Abstract

We present an optical-diffraction method for quantitative characterization of two-dimensional colloidal polycrystalline materials. From the angular-diffraction profile we can estimate both the average size of the crystalline grain and the defect density within the grains. Our statistical diffraction model shows that the diffraction line shape is close to a Lorentzian profile if a lot of defects exist within a grain, and it becomes close to Gaussian if the grains are essentially free of defects. This method is used for analyzing the quality of polystyrene colloidal crystals produced by the evaporation technique. The results are compared with direct statistical analysis of microscope images.

© 2000 Optical Society of America

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References

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  1. E. Yablonovitch, “Inhibited spontaneous emission in solid state physics and electronics,” Phys. Rev. Lett. 58, 2059 (1987).
    [CrossRef] [PubMed]
  2. J. Joannopoulos, R. Meade, and J. Winn, Photonic Crystals (Princeton Press, Princeton, NJ, 1995); J. D. Joannopoulos, P. R. Villenenve, and S. Fan, “Photonic crystals: putting a new twist on light,” Nature (London) 386, 143–149 (1997).
    [CrossRef]
  3. Y. Zhao and I. Avrutsky, “Two-dimensional colloidal crystal corrugated waveguide,” Opt. Lett. 24, 817–819 (1999); Y. Zhao, I. Avrutsky, and B. Li, “Optical coupling between monocrystalline colloidal crystals and a planar waveguide,” Appl. Phys. Lett. 75, 3596–3598 (1999).
    [CrossRef]
  4. O. D. Velev, T. A. Jede, R. F. Lobo, and A. M. Lenhoff, “Microstructured porous silica via colloidal crystallization,” Nature (London) 389, 447–448 (1997).
    [CrossRef]
  5. J. E. G. J. Wijnhoven and W. L. Vos, “Preparation of photonic crystals made of air spheres in Titania,” Science 281, 802–804 (1998).
    [CrossRef]
  6. B. T. Holland, C. F. Blanford, and A. Stein, “Synthesis of microporous minerals with highly ordered three-dimensional arrays of spheroidal voids,” Science 281, 538–540 (1998).
    [CrossRef] [PubMed]
  7. K. Busch and S. John, “Photonic band gap formation in certain self-organizing system,” Phys. Rev. E 58, 3896–3908 (1998).
    [CrossRef]
  8. O. D. Velev, T. A. Jede, R. F. Lobo, and A. M. Lenhoff, “Microstructureed porous silica obtained via colloidal crystal templates,” Chem. Mater. 10, 3597–3602 (1998).
    [CrossRef]
  9. N. D. Denkov, O. D. Velev, P. A. Kralchevsky, L. B. Ivanov, H. Yoshimura, and K. Nagayama, “Mechanism of formation of two-dimensional crystals from latex particles on substrates,” Langmuir 8, 3183–3190 (1992).
    [CrossRef]
  10. P. Pieranski, “Colloidal crystals,” Contemp. Phys. 24, 25–73 (1983).
    [CrossRef]
  11. S. H. Park, D. Qin, and Y. Xia, “Crystallization of mesoscale particles over large areas,” Adv. Mater. 10, 1028–1032 (1998).
    [CrossRef]
  12. R. Mayoral, J. Requena, J. S. Moya, C. Lopez, A. Cintas, H. Miguez, F. Meseguer, L. Vazquez, M. Holgado, and A. Blanco, “3D long-range ordering in an SiO2 submicrometer sphere sintered superstructures,” Adv. Mater. 9, 257–260 (1997).
    [CrossRef]
  13. A. E. Larsen and D. G. Grier, “Melting of metastable crystallites in charge-stabilized colloidal suspensions,” Phys. Rev. Lett. 76, 3862–3865 (1996).
    [CrossRef] [PubMed]
  14. S. Rakers, L. F. Chi, and H. Fuche, “Influence of the evaporation rate on the packing order of polydisperse latex monofilms,” Langmuir 13, 7121–7124 (1997).
    [CrossRef]
  15. A. N. Fitch, “High resolution powder diffraction studies of poly-crystalline materials,” Nucl. Instrum. Methods Phys. Res. 97, 63–69 (1995).
    [CrossRef]
  16. M. Born and E. Wolf, “Fraunhofer and Fresnel diffraction,” in Principles of Optics (Pergamon Press, New York, 1964), Section 8.3.3.

