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Structure transitions of polymer-mediated colloidal crystals
Physics Letters A 373 (2009) 3174–3181
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Physics Letters A www.elsevier.com/locate/pla
Structure transitions of polymer-mediated colloidal crystals C.R. Han a , J.G. Cao a , L.W. Zhou a,∗ , X.C. Pang b , J.L. Huang b,∗ , L.F. Zhang a , J.P. Huang a,∗,1 a b
Department of Physics and Surface Physics Laboratory (National Key Laboratory), Fudan University, Shanghai 200433, China The Key Laboratory of Molecular Engineering of Polymer, State Education Ministry of China, Department of Macromolecular Science, Fudan University, Shanghai 200433, China
a r t i c l e
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Article history: Received 31 January 2009 Received in revised form 25 June 2009 Accepted 27 June 2009 Available online 1 July 2009 Communicated by R. Wu PACS: 47.57.-s 42.70.Qs 47.57.J-
a b s t r a c t The self-assembled colloidal crystals with structure transitions were realized experimentally on substrates with polymer brushes. Such polymer-mediated colloidal crystals were found to experience a series of in-layer structure transitions including stable columnar, polycrystalline, mixed square-hexagonal, square, hexagonal structures when the mass ratio of colloidal particles to free polymers was adjusted from 0.125 to 10. Among these structures, a large area interfacial square structure was proved to be a body-centered tetragonal (BCT) structure in three dimensions in the polymer-mediated colloidal crystals. © 2009 Elsevier B.V. All rights reserved.
Keywords: Structures transitions Colloidal crystals Self-assembly Band structures
1. Introduction Colloidal crystals are materials with periodic structures typically made by the self-assembly of submicrometer colloidal particles. Self-assembled colloidal crystals present rich phase behaviours which can be used to fabricate nanomaterials with high performance in size-dependent optical, electrical, and mechanical properties [1–5], and especially in photonic band gaps [6–11]. It is highly desirable to fabricate colloidal crystals with various stable crystalline structures [12–14]. For instance, if such crystals were available, they could serve as photonic band gap materials [10,11,15–17], which is of particular importance in optoelectronic applications that require the manipulation of photons, e.g., as photonic components in telecommunication devices, lasers, and sensors. In monodisperse colloidal systems, the random hexagonal close-packed or face-centered-cubic structures with high symmetry can be formed by self-assembly methods [18,19]. Various techniques, such as dip coating [20], electrochemical growth [21], selfassembly with electrodes [22], evaporation [23] have been used for the fabrication of large colloidal crystals with a given structure and
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Corresponding authors. E-mail addresses: [email protected] (L.W. Zhou), [email protected] (J.L. Huang), [email protected] (J.P. Huang). 1 Tel.: +86 21 55665227; fax: +86 21 55665239. 0375-9601/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2009.06.050
fewer defects. All the methods mentioned above utilize a smooth and “hard” substrate to grow the colloidal crystals. The hard substrate served as the confinement to the system and permitted colloidal spheres to be assembled into highly ordered structures [24]. It has been reported that monodispersed colloidal spheres sequentially organized themselves into ordered 3D structures when they are subjected to physical confinement [25]. Using these methods, the three-dimensional colloidal crystal with given structures can be achieved. In recent years, several methods have been developed to tune the transition of the structures of the colloidal crystals. Different structures which were random hexagonal close packing, face-centered cubic, body-centered cubic, body-centered tetragonal, body-centered orthorhombic, and space-filling tetragonal were realized and the phase behaviours were controlled by varying the solvent salt concentration and external electric field [26,27]. Also the structure transformation from body-centered tetragonal to face-centered cubic had achieved by tuning the magnetic and electric fields [28]. The patterned substrates can also be used to direct the crystallization of the colloidal suspension and permit to tune the symmetry and the growth direction of the colloidal crystal [29–31]. In this Letter, we report an experimental self-assembly method to achieve colloidal crystal with various structures and structure transition and thus verify the previous theoretical findings by Ren and Ma [32]. The colloidal crystal is composed of colloidal particles
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and free polymers on a substrate with a soft polymer brush, and it is named as “polymer–colloidal particles–polymer brush” (PCB) colloidal crystal. This method of making PCB colloidal crystals and other colloidal structures is called a PCB method. Here the polymer brush [33–38] is used to modulate the softness of substrates. Using a polymer brush is an efficient way to control and stabilize the symmetry of the first layer of the colloidal structure. Different to previous “hard” or short range interactive substrate, it is found that “soft and flexible” polymer brush substrate may lead to a long range force between substrate and particles, and among particles. This long range force stabilizes the order of the nanometer sized particles and controls the symmetry of the structure formed in a desired direction. The free polymers are homogeneously dispersed in the solution which provides a depletion force (also called an entropic excluded volume force) [32,39–42]. It is known that the depletion force can cause a lot of phase separation behaviours in colloidal systems [43–46]. For example, when the colloidal–polymer mixture was coated onto a solid surface, columnar and lamellar structures can be found [4]. In our polymer–colloidal particles– polymer brush colloidal system, a brush-mediated long-range interaction between the colloidal particles competes with the steric packing effect of the particles themselves, and generates a series of ordered structures, which undergo symmetry-changing transitions [32]. In the experiment, we fixed the quantity of the free polymers and changed the mass ratio of colloidal particles to free polymers from 0.125 to 10. It was found that the colloidal particle structures undergo a series of transitions from amorphous, to polycrystalline, to columnar, to mixed square-hexagonal, to dense square, to mixed hexagonal-square, and to hexagonal structures on the polymer brush. The profiles of the polymer brush substrates were investigated by both the scanning probe microscope (SPM) and scanning electronic microscope (SEM) before colloidal particles were layered on and after they were rinsed off. As a result, the deformation of the polymer brush substrate shows the interaction of the polymer brush and the colloidal particles. The spectra of different phases of colloidal structures show different photonic band gaps as well.
