup where an inhomogeneous magnetic field is used to assemble the pyramids.
Figure 6. Ensemble of colloidal pyramids with different values of N.
Figure 7. Colloidal pyramids with N ) 3, 6, and 21.
the centers of the closest beads are positioned a distance x ≈ a from the domain wall. We also observed many individual pyramids as seen in Figure 7. Figure 7a shows two pyramids composed of N ) 3 and 6 beads (located next to each other), whereas Figure 7b shows a pyramid with N ) 21. In stronger magnetic fields, linear pearl chains are most often observed; see ref 12. It should also
be pointed out that in some cases we also observed pyramids with tails (i.e., with linear chains attached to the apex), but a physical explanation for this phenomenon has not yet been revealed. That is, we do not yet understand why these tails do not conform to increase the size of the pyramid in order to make it more stable. However, it should be emphasized that the tails are indeed of magnetic origin and are not caused by additional flow in the system. According to the theory presented, it should be possible to create larger and more stable pyramids by increasing the field (as long as pyramid formation is favorable). However, our experiments do not provide conclusive evidence that such a phenomenon occurs. Instead, we find pyramids with different values of N on the same domain wall in the same experiment. Only in some cases have we observed many small pyramids with the same N attached to the same domain wall (Figure 8). Our experiments suggest that pyramids with N in the range between 3 and 21 can exist as long as the external field is less than 1 kA/m. Although the theory presented here explains the qualitative features rather well, it also suggests critical fields an order of magnitude larger than the experimental values. This discrepancy could be related to the fact that we have neglected the beads’ influence on each others’ magnetic moments and that we have assumed that each bead is aligned only with the external magnetic field, thus neglecting the induced magnetic moment caused by the domain wall. Our calculations nonetheless correctly predict the qualitative behavior of the system and could prove useful in predicting the formation of colloidal pyramids. It should be noted that the critical field scales linearly with the strength of the domain wall (Msw), which could help us in tuning the structural arrangement of colloids. 4. Conclusions We visualized the formation of individual colloidal pyramids in different geometries. By tuning the strength of the magnetic field, we were able to obtain either pyramids or chains. Our results show that the formation of colloidal structures in strong magnetic is feasible. Moreover, it could be combined with other manipulation techniques (e.g., optical trapping), thus providing a new platform on which microscopic crystallites can be studied. Acknowledgment. I am very grateful to H.Z. Wen and T.M. Fischer for useful discussions and support. LA051140V