1998

J. E. G. J. Wijnhoven and W. L. Vos, “Preparation of photonic crystals made of air spheres in Titania,” Science 281, 802–804 (1998).
[CrossRef]

B. T. Holland, C. F. Blanford, and A. Stein, “Synthesis of microporous minerals with highly ordered three-dimensional arrays of spheroidal voids,” Science 281, 538–540 (1998).
[CrossRef] [PubMed]

K. Busch and S. John, “Photonic band gap formation in certain self-organizing system,” Phys. Rev. E 58, 3896–3908 (1998).
[CrossRef]

O. D. Velev, T. A. Jede, R. F. Lobo, and A. M. Lenhoff, “Microstructureed porous silica obtained via colloidal crystal templates,” Chem. Mater. 10, 3597–3602 (1998).
[CrossRef]

S. H. Park, D. Qin, and Y. Xia, “Crystallization of mesoscale particles over large areas,” Adv. Mater. 10, 1028–1032 (1998).
[CrossRef]

1997

R. Mayoral, J. Requena, J. S. Moya, C. Lopez, A. Cintas, H. Miguez, F. Meseguer, L. Vazquez, M. Holgado, and A. Blanco, “3D long-range ordering in an SiO2 submicrometer sphere sintered superstructures,” Adv. Mater. 9, 257–260 (1997).
[CrossRef]

O. D. Velev, T. A. Jede, R. F. Lobo, and A. M. Lenhoff, “Microstructured porous silica via colloidal crystallization,” Nature (London) 389, 447–448 (1997).
[CrossRef]

S. Rakers, L. F. Chi, and H. Fuche, “Influence of the evaporation rate on the packing order of polydisperse latex monofilms,” Langmuir 13, 7121–7124 (1997).
[CrossRef]

1996

A. E. Larsen and D. G. Grier, “Melting of metastable crystallites in charge-stabilized colloidal suspensions,” Phys. Rev. Lett. 76, 3862–3865 (1996).
[CrossRef] [PubMed]

1995

A. N. Fitch, “High resolution powder diffraction studies of poly-crystalline materials,” Nucl. Instrum. Methods Phys. Res. 97, 63–69 (1995).
[CrossRef]

1992

N. D. Denkov, O. D. Velev, P. A. Kralchevsky, L. B. Ivanov, H. Yoshimura, and K. Nagayama, “Mechanism of formation of two-dimensional crystals from latex particles on substrates,” Langmuir 8, 3183–3190 (1992).
[CrossRef]

1987

E. Yablonovitch, “Inhibited spontaneous emission in solid state physics and electronics,” Phys. Rev. Lett. 58, 2059 (1987).
[CrossRef] [PubMed]

1983

P. Pieranski, “Colloidal crystals,” Contemp. Phys. 24, 25–73 (1983).
[CrossRef]

Blanco, A.

R. Mayoral, J. Requena, J. S. Moya, C. Lopez, A. Cintas, H. Miguez, F. Meseguer, L. Vazquez, M. Holgado, and A. Blanco, “3D long-range ordering in an SiO2 submicrometer sphere sintered superstructures,” Adv. Mater. 9, 257–260 (1997).
[CrossRef]

Blanford, C. F.

B. T. Holland, C. F. Blanford, and A. Stein, “Synthesis of microporous minerals with highly ordered three-dimensional arrays of spheroidal voids,” Science 281, 538–540 (1998).
[CrossRef] [PubMed]

Busch, K.

K. Busch and S. John, “Photonic band gap formation in certain self-organizing system,” Phys. Rev. E 58, 3896–3908 (1998).
[CrossRef]

Chi, L. F.

S. Rakers, L. F. Chi, and H. Fuche, “Influence of the evaporation rate on the packing order of polydisperse latex monofilms,” Langmuir 13, 7121–7124 (1997).
[CrossRef]

Cintas, A.