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CuBr (71 mg, 0.5 mmol), bipyridine (156 mg, 1 mmol), degassed H2 O (1 ml), and MeOH (4 ml). After freeze-drawing for three times, the ampule were put into a thermostatic oil bath at 30 ◦ C for 22 h. After completion of the polymerization, the substrate was cleaned by sonication in tetrahydrofuran (THF) for 20 min, and dried in a nitrogen flow. The polymer brush thicknesses of 400 nm were determined by ellipsometer and the step measurement, which are consistent with each other. This thickness is comparable to the diameter of the colloidal particles. The sample preparation process of PCB colloidal crystals is the following: firstly, 0.01 g PEO was dissolved in 20 ml methanol solvent (methanol/water volume ratio of 4:1) to make free polymer solution A. The colloidal PS particles were then added into 2 ml of free polymer solution A according to the PS/PEO mass ratio from 0.125 to 10. The mixtures were made uniform by ultrasonic dispersion for 10 min. The polymer-brush-attached square silicon chips were immersed into the methanol solvent (methanol/water volume ratio of 4:1) for 20 min at 30 ◦ C for pre-inteneration of the polymer brushes. The chips were then immersed vertically into the colloidal-polymer mixture with different PS/PEO mass ratio for 20 h. After the solvent was evaporated at 40 ◦ C, several different colloidal structures were found such as the hexagonal, hexagonal-square mixture, square or columnar. In the preparation of PCB colloidal crystals, the mass of PEO was kept constant while the concentration of the PS particles was varied according to PS/PEO mass ratios ranging from 0.125 to 10. The transition of the colloid structures occurs with the increase of the nanoparticle concentration. After the solvent was completely evaporated, the two-dimensional filling fraction of columnar, square, and hexagonal structures were calculated by Image-Pro, and found increasing from columnar (0.3516), via square (0.6333), to hexagonal (0.8104) structures. The ability to control structures of colloidal crystals using a polymer brush layer and free polymer is quite unique in comparison with other self-assembly methods. The methanol/water solvent used in our experiment is the same as that used in PMMA polymerization [48]. From this we could deduce that PMMA could be dissolved in the methanol/water solvent, and the polymer brush is soft in the solvent at 40 ◦ C.
2. Experiment 3. Results and discussion The single- or multi-layered PCB colloidal crystals are composed of colloidal particles, free polymers, and polymer brush by using a self-assembly method. The colloidal particles are polystyrene (PS) nanometer-sized spheres 5030A (Duke Scientific Corporation, Fremont USA), which have an average diameter (D) of 300 nm. The size uniformity (The standard deviation of the mean diameter expressed as a percentage of the mean diameter) is less than 3%. The free polymer is poly(ethylene oxide) (PEO) which has a molecular weight of about 20,000. The polymer brush is poly(methyl methacrylate) (PMMA) and was prepared by the surface-initiated atom transfer radical polymerization method [47]. The silane initiator used in this polymerization was (11-(2-bromo2-methyl)propionyloxy)undecyltrichlorosilane [48]. The details of the preparation procedure of the PMMA polymer brush contain three steps. Step 1: the 1 cm × 1 cm silicon wafers were cleaned consequently by de-ionized water ( min), acetone (20 min), methanol (5 min), H2 SO4 /H2 O2 (volume ratio is 7/3 for 1 h), and de-ionized H2 O (10 min), and then dried in a nitrogen flow. Step 2: pre-cleaned silicon wafers were put into a silane/toluene solution (a mixture of 10–20 μl silane initiator and 10 ml toluene solvent) at 15 ◦ C for 15 h. Then the siliconwafers-grafted initiator was removed from the solution and rinsed by toluene, acetone, methanol, and de-ionized water. Finally, they were dried with nitrogen and deposited in a drying box [48]. Step 3: The initiator wafers were immersed in a homogeneous maroon solution composed of methyl methacrylate (MMA) (5 g, 50 mmol),
The SEM observation is an intuitive way to determine the structures of materials in this study. Fig. 1 shows self-assembled colloidal crystals on substrates with polymer brush thickness of 400 nm and the transition of structures from amorphous, via columnar, polycrystalline, mixed square-hexagonal, dense square, mixed hexagonal-square, and to hexagonal structures. In the experiment, it is found that the transition of these structures can be controlled by varying the mass ratio of the colloidal PS particles to the free PEO polymers. When the PS/PEO mass ratio ranges from 0 to 0.2, the distribution of colloidal particles is random. When the ratio was increased from 0.2 to 1, the columnar structures were discovered. Especially, when the ratio was 0.3, the columnar structure reached its maximum. The maximum area of the columnar structures can be as large as 10 000 μm2 . We can see the colloidal particles arranged along nearly one direction to form the columnar structure on the 1 cm × 1 cm polymer brush substrate. This suggests that the polymer brush provides the colloidal particles a long-range anisotropic force. Here, the anisotropy means the organization direction of the particles on the surface of the polymer brush. The anisotropic interaction originates from the stretching energy cost of polymer brush chains which depends strongly on the aggregation style of the colloidal particles. To minimize the free energy, the nanoparticles aggregate at the polymer brush-air interface choosing a baguette shape and growing anisotropically along a one-dimensional direction, resulting in columnar structures [49].
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Fig. 1. (Color online.) Structure transition of colloidal crystals with increasing ratio of PS/PEO from 0.125 to 10. From left to right, the colloidal crystals transform from amorphous, to columnar, to polycrystalline, to mixed square-hexagonal, to dense square, to mixed hexagonal-square, and to hexagonal structures. In this figure, A denotes “amorphous”, C denotes “columnar”, S denotes “square”, S+H denotes “mixed square-hexagonal” (the area of square structure is larger than that of hexagonal one), H+S denotes “mixed hexagonal-square” (hexagonal area is more than square one), and H denotes “hexagonal”. The ratios of 0.3, 4 and 10 are the places where the corresponding colloidal structures reach their maxima.
When the ratio increased from 1 to 4, the short-ranged ordered polycrystalline structures changed to the mixed square-hexagonal. During this change, both the square and hexagonal structures increased with the mass ratio. However, the area of square structure increased faster than that of hexagonal one and reached the maximum at a PS/PEO mass ratio of 4. At this optimal ratio we found that the maximum region of the square structure of the body centered tetragonal (bct) can be over 10 000 μm2 . The inset in Fig. 1 (S) is the magnification of the square structure, which is the (100) face of the BCT structure. Fig. 2A represents the large region of squared structure when the PS/PEO ratio is 4. The inset on the right corner of Fig. 2 is the magnification of the corresponding structures. When the PS/PEO mass ratio increased from 4 to 10, the mixed hexagonal-square structure was occupied by increasing hexagonal structure (H). When the ratio is larger than 10, most of the area is covered by the hexagonal structure. The maximum region of the hexagonal structures is 0.1 cm × 0.1 cm in our experiment. Fig. 2B shows the large area hexagonal structure. The area could be even larger if we keep the extra circumstances (temperature, stability, uniformity of the polymer brush) more stable. Such transitions conform to the sequence from disorder to ordered structures, and from short-range to long-range orders. The proportions of different structures in the colloidal crystal films with area of 4 mm × 8 mm were shown in Fig. 3. At the PS/PEO mass ratio of 0.3, the proportion of the column structure is 40%. At the ratio of PS/PEO of 4, 55% of the area of 4 mm × 8 mm is square structure and 35% is hexagonal. The largest region of the square structure is 100 μm × 100 μm (see Fig. 2A for the morphology of a large region of the square structure). The small square region is several micrometer-size. At the ratio of 10, nearly 80% of the film is covered by hexagonal structure.
