## Orientation control of micro-domains in anisotropic laser ceramics |

Optical Materials Express, Vol. 3, Issue 6, pp. 829-841 (2013)

http://dx.doi.org/10.1364/OME.3.000829

Acrobat PDF (3114 KB)

### Abstract

We present theoretical studies on the orientation control of micro-domains in anisotropic laser ceramics, and produce the distribution function of the crystal orientation in micro-domains including anisotropic laser ceramics. Also the improvement in the orientation distribution caused by preferential grain growth is discussed, where our theoretical analyses were applied to several different Nd:FAP ceramics. Detailed XRD analyses based on this distribution function show that the preferential grain growth improved the orientation distribution of the green body that was slip-casted under magnetic field.

© 2013 OSA

## 1. Introduction

1. I. Shoji, S. Kurimura, Y. Sato, T. Taira, A. Ikesue, and K. Yoshida, “Optical properties and laser characteristics of highly Nd^{3+}-doped Y_{3}Al_{5}O_{12} ceramics,” Appl. Phys. Lett. **77**(7), 939–941 (2000). [CrossRef]

2. A. A. Kaminskii, M. S. Akchurin, R. V. Gainutdinov, K. Takaichi, A. Shirakava, H. Yagi, T. Yanagitani, and K. Ueda, “Microhardness and fracture toughness of Y_{2}O_{3}- and Y_{3}Al_{5}O_{12}-based nanocrystalline laser ceramics,” Crystallogr. Rep. **50**(5), 869–873 (2005). [CrossRef]

3. S. J. McNaught, H. Komine, S. B. Weiss, R. Simpson, A. M. F. Johnson, J. Machan, C. P. Asman, M. Weber, G. C. Jones, M. M. Valley, A. Jankevics, D. Burchman, M. McClellan, J. Sollee, J. Marmo, and H. Injeyan, “100 kW coherently combined slab MOPAs,” in *Conference on Lasers and Electro-Optics/International Quantum Electronics Conference*, OSA Technical Digest (CD) (Optical Society of America, 2009), paper CThA1. [CrossRef]

4. Y. Sato, J. Saikawa, T. Taira, and A. Ikesue, “Characteristics of Nd^{3+}-doped Y_{3}ScAl_{4}O_{12} ceramic laser,” Opt. Mater. **29**(10), 1277–1282 (2007). [CrossRef]

5. Y. Sato, A. Ikesue, and T. Taira, “Tailored spectral designing of layer-by-layer type composite Nd:Y_{3}ScAl_{4}O_{12}/Nd:Y_{3}Al_{5}O_{12} ceramics,” IEEE J. Sel. Top. Quantum Electron. **13**(3), 838–843 (2007). [CrossRef]

1. I. Shoji, S. Kurimura, Y. Sato, T. Taira, A. Ikesue, and K. Yoshida, “Optical properties and laser characteristics of highly Nd^{3+}-doped Y_{3}Al_{5}O_{12} ceramics,” Appl. Phys. Lett. **77**(7), 939–941 (2000). [CrossRef]

6. Y. Sato, T. Taira, N. Pavel, and V. Lupei, “Laser operation with near quantum-defect slope efficiency in Nd:YVO_{4} under direct pumping into emitting level,” Appl. Phys. Lett. **82**(6), 844–846 (2003). [CrossRef]

7. S. A. Payne, L. K. Smith, L. D. Deloach, W. L. Kway, J. B. Tassano, and W. F. Krupke, “Laser, optical, and thermomechanical properties of Yb-doped fluoroapatite,” IEEE J. Quantum Electron. **30**(1), 170–179 (1994). [CrossRef]

8. T. Taira, “Domain-controlled laser ceramics toward Giant Micro-photonics [Invited],” Opt. Mater. Express **1**(5), 1040–1050 (2011). [CrossRef]

*k*-th domain

*G*aswhere

_{k}*T*,

*γ*,

_{g}*σ*,

_{ij}*ε*,

_{ij}**,**

*E***,**

*B**V*,

_{k}*S*,

_{k}*μ*,

_{k}*N*,

_{k}*r*,

_{k}

*P**, and*

_{k}

*M**are temperature, boundary energy of the grain, stress tensor, strain tensor, external electric field, applied magnetic flux density, volume, entropy, chemical potential, molecular number, curvature radius of the surface, dielectric polarization, and magnetization of*