R. Mayoral, J. Requena, J. S. Moya, C. Lopez, A. Cintas, H. Miguez, F. Meseguer, L. Vazquez, M. Holgado, and A. Blanco, “3D long-range ordering in an SiO2 submicrometer sphere sintered superstructures,” Adv. Mater. 9, 257–260 (1997).
[CrossRef]

Denkov, N. D.

N. D. Denkov, O. D. Velev, P. A. Kralchevsky, L. B. Ivanov, H. Yoshimura, and K. Nagayama, “Mechanism of formation of two-dimensional crystals from latex particles on substrates,” Langmuir 8, 3183–3190 (1992).
[CrossRef]

Fitch, A. N.

A. N. Fitch, “High resolution powder diffraction studies of poly-crystalline materials,” Nucl. Instrum. Methods Phys. Res. 97, 63–69 (1995).
[CrossRef]

Fuche, H.

S. Rakers, L. F. Chi, and H. Fuche, “Influence of the evaporation rate on the packing order of polydisperse latex monofilms,” Langmuir 13, 7121–7124 (1997).
[CrossRef]

Grier, D. G.

A. E. Larsen and D. G. Grier, “Melting of metastable crystallites in charge-stabilized colloidal suspensions,” Phys. Rev. Lett. 76, 3862–3865 (1996).
[CrossRef] [PubMed]

Holgado, M.

R. Mayoral, J. Requena, J. S. Moya, C. Lopez, A. Cintas, H. Miguez, F. Meseguer, L. Vazquez, M. Holgado, and A. Blanco, “3D long-range ordering in an SiO2 submicrometer sphere sintered superstructures,” Adv. Mater. 9, 257–260 (1997).
[CrossRef]

Holland, B. T.

B. T. Holland, C. F. Blanford, and A. Stein, “Synthesis of microporous minerals with highly ordered three-dimensional arrays of spheroidal voids,” Science 281, 538–540 (1998).
[CrossRef] [PubMed]

Ivanov, L. B.

N. D. Denkov, O. D. Velev, P. A. Kralchevsky, L. B. Ivanov, H. Yoshimura, and K. Nagayama, “Mechanism of formation of two-dimensional crystals from latex particles on substrates,” Langmuir 8, 3183–3190 (1992).
[CrossRef]

Jede, T. A.

O. D. Velev, T. A. Jede, R. F. Lobo, and A. M. Lenhoff, “Microstructureed porous silica obtained via colloidal crystal templates,” Chem. Mater. 10, 3597–3602 (1998).
[CrossRef]

O. D. Velev, T. A. Jede, R. F. Lobo, and A. M. Lenhoff, “Microstructured porous silica via colloidal crystallization,” Nature (London) 389, 447–448 (1997).
[CrossRef]

John, S.

K. Busch and S. John, “Photonic band gap formation in certain self-organizing system,” Phys. Rev. E 58, 3896–3908 (1998).
[CrossRef]

Kralchevsky, P. A.

N. D. Denkov, O. D. Velev, P. A. Kralchevsky, L. B. Ivanov, H. Yoshimura, and K. Nagayama, “Mechanism of formation of two-dimensional crystals from latex particles on substrates,” Langmuir 8, 3183–3190 (1992).
[CrossRef]

Larsen, A. E.

A. E. Larsen and D. G. Grier, “Melting of metastable crystallites in charge-stabilized colloidal suspensions,” Phys. Rev. Lett. 76, 3862–3865 (1996).
[CrossRef] [PubMed]

Lenhoff, A. M.

O. D. Velev, T. A. Jede, R. F. Lobo, and A. M. Lenhoff, “Microstructureed porous silica obtained via colloidal crystal templates,” Chem. Mater. 10, 3597–3602 (1998).
[CrossRef]

O. D. Velev, T. A. Jede, R. F. Lobo, and A. M. Lenhoff, “Microstructured porous silica via colloidal crystallization,” Nature (London) 389, 447–448 (1997).
[CrossRef]

Lobo, R. F.

O. D. Velev, T. A. Jede, R. F. Lobo, and A. M. Lenhoff, “Microstructureed porous silica obtained via colloidal crystal templates,” Chem. Mater. 10, 3597–3602 (1998).
[CrossRef]

O. D. Velev, T. A. Jede, R. F. Lobo, and A. M. Lenhoff, “Microstructured porous silica via colloidal crystallization,” Nature (London) 389, 447–448 (1997).
[CrossRef]

Lopez, C.