Fig. 4 shows the topography of the polymer brush which was prepared by surface-initiated atom transfer radical polymerization. Panel A is the SPM topography of the polymer brush by the step measurement which denotes the polymer brush thickness (400 nm). It is consistent with the result in panel B which shows the SEM image of the cross-section of PMMA polymer brush. To reveal the function of both polymer brush and free polymer in the formation of structure-changeable colloidal crystals, we have compared four different combinations of the PEO polymers and the polymer brush with the same concentration of PS particles, see Fig. 5. This figure shows four different combinations of PEO polymers and polymer brush in order to illuminate the interaction in colloidal structures. Fig. 5a shows the closely packed hexagonal structure assembled from a monodispersed colloidal suspension on a silicon substrate without polymer brush and free polymers. In this case, the structures other than the hexagonal ones are difficult to find. The small areas of the hexagonal and square structures as shown in Fig. 5b were formed with a monodispersed colloidal suspension with free polymers, but without polymer brush. There are a few short range phases in Fig. 5b but only one phase in Fig. 5a; this is due to the depletion effect between colloidal particles and the free polymers [39–42]. Fig. 5c is a sample made by using the polymer brush but without free polymers. Both long range ordered hexagonal and square structures are found in this sample, but it is clear that the closely packed hexagonal structure covers a wider area than the square structure does. The long-range ordered structures are due to the effect of the soft polymer brush substrate. Fig. 5d shows the sample made with both polymer brush and free polymers, and several large areas of square structures were formed. The structures in Figs. 5c and d are more ordered than that in Fig. 5b. It is because the polymer brush has a long range
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Fig. 2. (Color online.) (A) The SEM image of the large region square structure with the PS/PEO ratio of 4. (B) The SEM image of the large region hexagonal structure with the PS/PEO ratio of 10. The insets on the right corner are the magnification of the corresponding structures.
effect on the colloidal particles and stabilizes the ordered structure formed in the free polymer-colloidal mixtures. Fig. 5d is the result of two interactions which are the attractive depletion force due to the large entropy of polymer brushes and free polymers, and the repulsive steric force between colloidal particles. PS/PEO mass ratio of 4 was kept in Figs. 5b and d. It shows that the free polymers have an important effect on enriching the phase behaviour of the colloidal particle structures. However, the square structures are much easier to be stabilized on a soft substrate (Fig. 5d) than on a hard one (Fig. 5b). To understand the formation process of colloidal crystal structure, we have also paid attention to the first layer structures by viewing the deformation of the polymer brush after the PS particles were rinsed off. The morphology of polymer brush was made by immersing the PCB colloidal crystals into the THF solvent for 4 days at 20 ◦ C to dissolve PS particles, plus a sonication in the same THF solution for 1 min to remove traces of the PS particles. Figs. 6a and b are the SEM top views of the morphology of polymer brushes grafted on the silicon substrate after colloidal particles have been removed. From each of these pictures, we find the profiles of the first layer of colloidal particles left on the polymer brush and these are consistent with phase structures of the corresponding three-dimensional colloidal crystals. Fig. 6a
Fig. 3. (Color online.) The proportion of different structures in the colloidal crystal films with area of 4 mm × 8 mm corresponds to the mass ratio of PS/PEO from 1 to 10 when the thickness of polymer brush is comparable to the size of the PS particles. Solid curves are a guide for the eye.
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Fig. 4. (Color online.) Patterns of the polymer brush. (A) The step measurement by SPM which indicates that the thickness of the polymer brush is about 400 nm. (B) SEM image of cross-section of PMMA polymer brush.
Fig. 5. (Color online.) SEM pictures of the morphology of colloidal crystal structures with different combinations of the free polymer (PEO) and the polymer brush. (a) The colloidal crystal sample without free polymer and polymer brush. (b) The colloidal crystal with free polymer, but without polymer brush. (c) The colloidal crystal sample with polymer brush, but without free polymer. (d) The sample with both polymer brush and free polymer. Upper-right corner shows an enlargement of a part of the square structure.
Fig. 6. (Color online.) The SEM pictures of the morphology of polymer brush grafted on silicon substrates after the colloidal particles were rinsed out, for (a) a hexagonal structure and (b) square structure of colloidal crystals.
shows that the first layer is the hexagonal structure, and the threedimensional structures are closely packed. Fig. 6b displays that the first layer is a square structure, and the three-dimensional structure is BCT. Different phase profiles also confirm that the polymer brush can interact with the colloidal particles to control the symmetry of the particle structures and stabilize the in-plane structures of the colloidal crystals.
The PS particles were removed by THF solvent, which is also a good solvent for both PMMA and PS, but leaves visible features on the surface of the PMMA as if there were no swelling of the PMMA surface with the THF. Three possible reasons are list below. First, the PS particles are not cross-linked, and they could be dissolved by the THF. Second, there are layers of Au on the surface of the samples which could prevent the THF from immersing into
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the samples. This is because the samples had made the SEM observation before they were immersed into the THF. Third, the free polymer PEO which could not dissolve well in THF may protect the surface of PMMA. Clearly, it is not easy to remove the PS particles thoroughly. After being immersed into the THF solvent for 4 days at 20 ◦ C, there were still a few layers of PS colloids on the polymer brush. Therefore we placed the samples in THF solvent for sonication (for 1 min) and found the particles were shaken away from the surfaces of polymer brushes. We subsequently produced the morphology of the polymer brush by SEM, see Figs. 6a and b. It has reported that the BCT structure of the colloidal crystal had been realized by sedimentation of the colloidal dispersion of micrometer spheres in the electrical field [26,27,50]. Both Meng et al. [51] and Brevo et al. [52] have talked about the crystallization behaviour of the colloidal particles at the transition regions from n to n + 1 layers. The formation of square structure occurred in the layer transition region during the increase of the thickness. That is to say, in their systems, the position of the square struc-
Fig. 7. (Color online.) Experimentally measured reflectance spectra of the hexagonal and square colloidal crystals.