_{k}*k*-th domain, respectively. The first and second terms of the right side in Eq. (1) express the crystallization from liquid phase in crystal growth. The second term also generates the motive force of the grain growth by the solid-state reaction [9

9. A. Ikesue, T. Kinoshita, K. Kamata, and K. Yoshida, “Fabrication and optical properties of high performance polycrystalline Nd:YAG ceramics for solid-state lasers,” J. Am. Ceram. Soc. **78**(4), 1033–1040 (1995). [CrossRef]

11. M. Harada, K. Muramatsu, Y. Iwasaki, S. Kurimura, and T. Taira, “Periodic twinning in crystal quartz for optical quasi-phase matched secondary harmonic conversion,” J. Mater. Res. **19**(04), 969–972 (2004). [CrossRef]

12. H. Ishizuki and T. Taira, “Half-joule output optical-parametric oscillation by using 10-mm-thick periodically poled Mg-doped congruent LiNbO_{3.},” Opt. Express **20**(18), 20002–20010 (2012). [CrossRef] [PubMed]

## 2. Theory

### 2.1 Magnetostatic potential of the anisotropic micro-domain under magnetic field

*T**onto the*

_{k}*k*-th anisotropic micro-domain that reduces the angle

*θ*between the control axis and the direction of easy magnetization axis in

_{k}*k*-th micro-domain.

*φ*is the angle describing a precession movement of the micro-domain, and is free from

_{k}

*T**, and fluctuates by the thermal disturbance depending on the product of*

_{k}*T*and Boltzmann’s constant

*k*. By use of the magnetic susceptibility tensor

**of the media,**

*χ*

*M**is given bywhere*

_{k}*μ*is the magnetic permeability in vacuum. In the case of a crystal belonging to the uni-axial crystal system,

**is a diagonal tensor with two independent elements. The crystal direction with larger element**

*χ**χ*

_{e}is called as easy magnetization axis, of and the direction with smaller element

*χ*

_{h}is difficult magnetization axis. When one of crystal axes is easy magnetization axis and other two axes are hard magnetization axes, the easy magnetization axis can be aligned along applied static magnetic field [8

8. T. Taira, “Domain-controlled laser ceramics toward Giant Micro-photonics [Invited],” Opt. Mater. Express **1**(5), 1040–1050 (2011). [CrossRef]

### 2.2 Figure of merit for the orientation control

### 2.3 Distribution function for the orientation angle under thermal equilibrium

*f*

_{eq}(

*θ*,

*φ*,

*V*,

*B*) for

*N*micro-domains with volume of

_{V}*V*under magnetic flux density

*B*expressed by Boltzmann distribution aswhere

*Ω*is a solid angle and equals to sin

*θdθdφ*. By use of the simple approximation,

*f*

_{eq}(

*θ*,

*φ*,

*V*,

*B*)

*dΩ*can be given bywhere

*B*

_{min}is the minimum magnetic flux density for orientation control which is defined so that the anisotropy in magnetostatic potential is equal to

*kT*.

*B*

_{min}is given byIn the case of diffusion phenomena, the distribution function is approximated by Gaussian in general. The approximation at the derivation Eq. (7) is convenient to calculate, especially by use of polar coordinates, and it is comparable to Gaussian distribution in the order of approximation.

### 2.4 Improvement of the orientation distribution by preferential growth

*G*during initial condition in sintering process is given bywith the bounding condition of the conservation in total volume asHere we consider the interaction between 2 grains. In this case Eq. (9) can be simplified toEquation (11) shows the preferential grain growth in larger micro-domains, where smaller grains tend to reduce their volume and larger grains to increase their volume. It also indicates that grain growth will stop after almost uniform grain size can be established.

*D*is proportional to its volume, we can find that the distribution function

*f*

_{eq}for micro-domains of

*D*larger than the mean diameter of all micro-domains in slurry

*D*

_{m}is drastically improved. Even under

*B*

_{min}

^{m}that is minimum magnetic flux density for micro-domains with

*D*

_{m}is given byEquation (12) shows that micro-domains with larger domain size can be well aligned as a core particle in preferential grain growth if we can realize the appropriate distribution of the micro-domain size in the slurry under slip casting process.