R. Mayoral, J. Requena, J. S. Moya, C. Lopez, A. Cintas, H. Miguez, F. Meseguer, L. Vazquez, M. Holgado, and A. Blanco, “3D long-range ordering in an SiO2 submicrometer sphere sintered superstructures,” Adv. Mater. 9, 257–260 (1997).
[CrossRef]

Mayoral, R.

R. Mayoral, J. Requena, J. S. Moya, C. Lopez, A. Cintas, H. Miguez, F. Meseguer, L. Vazquez, M. Holgado, and A. Blanco, “3D long-range ordering in an SiO2 submicrometer sphere sintered superstructures,” Adv. Mater. 9, 257–260 (1997).
[CrossRef]

Meseguer, F.

R. Mayoral, J. Requena, J. S. Moya, C. Lopez, A. Cintas, H. Miguez, F. Meseguer, L. Vazquez, M. Holgado, and A. Blanco, “3D long-range ordering in an SiO2 submicrometer sphere sintered superstructures,” Adv. Mater. 9, 257–260 (1997).
[CrossRef]

Miguez, H.

R. Mayoral, J. Requena, J. S. Moya, C. Lopez, A. Cintas, H. Miguez, F. Meseguer, L. Vazquez, M. Holgado, and A. Blanco, “3D long-range ordering in an SiO2 submicrometer sphere sintered superstructures,” Adv. Mater. 9, 257–260 (1997).
[CrossRef]

Moya, J. S.

R. Mayoral, J. Requena, J. S. Moya, C. Lopez, A. Cintas, H. Miguez, F. Meseguer, L. Vazquez, M. Holgado, and A. Blanco, “3D long-range ordering in an SiO2 submicrometer sphere sintered superstructures,” Adv. Mater. 9, 257–260 (1997).
[CrossRef]

Nagayama, K.

N. D. Denkov, O. D. Velev, P. A. Kralchevsky, L. B. Ivanov, H. Yoshimura, and K. Nagayama, “Mechanism of formation of two-dimensional crystals from latex particles on substrates,” Langmuir 8, 3183–3190 (1992).
[CrossRef]

Park, S. H.

S. H. Park, D. Qin, and Y. Xia, “Crystallization of mesoscale particles over large areas,” Adv. Mater. 10, 1028–1032 (1998).
[CrossRef]

Pieranski, P.

P. Pieranski, “Colloidal crystals,” Contemp. Phys. 24, 25–73 (1983).
[CrossRef]

Qin, D.

S. H. Park, D. Qin, and Y. Xia, “Crystallization of mesoscale particles over large areas,” Adv. Mater. 10, 1028–1032 (1998).
[CrossRef]

Rakers, S.

S. Rakers, L. F. Chi, and H. Fuche, “Influence of the evaporation rate on the packing order of polydisperse latex monofilms,” Langmuir 13, 7121–7124 (1997).
[CrossRef]

Requena, J.

R. Mayoral, J. Requena, J. S. Moya, C. Lopez, A. Cintas, H. Miguez, F. Meseguer, L. Vazquez, M. Holgado, and A. Blanco, “3D long-range ordering in an SiO2 submicrometer sphere sintered superstructures,” Adv. Mater. 9, 257–260 (1997).
[CrossRef]

Stein, A.

B. T. Holland, C. F. Blanford, and A. Stein, “Synthesis of microporous minerals with highly ordered three-dimensional arrays of spheroidal voids,” Science 281, 538–540 (1998).
[CrossRef] [PubMed]

Vazquez, L.

R. Mayoral, J. Requena, J. S. Moya, C. Lopez, A. Cintas, H. Miguez, F. Meseguer, L. Vazquez, M. Holgado, and A. Blanco, “3D long-range ordering in an SiO2 submicrometer sphere sintered superstructures,” Adv. Mater. 9, 257–260 (1997).
[CrossRef]

Velev, O. D.