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ture was between the nth and (n + 1)th layers. While both the nth and (n + 1)th layers were hexagonal regions. Our work is different from their works. Firstly, the electrostatic interaction in our system is small. The polystyrene spheres with a small quantity of sulfate radicals on them are negatively charged, but the quantity of the charge is very little. Most part of the surface of the spheres is neutral. So the depletion force which induced by the electrostatic effect is small enough to be ignored. Secondly, the large region of the square structure is located in the surface of the same layer but not in the layer transition region. The formation of the square structure is mainly induced by the entropic effect of the polymer brush and the steric packing effect of the colloidal particles. If the brush-mediated interaction between colloidal particles dominates over the steric packing effect of the colloids, the stable square structure forms for possibly large conformational entropy of polymer brush. If the steric packing effect dominates, the stable hexagonal structure forms. It is well known that, on a hard substrate, a square-structured colloidal crystal is not stable, while a hexagon-structured one is. This is due to the fact that the latter has lower free energy than the former. However, on a polymer brush-mediated soft substrate, square-structured colloidal crystal can be stable under certain concentration of colloidal particles (Fig. 5c and Fig. 5d). This is because, on the soft surface of the polymer brush, if the interaction of different components and the entropy of free polymer chains are omitted, the approximate total free energy density consists of two parts: repulsive steric energy of colloidal particle and energy due to the entropy of the polymer brush. Here the latter may lead to an attractive interaction between colloidal particles (depletion effect). Namely, the total free energy density per unit area, F = N c F st − N br T S br , where F st is the steric energy of a particle, S br is the entropy of a single chain of the polymer brush, N c and N br are the numbers of colloidal particles and chains of
Fig. 8. (Color online.) Numerically simulated band structures of (a) BCT and (b) FCC structures. The first Brillouin zones of the (c) BCT and (d) FCC structures. The red arrows in (c) and (d) display the measurement direction of the reflectance spectra of square and hexagonal colloidal crystal, respectively. Parameters: D = 300 nm. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this or Letter.)
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polymer brushes in a unit area, respectively [32]. We owe the transitions of the structures (Fig. 1) to the competition between the entropy effect (depletion effect) of the polymer brush and the effect of repulsive particle steric energy. S br and F st are calculated in Ref. [3,6], and it is found that both of them are dependent on the colloidal particle concentration; namely, they would be different when the colloidal structure is different. It is found from the calculation that the entropy of the polymer brush of square-structure (Fig. 6b) is about 1.1k B ; but the entropy of the polymer brush of hexagonal structure is −0.2k B . In this work, the particle concentration is defined as the mass ratio of the colloidal particles to the free polymers on an adjustable polymer brush. When the particle concentration is about 4, although the repulsive steric energy for the square structure is higher, the total free energy of the square structure (or BCT structure in 3D) is low, and the corresponding structure is stable. This is because a polymer brush of square structure has very high entropy. Similarly, when PS/PEO = 10, although the entropy of the polymer brush of the hexagonal structure is very low, the repulsive steric energy for the hexagonal structure is also low, thus the total free energy of the hexagonal structure is low, and the corresponding structure is stable. To distinguish the square phase from the hexagonal one, the reflectance spectra of the PCB colloidal crystals were measured. A deuterium–tungsten light source was used to emit light of 1 mW with wavelength varying from 400 to 1100 nm. The diameter of the light beam was 1 mm. The spectrum of the silicon substrate was first measured as a reference. The samples were then illuminated by the collimated light source and the incident direction was inclined to the normal direction of the samples. The reflected light by the samples was analyzed by a monochromator with a resolving wavelength varying from 400 nm to 850 nm. The spectra of the reflected light were obtained as shown in Fig. 7. We experimentally observed two peaks in the reflectance spectra: One is located at 657 nm for the square structure sample with the PS/PEO mass ratio of 4, the other is at 710 nm for the hexagonal structure sample with the PS/PEO mass ratio of 10. Further, we used the commercial R-soft software to calculate the band structures of the BCT (Fig. 8a) and FCC (Fig. 8b) crystal lattices. In Figs. 8a and b, the arrows show that there are some local frequency-gaps in a certain wave vector region. The red arrows in Figs. 8c and d show the measurement direction of different structures. The calculated frequency-gap corresponds to reflectance wavelengths of 660 nm and 707 nm for the BCT and FCC lattices, respectively. They agree with the experimental result, namely, 657 nm and 710 nm for square and hexagonal structures, respectively. The agreement between our experiment and theory shows that the (interfacial) square structure observed in our experiments has a BCT structure in three dimensions, and that the (interfacial) hexagonal structure accords with the FCC structure in three dimensions. We have so far proved that the square phase of colloidal crystals is BCT structure by a comparison between experimental reflectance spectra and their model calculations. The threedimensional structures other than BCT were proven to be impossible because of giving wrong positions of reflection peaks. We are thus convinced that the square-phased colloidal crystals assembled via polymer-mediated self-organization has a three-dimensional BCT structure. 4. Conclusion By using free polymers and polymer brushes, we have experimentally demonstrated a theoretically proposed method [32] for fabricating a class of colloidal crystals with stable tunable crystalline structures. Such polymer-mediated colloidal crystals were found to experience a series of in-layer structure transitions including a stable square structure when we adjusted the mass ratio
of colloidal particles to free polymers. This work presents a new experimental method to fabricate colloidal crystals with structures transition, which have practical applications in various fields.
Acknowledgements
We thank R.K. Jing and P.X. Chen for their fruitful discussion and support, and L. Xu, Q.M. Zhang, L. Zhou, J.M. Hao, and G. Wang for their expertise. This work was supported by the National Natural Science Foundation of China under Grant Nos. 10334020, 10574027, 10604014, and 10874025. L.F.Z. and J.P.H. also acknowledge the financial support by the Pujiang Talent Project (No. 06PJ14006) of the Science and Technology Commission of Shanghai Municipality, by the Shanghai Education Committee and the Shanghai Education Development Foundation (“Shu Guang” project under Grant No. 05SG01), and by CNKBRSF under Grant No. 2006CB921706.
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Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Structure transitions of polymer-mediated colloidal crystals C.R. Han a , J.G. Cao a , L.W. Zhou a,∗ , X.C. Pang b , J.L. Huang b,∗ , L.F. Zhang a , J.P. Huang a,∗,1 a b
Department of Physics and Surface Physics Laboratory (National Key Laboratory), Fudan University, Shanghai 200433, China The Key Laboratory of Molecular Engineering of Polymer, State Education Ministry of China, Department of Macromolecular Science, Fudan University, Shanghai 200433, China
a r t i c l e
i n f o
Article history: Received 31 January 2009 Received in revised form 25 June 2009 Accepted 27 June 2009 Available online 1 July 2009 Communicated by R. Wu PACS: 47.57.-s 42.70.Qs 47.57.J-
a b s t r a c t The self-assembled colloidal crystals with structure transitions were realized experimentally on substrates with polymer brushes. Such polymer-mediated colloidal crystals were found to experience a series of in-layer structure transitions including stable columnar, polycrystalline, mixed square-hexagonal, square, hexagonal structures when the mass ratio of colloidal particles to free polymers was adjusted from 0.125 to 10. Among these structures, a large area interfacial square structure was proved to be a body-centered tetragonal (BCT) structure in three dimensions in the polymer-mediated colloidal crystals. © 2009 Elsevier B.V. All rights reserved.
Keywords: Structures transitions Colloidal crystals Self-assembly Band structures
1. Introduction Colloidal crystals are materials with periodic structures typically made by the self-assembly of submicrometer colloidal particles. Self-assembled colloidal crystals present rich phase behaviours which can be used to fabricate nanomaterials with high performance in size-dependent optical, electrical, and mechanical properties [1–5], and especially in photonic band gaps [6–11]. It is highly desirable to fabricate colloidal crystals with various stable crystalline structures [12–14]. For instance, if such crystals were available, they could serve as photonic band gap materials [10,11,15–17], which is of particular importance in optoelectronic applications that require the manipulation of photons, e.g., as photonic components in telecommunication devices, lasers, and sensors. In monodisperse colloidal systems, the random hexagonal close-packed or face-centered-cubic structures with high symmetry can be formed by self-assembly methods [18,19]. Various techniques, such as dip coating [20], electrochemical growth [21], selfassembly with electrodes [22], evaporation [23] have been used for the fabrication of large colloidal crystals with a given structure and
*
Corresponding authors. E-mail addresses: [email protected] (L.W. Zhou), [email protected] (J.L. Huang), [email protected] (J.P. Huang). 1 Tel.: +86 21 55665227; fax: +86 21 55665239. 0375-9601/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2009.06.050
fewer defects. All the methods mentioned above utilize a smooth and “hard” substrate to grow the colloidal crystals. The hard substrate served as the confinement to the system and permitted colloidal spheres to be assembled into highly ordered structures [24]. It has been reported that monodispersed colloidal spheres sequentially organized themselves into ordered 3D structures when they are subjected to physical confinement [25]. Using these methods, the three-dimensional colloidal crystal with given structures can be achieved. In recent years, several methods have been developed to tune the transition of the structures of the colloidal crystals. Different structures which were random hexagonal close packing, face-centered cubic, body-centered cubic, body-centered tetragonal, body-centered orthorhombic, and space-filling tetragonal were realized and the phase behaviours were controlled by varying the solvent salt concentration and external electric field [26,27]. Also the structure transformation from body-centered tetragonal to face-centered cubic had achieved by tuning the magnetic and electric fields [28]. The patterned substrates can also be used to direct the crystallization of the colloidal suspension and permit to tune the symmetry and the growth direction of the colloidal crystal [29–31]. In this Letter, we report an experimental self-assembly method to achieve colloidal crystal with various structures and structure transition and thus verify the previous theoretical findings by Ren and Ma [32]. The colloidal crystal is composed of colloidal particles
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and free polymers on a substrate with a soft polymer brush, and it is named as “polymer–colloidal particles–polymer brush” (PCB) colloidal crystal. This method of making PCB colloidal crystals and other colloidal structures is called a PCB method. Here the polymer brush [33–38] is used to modulate the softness of substrates. Using a polymer brush is an efficient way to control and stabilize the symmetry of the first layer of the colloidal structure. Different to previous “hard” or short range interactive substrate, it is found that “soft and flexible” polymer brush substrate may lead to a long range force between substrate and particles, and among particles. This long range force stabilizes the order of the nanometer sized particles and controls the symmetry of the structure formed in a desired direction. The free polymers are homogeneously dispersed in the solution which provides a depletion force (also called an entropic excluded volume force) [32,39–42]. It is known that the depletion force can cause a lot of phase separation behaviours in colloidal systems [43–46]. For example, when the colloidal–polymer mixture was coated onto a solid surface, columnar and lamellar structures can be found [4]. In our polymer–colloidal particles– polymer brush colloidal system, a brush-mediated long-range interaction between the colloidal particles competes with the steric packing effect of the particles themselves, and generates a series of ordered structures, which undergo symmetry-changing transitions [32]. In the experiment, we fixed the quantity of the free polymers and changed the mass ratio of colloidal particles to free polymers from 0.125 to 10. It was found that the colloidal particle structures undergo a series of transitions from amorphous, to polycrystalline, to columnar, to mixed square-hexagonal, to dense square, to mixed hexagonal-square, and to hexagonal structures on the polymer brush. The profiles of the polymer brush substrates were investigated by both the scanning probe microscope (SPM) and scanning electronic microscope (SEM) before colloidal particles were layered on and after they were rinsed off. As a result, the deformation of the polymer brush substrate shows the interaction of the polymer brush and the colloidal particles. The spectra of different phases of colloidal structures show different photonic band gaps as well.