O. D. Velev, T. A. Jede, R. F. Lobo, and A. M. Lenhoff, “Microstructureed porous silica obtained via colloidal crystal templates,” Chem. Mater. 10, 3597–3602 (1998).
[CrossRef]

O. D. Velev, T. A. Jede, R. F. Lobo, and A. M. Lenhoff, “Microstructured porous silica via colloidal crystallization,” Nature (London) 389, 447–448 (1997).
[CrossRef]

N. D. Denkov, O. D. Velev, P. A. Kralchevsky, L. B. Ivanov, H. Yoshimura, and K. Nagayama, “Mechanism of formation of two-dimensional crystals from latex particles on substrates,” Langmuir 8, 3183–3190 (1992).
[CrossRef]

Vos, W. L.

J. E. G. J. Wijnhoven and W. L. Vos, “Preparation of photonic crystals made of air spheres in Titania,” Science 281, 802–804 (1998).
[CrossRef]

Wijnhoven, J. E. G. J.

J. E. G. J. Wijnhoven and W. L. Vos, “Preparation of photonic crystals made of air spheres in Titania,” Science 281, 802–804 (1998).
[CrossRef]

Xia, Y.

S. H. Park, D. Qin, and Y. Xia, “Crystallization of mesoscale particles over large areas,” Adv. Mater. 10, 1028–1032 (1998).
[CrossRef]

Yablonovitch, E.

E. Yablonovitch, “Inhibited spontaneous emission in solid state physics and electronics,” Phys. Rev. Lett. 58, 2059 (1987).
[CrossRef] [PubMed]

Yoshimura, H.

N. D. Denkov, O. D. Velev, P. A. Kralchevsky, L. B. Ivanov, H. Yoshimura, and K. Nagayama, “Mechanism of formation of two-dimensional crystals from latex particles on substrates,” Langmuir 8, 3183–3190 (1992).
[CrossRef]

Adv. Mater.

S. H. Park, D. Qin, and Y. Xia, “Crystallization of mesoscale particles over large areas,” Adv. Mater. 10, 1028–1032 (1998).
[CrossRef]

R. Mayoral, J. Requena, J. S. Moya, C. Lopez, A. Cintas, H. Miguez, F. Meseguer, L. Vazquez, M. Holgado, and A. Blanco, “3D long-range ordering in an SiO2 submicrometer sphere sintered superstructures,” Adv. Mater. 9, 257–260 (1997).
[CrossRef]

Chem. Mater.

O. D. Velev, T. A. Jede, R. F. Lobo, and A. M. Lenhoff, “Microstructureed porous silica obtained via colloidal crystal templates,” Chem. Mater. 10, 3597–3602 (1998).
[CrossRef]

Contemp. Phys.

P. Pieranski, “Colloidal crystals,” Contemp. Phys. 24, 25–73 (1983).
[CrossRef]

Langmuir

N. D. Denkov, O. D. Velev, P. A. Kralchevsky, L. B. Ivanov, H. Yoshimura, and K. Nagayama, “Mechanism of formation of two-dimensional crystals from latex particles on substrates,” Langmuir 8, 3183–3190 (1992).
[CrossRef]

S. Rakers, L. F. Chi, and H. Fuche, “Influence of the evaporation rate on the packing order of polydisperse latex monofilms,” Langmuir 13, 7121–7124 (1997).
[CrossRef]

Nature (London)

O. D. Velev, T. A. Jede, R. F. Lobo, and A. M. Lenhoff, “Microstructured porous silica via colloidal crystallization,” Nature (London) 389, 447–448 (1997).
[CrossRef]

Nucl. Instrum. Methods Phys. Res.

A. N. Fitch, “High resolution powder diffraction studies of poly-crystalline materials,” Nucl. Instrum. Methods Phys. Res. 97, 63–69 (1995).
[CrossRef]

Phys. Rev. E

K. Busch and S. John, “Photonic band gap formation in certain self-organizing system,” Phys. Rev. E 58, 3896–3908 (1998).
[CrossRef]

Phys. Rev. Lett.