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CuBr (71 mg, 0.5 mmol), bipyridine (156 mg, 1 mmol), degassed H2 O (1 ml), and MeOH (4 ml). After freeze-drawing for three times, the ampule were put into a thermostatic oil bath at 30 ◦ C for 22 h. After completion of the polymerization, the substrate was cleaned by sonication in tetrahydrofuran (THF) for 20 min, and dried in a nitrogen flow. The polymer brush thicknesses of 400 nm were determined by ellipsometer and the step measurement, which are consistent with each other. This thickness is comparable to the diameter of the colloidal particles. The sample preparation process of PCB colloidal crystals is the following: firstly, 0.01 g PEO was dissolved in 20 ml methanol solvent (methanol/water volume ratio of 4:1) to make free polymer solution A. The colloidal PS particles were then added into 2 ml of free polymer solution A according to the PS/PEO mass ratio from 0.125 to 10. The mixtures were made uniform by ultrasonic dispersion for 10 min. The polymer-brush-attached square silicon chips were immersed into the methanol solvent (methanol/water volume ratio of 4:1) for 20 min at 30 ◦ C for pre-inteneration of the polymer brushes. The chips were then immersed vertically into the colloidal-polymer mixture with different PS/PEO mass ratio for 20 h. After the solvent was evaporated at 40 ◦ C, several different colloidal structures were found such as the hexagonal, hexagonal-square mixture, square or columnar. In the preparation of PCB colloidal crystals, the mass of PEO was kept constant while the concentration of the PS particles was varied according to PS/PEO mass ratios ranging from 0.125 to 10. The transition of the colloid structures occurs with the increase of the nanoparticle concentration. After the solvent was completely evaporated, the two-dimensional filling fraction of columnar, square, and hexagonal structures were calculated by Image-Pro, and found increasing from columnar (0.3516), via square (0.6333), to hexagonal (0.8104) structures. The ability to control structures of colloidal crystals using a polymer brush layer and free polymer is quite unique in comparison with other self-assembly methods. The methanol/water solvent used in our experiment is the same as that used in PMMA polymerization [48]. From this we could deduce that PMMA could be dissolved in the methanol/water solvent, and the polymer brush is soft in the solvent at 40 ◦ C.
2. Experiment 3. Results and discussion The single- or multi-layered PCB colloidal crystals are composed of colloidal particles, free polymers, and polymer brush by using a self-assembly method. The colloidal particles are polystyrene (PS) nanometer-sized spheres 5030A (Duke Scientific Corporation, Fremont USA), which have an average diameter (D) of 300 nm. The size uniformity (The standard deviation of the mean diameter expressed as a percentage of the mean diameter) is less than 3%. The free polymer is poly(ethylene oxide) (PEO) which has a molecular weight of about 20,000. The polymer brush is poly(methyl methacrylate) (PMMA) and was prepared by the surface-initiated atom transfer radical polymerization method [47]. The silane initiator used in this polymerization was (11-(2-bromo2-methyl)propionyloxy)undecyltrichlorosilane [48]. The details of the preparation procedure of the PMMA polymer brush contain three steps. Step 1: the 1 cm × 1 cm silicon wafers were cleaned consequently by de-ionized water ( min), acetone (20 min), methanol (5 min), H2 SO4 /H2 O2 (volume ratio is 7/3 for 1 h), and de-ionized H2 O (10 min), and then dried in a nitrogen flow. Step 2: pre-cleaned silicon wafers were put into a silane/toluene solution (a mixture of 10–20 μl silane initiator and 10 ml toluene solvent) at 15 ◦ C for 15 h. Then the siliconwafers-grafted initiator was removed from the solution and rinsed by toluene, acetone, methanol, and de-ionized water. Finally, they were dried with nitrogen and deposited in a drying box [48]. Step 3: The initiator wafers were immersed in a homogeneous maroon solution composed of methyl methacrylate (MMA) (5 g, 50 mmol),
The SEM observation is an intuitive way to determine the structures of materials in this study. Fig. 1 shows self-assembled colloidal crystals on substrates with polymer brush thickness of 400 nm and the transition of structures from amorphous, via columnar, polycrystalline, mixed square-hexagonal, dense square, mixed hexagonal-square, and to hexagonal structures. In the experiment, it is found that the transition of these structures can be controlled by varying the mass ratio of the colloidal PS particles to the free PEO polymers. When the PS/PEO mass ratio ranges from 0 to 0.2, the distribution of colloidal particles is random. When the ratio was increased from 0.2 to 1, the columnar structures were discovered. Especially, when the ratio was 0.3, the columnar structure reached its maximum. The maximum area of the columnar structures can be as large as 10 000 μm2 . We can see the colloidal particles arranged along nearly one direction to form the columnar structure on the 1 cm × 1 cm polymer brush substrate. This suggests that the polymer brush provides the colloidal particles a long-range anisotropic force. Here, the anisotropy means the organization direction of the particles on the surface of the polymer brush. The anisotropic interaction originates from the stretching energy cost of polymer brush chains which depends strongly on the aggregation style of the colloidal particles. To minimize the free energy, the nanoparticles aggregate at the polymer brush-air interface choosing a baguette shape and growing anisotropically along a one-dimensional direction, resulting in columnar structures [49].
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Fig. 1. (Color online.) Structure transition of colloidal crystals with increasing ratio of PS/PEO from 0.125 to 10. From left to right, the colloidal crystals transform from amorphous, to columnar, to polycrystalline, to mixed square-hexagonal, to dense square, to mixed hexagonal-square, and to hexagonal structures. In this figure, A denotes “amorphous”, C denotes “columnar”, S denotes “square”, S+H denotes “mixed square-hexagonal” (the area of square structure is larger than that of hexagonal one), H+S denotes “mixed hexagonal-square” (hexagonal area is more than square one), and H denotes “hexagonal”. The ratios of 0.3, 4 and 10 are the places where the corresponding colloidal structures reach their maxima.
When the ratio increased from 1 to 4, the short-ranged ordered polycrystalline structures changed to the mixed square-hexagonal. During this change, both the square and hexagonal structures increased with the mass ratio. However, the area of square structure increased faster than that of hexagonal one and reached the maximum at a PS/PEO mass ratio of 4. At this optimal ratio we found that the maximum region of the square structure of the body centered tetragonal (bct) can be over 10 000 μm2 . The inset in Fig. 1 (S) is the magnification of the square structure, which is the (100) face of the BCT structure. Fig. 2A represents the large region of squared structure when the PS/PEO ratio is 4. The inset on the right corner of Fig. 2 is the magnification of the corresponding structures. When the PS/PEO mass ratio increased from 4 to 10, the mixed hexagonal-square structure was occupied by increasing hexagonal structure (H). When the ratio is larger than 10, most of the area is covered by the hexagonal structure. The maximum region of the hexagonal structures is 0.1 cm × 0.1 cm in our experiment. Fig. 2B shows the large area hexagonal structure. The area could be even larger if we keep the extra circumstances (temperature, stability, uniformity of the polymer brush) more stable. Such transitions conform to the sequence from disorder to ordered structures, and from short-range to long-range orders. The proportions of different structures in the colloidal crystal films with area of 4 mm × 8 mm were shown in Fig. 3. At the PS/PEO mass ratio of 0.3, the proportion of the column structure is 40%. At the ratio of PS/PEO of 4, 55% of the area of 4 mm × 8 mm is square structure and 35% is hexagonal. The largest region of the square structure is 100 μm × 100 μm (see Fig. 2A for the morphology of a large region of the square structure). The small square region is several micrometer-size. At the ratio of 10, nearly 80% of the film is covered by hexagonal structure.