E. Yablonovitch, “Inhibited spontaneous emission in solid state physics and electronics,” Phys. Rev. Lett. 58, 2059 (1987).
[CrossRef] [PubMed]

A. E. Larsen and D. G. Grier, “Melting of metastable crystallites in charge-stabilized colloidal suspensions,” Phys. Rev. Lett. 76, 3862–3865 (1996).
[CrossRef] [PubMed]

Science

J. E. G. J. Wijnhoven and W. L. Vos, “Preparation of photonic crystals made of air spheres in Titania,” Science 281, 802–804 (1998).
[CrossRef]

B. T. Holland, C. F. Blanford, and A. Stein, “Synthesis of microporous minerals with highly ordered three-dimensional arrays of spheroidal voids,” Science 281, 538–540 (1998).
[CrossRef] [PubMed]

Other

J. Joannopoulos, R. Meade, and J. Winn, Photonic Crystals (Princeton Press, Princeton, NJ, 1995); J. D. Joannopoulos, P. R. Villenenve, and S. Fan, “Photonic crystals: putting a new twist on light,” Nature (London) 386, 143–149 (1997).
[CrossRef]

Y. Zhao and I. Avrutsky, “Two-dimensional colloidal crystal corrugated waveguide,” Opt. Lett. 24, 817–819 (1999); Y. Zhao, I. Avrutsky, and B. Li, “Optical coupling between monocrystalline colloidal crystals and a planar waveguide,” Appl. Phys. Lett. 75, 3596–3598 (1999).
[CrossRef]

M. Born and E. Wolf, “Fraunhofer and Fresnel diffraction,” in Principles of Optics (Pergamon Press, New York, 1964), Section 8.3.3.

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Figures (7)

Fig. 1
Fig. 1

AFM image of the colloidal crystal. Arrows point to the crystal defects. The inset below the image shows mutual orientation of the crystal and the wave vectors in the inverse space.

Fig. 2
Fig. 2

Measurements setup for the angular diffraction spectrum.

Fig. 3
Fig. 3

(a) Photographic picture of the diffraction in the case of a polycrystalline sample and a laser beam width of 3 mm, and (b) the laser beam focused a single crystallite 20 µm in diameter.

Fig. 4
Fig. 4

Entire measured diffraction spectrum (top), and a single lobe with a Lorentzian fit (bottom).

Fig. 5
Fig. 5

Major crystal imperfections. The top image shows both crystal defects and crystal boundaries (dashed lines). The bottom image shows a structure that maintains the crystal orientation throughout the entire area, so it is a monocrystal with a large number of defects.

Fig. 6
Fig. 6

Calculated diffraction spectrum for different crystallite size, defect density, and defect distribution: (a) D¯=50 µm, 1/σ¯=10 µm, and the Poisson distribution of Dl; (b) D¯=10 µm, 1/σ¯=10 µm, and the Poisson distribution of Dl; (c) D¯=50 µm, 1/σ¯=10 µm, and the exponential distribution of Dl; (d) D¯=10 µm, 1/σ¯=10 µm, and the exponential distribution of Dl; (e) D¯=10 µm, 1/σ¯=50 µm, and the Poisson distribution of Dl.

Fig. 7
Fig. 7

Measured data fitted with the model curve.

Tables (1)

Tables Icon

Table 1 Colloidal Crystal Parametersa

Equations (11)

Equations on this page are rendered with MathJax. Learn more.

|ρ|=(2π/λ)sin(θ),
Ts(x; y)
=lAl(y)(x-xl)×m=0MlPm(l)(x-xm(l))exp[j2π(ξ0x)+jδm(l)].
M¯l=σD¯l.
G(ξ,η)=F [Ts(x; y)]=-dy-dx{Ts(x; y)×exp[-j(ξx+ηy)]}.
S(ξ,η)=limL E¯{|G(ξ,η)|2}4L2=F [Rxy(τx,τy)]=F [Rxy(τx,τy)],
Rxy(τx,τy)=Rx(τx)Ry(τy)/Rxy(0, 0),
Rx(τx)=E¯{Ts(x; y)Ts(x+τx; y)}
S(ξ,η)=F [Rx(τx)]F [Ry(τy)]/Rxy(0,0)=Sx(ξ)Sy(η)/Rxy(0, 0).
U(θ)=Ak0 cos(θ)Sx[k0 sin(θ)],
Sx(ξ)=E¯- dx[Ts(x; y)exp(-j2πξx)]2.

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