Fig. 4 shows the topography of the polymer brush which was prepared by surface-initiated atom transfer radical polymerization. Panel A is the SPM topography of the polymer brush by the step measurement which denotes the polymer brush thickness (400 nm). It is consistent with the result in panel B which shows the SEM image of the cross-section of PMMA polymer brush. To reveal the function of both polymer brush and free polymer in the formation of structure-changeable colloidal crystals, we have compared four different combinations of the PEO polymers and the polymer brush with the same concentration of PS particles, see Fig. 5. This figure shows four different combinations of PEO polymers and polymer brush in order to illuminate the interaction in colloidal structures. Fig. 5a shows the closely packed hexagonal structure assembled from a monodispersed colloidal suspension on a silicon substrate without polymer brush and free polymers. In this case, the structures other than the hexagonal ones are difficult to find. The small areas of the hexagonal and square structures as shown in Fig. 5b were formed with a monodispersed colloidal suspension with free polymers, but without polymer brush. There are a few short range phases in Fig. 5b but only one phase in Fig. 5a; this is due to the depletion effect between colloidal particles and the free polymers [39–42]. Fig. 5c is a sample made by using the polymer brush but without free polymers. Both long range ordered hexagonal and square structures are found in this sample, but it is clear that the closely packed hexagonal structure covers a wider area than the square structure does. The long-range ordered structures are due to the effect of the soft polymer brush substrate. Fig. 5d shows the sample made with both polymer brush and free polymers, and several large areas of square structures were formed. The structures in Figs. 5c and d are more ordered than that in Fig. 5b. It is because the polymer brush has a long range
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Fig. 2. (Color online.) (A) The SEM image of the large region square structure with the PS/PEO ratio of 4. (B) The SEM image of the large region hexagonal structure with the PS/PEO ratio of 10. The insets on the right corner are the magnification of the corresponding structures.
effect on the colloidal particles and stabilizes the ordered structure formed in the free polymer-colloidal mixtures. Fig. 5d is the result of two interactions which are the attractive depletion force due to the large entropy of polymer brushes and free polymers, and the repulsive steric force between colloidal particles. PS/PEO mass ratio of 4 was kept in Figs. 5b and d. It shows that the free polymers have an important effect on enriching the phase behaviour of the colloidal particle structures. However, the square structures are much easier to be stabilized on a soft substrate (Fig. 5d) than on a hard one (Fig. 5b). To understand the formation process of colloidal crystal structure, we have also paid attention to the first layer structures by viewing the deformation of the polymer brush after the PS particles were rinsed off. The morphology of polymer brush was made by immersing the PCB colloidal crystals into the THF solvent for 4 days at 20 ◦ C to dissolve PS particles, plus a sonication in the same THF solution for 1 min to remove traces of the PS particles. Figs. 6a and b are the SEM top views of the morphology of polymer brushes grafted on the silicon substrate after colloidal particles have been removed. From each of these pictures, we find the profiles of the first layer of colloidal particles left on the polymer brush and these are consistent with phase structures of the corresponding three-dimensional colloidal crystals. Fig. 6a
Fig. 3. (Color online.) The proportion of different structures in the colloidal crystal films with area of 4 mm × 8 mm corresponds to the mass ratio of PS/PEO from 1 to 10 when the thickness of polymer brush is comparable to the size of the PS particles. Solid curves are a guide for the eye.
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Fig. 4. (Color online.) Patterns of the polymer brush. (A) The step measurement by SPM which indicates that the thickness of the polymer brush is about 400 nm. (B) SEM image of cross-section of PMMA polymer brush.
Fig. 5. (Color online.) SEM pictures of the morphology of colloidal crystal structures with different combinations of the free polymer (PEO) and the polymer brush. (a) The colloidal crystal sample without free polymer and polymer brush. (b) The colloidal crystal with free polymer, but without polymer brush. (c) The colloidal crystal sample with polymer brush, but without free polymer. (d) The sample with both polymer brush and free polymer. Upper-right corner shows an enlargement of a part of the square structure.
Fig. 6. (Color online.) The SEM pictures of the morphology of polymer brush grafted on silicon substrates after the colloidal particles were rinsed out, for (a) a hexagonal structure and (b) square structure of colloidal crystals.
shows that the first layer is the hexagonal structure, and the threedimensional structures are closely packed. Fig. 6b displays that the first layer is a square structure, and the three-dimensional structure is BCT. Different phase profiles also confirm that the polymer brush can interact with the colloidal particles to control the symmetry of the particle structures and stabilize the in-plane structures of the colloidal crystals.
The PS particles were removed by THF solvent, which is also a good solvent for both PMMA and PS, but leaves visible features on the surface of the PMMA as if there were no swelling of the PMMA surface with the THF. Three possible reasons are list below. First, the PS particles are not cross-linked, and they could be dissolved by the THF. Second, there are layers of Au on the surface of the samples which could prevent the THF from immersing into
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the samples. This is because the samples had made the SEM observation before they were immersed into the THF. Third, the free polymer PEO which could not dissolve well in THF may protect the surface of PMMA. Clearly, it is not easy to remove the PS particles thoroughly. After being immersed into the THF solvent for 4 days at 20 ◦ C, there were still a few layers of PS colloids on the polymer brush. Therefore we placed the samples in THF solvent for sonication (for 1 min) and found the particles were shaken away from the surfaces of polymer brushes. We subsequently produced the morphology of the polymer brush by SEM, see Figs. 6a and b. It has reported that the BCT structure of the colloidal crystal had been realized by sedimentation of the colloidal dispersion of micrometer spheres in the electrical field [26,27,50]. Both Meng et al. [51] and Brevo et al. [52] have talked about the crystallization behaviour of the colloidal particles at the transition regions from n to n + 1 layers. The formation of square structure occurred in the layer transition region during the increase of the thickness. That is to say, in their systems, the position of the square struc-
Fig. 7. (Color online.) Experimentally measured reflectance spectra of the hexagonal and square colloidal crystals.
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ture was between the nth and (n + 1)th layers. While both the nth and (n + 1)th layers were hexagonal regions. Our work is different from their works. Firstly, the electrostatic interaction in our system is small. The polystyrene spheres with a small quantity of sulfate radicals on them are negatively charged, but the quantity of the charge is very little. Most part of the surface of the spheres is neutral. So the depletion force which induced by the electrostatic effect is small enough to be ignored. Secondly, the large region of the square structure is located in the surface of the same layer but not in the layer transition region. The formation of the square structure is mainly induced by the entropic effect of the polymer brush and the steric packing effect of the colloidal particles. If the brush-mediated interaction between colloidal particles dominates over the steric packing effect of the colloids, the stable square structure forms for possibly large conformational entropy of polymer brush. If the steric packing effect dominates, the stable hexagonal structure forms. It is well known that, on a hard substrate, a square-structured colloidal crystal is not stable, while a hexagon-structured one is. This is due to the fact that the latter has lower free energy than the former. However, on a polymer brush-mediated soft substrate, square-structured colloidal crystal can be stable under certain concentration of colloidal particles (Fig. 5c and Fig. 5d). This is because, on the soft surface of the polymer brush, if the interaction of different components and the entropy of free polymer chains are omitted, the approximate total free energy density consists of two parts: repulsive steric energy of colloidal particle and energy due to the entropy of the polymer brush. Here the latter may lead to an attractive interaction between colloidal particles (depletion effect). Namely, the total free energy density per unit area, F = N c F st − N br T S br , where F st is the steric energy of a particle, S br is the entropy of a single chain of the polymer brush, N c and N br are the numbers of colloidal particles and chains of
Fig. 8. (Color online.) Numerically simulated band structures of (a) BCT and (b) FCC structures. The first Brillouin zones of the (c) BCT and (d) FCC structures. The red arrows in (c) and (d) display the measurement direction of the reflectance spectra of square and hexagonal colloidal crystal, respectively. Parameters: D = 300 nm. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this or Letter.)
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polymer brushes in a unit area, respectively [32]. We owe the transitions of the structures (Fig. 1) to the competition between the entropy effect (depletion effect) of the polymer brush and the effect of repulsive particle steric energy. S br and F st are calculated in Ref. [3,6], and it is found that both of them are dependent on the colloidal particle concentration; namely, they would be different when the colloidal structure is different. It is found from the calculation that the entropy of the polymer brush of square-structure (Fig. 6b) is about 1.1k B ; but the entropy of the polymer brush of hexagonal structure is −0.2k B . In this work, the particle concentration is defined as the mass ratio of the colloidal particles to the free polymers on an adjustable polymer brush. When the particle concentration is about 4, although the repulsive steric energy for the square structure is higher, the total free energy of the square structure (or BCT structure in 3D) is low, and the corresponding structure is stable. This is because a polymer brush of square structure has very high entropy. Similarly, when PS/PEO = 10, although the entropy of the polymer brush of the hexagonal structure is very low, the repulsive steric energy for the hexagonal structure is also low, thus the total free energy of the hexagonal structure is low, and the corresponding structure is stable. To distinguish the square phase from the hexagonal one, the reflectance spectra of the PCB colloidal crystals were measured. A deuterium–tungsten light source was used to emit light of 1 mW with wavelength varying from 400 to 1100 nm. The diameter of the light beam was 1 mm. The spectrum of the silicon substrate was first measured as a reference. The samples were then illuminated by the collimated light source and the incident direction was inclined to the normal direction of the samples. The reflected light by the samples was analyzed by a monochromator with a resolving wavelength varying from 400 nm to 850 nm. The spectra of the reflected light were obtained as shown in Fig. 7. We experimentally observed two peaks in the reflectance spectra: One is located at 657 nm for the square structure sample with the PS/PEO mass ratio of 4, the other is at 710 nm for the hexagonal structure sample with the PS/PEO mass ratio of 10. Further, we used the commercial R-soft software to calculate the band structures of the BCT (Fig. 8a) and FCC (Fig. 8b) crystal lattices. In Figs. 8a and b, the arrows show that there are some local frequency-gaps in a certain wave vector region. The red arrows in Figs. 8c and d show the measurement direction of different structures. The calculated frequency-gap corresponds to reflectance wavelengths of 660 nm and 707 nm for the BCT and FCC lattices, respectively. They agree with the experimental result, namely, 657 nm and 710 nm for square and hexagonal structures, respectively. The agreement between our experiment and theory shows that the (interfacial) square structure observed in our experiments has a BCT structure in three dimensions, and that the (interfacial) hexagonal structure accords with the FCC structure in three dimensions. We have so far proved that the square phase of colloidal crystals is BCT structure by a comparison between experimental reflectance spectra and their model calculations. The threedimensional structures other than BCT were proven to be impossible because of giving wrong positions of reflection peaks. We are thus convinced that the square-phased colloidal crystals assembled via polymer-mediated self-organization has a three-dimensional BCT structure. 4. Conclusion By using free polymers and polymer brushes, we have experimentally demonstrated a theoretically proposed method [32] for fabricating a class of colloidal crystals with stable tunable crystalline structures. Such polymer-mediated colloidal crystals were found to experience a series of in-layer structure transitions including a stable square structure when we adjusted the mass ratio
of colloidal particles to free polymers. This work presents a new experimental method to fabricate colloidal crystals with structures transition, which have practical applications in various fields.
Acknowledgements
We thank R.K. Jing and P.X. Chen for their fruitful discussion and support, and L. Xu, Q.M. Zhang, L. Zhou, J.M. Hao, and G. Wang for their expertise. This work was supported by the National Natural Science Foundation of China under Grant Nos. 10334020, 10574027, 10604014, and 10874025. L.F.Z. and J.P.H. also acknowledge the financial support by the Pujiang Talent Project (No. 06PJ14006) of the Science and Technology Commission of Shanghai Municipality, by the Shanghai Education Committee and the Shanghai Education Development Foundation (“Shu Guang” project under Grant No. 05SG01), and by CNKBRSF under Grant No. 2006CB921706.
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