### 2.5 Relation between Lotgering factor and orientation distribution

*f*and the distribution of the crystal orientation. X-ray diffraction intensity

*I*

_{r}(

*hkl*) is proportional to the amount of the micro-domains that has (

*hkl*)-plane perpendicular to incident X-ray. The angle between

*c*-axis and control axis

*θ*(

*hkl*) of these micro-domains are expressed bywhere

*a*,

*c*, and

*δ*is a lattice constant for two independent crystal axes in uni-axial crystals, and the index of crystal structure which is 1 for hexagonal crystals and 0 for tetragonal crystals. By use of

*θ*(

*hkl*) the branching ratio

*ρ*

_{s}(

*hkl*) for the orientation controlled anisotropic ceramics can be expected to be comparable toFinally, the theoretical relation between distribution function and Lotgering factor is given by

*P*

_{eq}of micro-domains in slurry under thermal equilibrium from Eq. (7). By use of a certain

*θ*

_{0}below

*π*/2,

*P*

_{eq}[0 ≤

*θ*≤

*θ*

_{0}] for the distribution probability of the crystal orientation within the range of

*θ*from 0 to

*θ*

_{0}can be calculated as

## 3. Experimental setup and results

### 3.1 Fabrication of transparent Nd:FAP ceramics

7. S. A. Payne, L. K. Smith, L. D. Deloach, W. L. Kway, J. B. Tassano, and W. F. Krupke, “Laser, optical, and thermomechanical properties of Yb-doped fluoroapatite,” IEEE J. Quantum Electron. **30**(1), 170–179 (1994). [CrossRef]

^{3+}in these ceramics was 2at.% which examined by X-ray fluorescence analyzer (JSX-3400RII, JEOL).

### 3.2 XRD observation of Nd:FAP ceramics

^{−1}. XRD patterns of ceramic specimens were detected from the surface perpendicular to the direction of applied magnetic field. All these diffraction peaks can be assigned to the standard data in ICDD - #00-015-0876 card for flourapatite.

16. J. Akiyama, Y. Sato, and T. Taira, “Laser demonstration of diode-pumped Nd^{3+}-doped fluorapatite anisotropic ceramics,” Appl. Phys. Express **4**(2), 022703 (2011). [CrossRef]

## 4. Discussions

### 4.1 Lotgering factor of Nd:FAP ceramics

*c*-axis): (100), (110), (200), (210), (300), (310), (400), (320), (410), and (330). By use of XRD pattern from Nd:FAP raw powder as a reference, Lotgering factors of Nd:FAP ceramics-1 and Nd:FAP ceramics-2 are calculated to be 0.96 and 0.93, respectively. Lotgering factor suggests that the quality of Nd:FAP ceramics-1 as anisotropic laser ceramics is higher than that of Nd:FAP ceramics-2.

### 4.2 Calculation of distribution probability

*P*

_{eq}for primary particles aligned within the range of

*θ*from 0 to

*θ*

_{0}that is given by Eq. (16). Even though we are satisfied by the rough alignment within ± 30 degree, 4 times larger magnetic field than

*B*

_{min}is required to realize this situation without preferential grain growth. If we want to obtain powder compact where more than 95% of primary particles are aligned within ± 1 degree, more than 99

*B*

_{min}is required, which considered to be more than 100 tesla of magnetic flux density from our previous report [19

19. Y. Sato, J. Akiyama, and T. Taira, “Fundamental investigations in orientation control process for anisotropic laser ceramics,” Phys. Status Solidi C (2013), doi:. [CrossRef]

*D*that is larger than

*D*

_{m}under

*B*

_{min}

^{m}. Even if the size distribution of raw powder is quite controlled to be homogeneous, 99% of micro-domains with only 4 times larger size than mean particles can be aligned within ± 15 degree, where only 10% of mean particles are aligned within ± 15 degree. In the case of core micro-domains with 25 times larger diameter than mean particle, more than 99% of these core micro-domains can be aligned within the range of ± 1 degree along

*c*-axis. This calculation indicates that nearly perfect orientation control is capable if we can make the preferential grain growth performed well.

### 4.3 Relation between distribution of the orientation and Lorgering factor in Nd:FAP ceramics

20. N. Leroy and E. Bres, “Structure and substitutions in fluorapatite,” Eur. Cell. Mater. **2**, 36–48 (2001). [PubMed]

*B*

_{min}), 99% of micro-domains have the angle between

*c*-axis and control axis within ± 78 degree. On the contrary, 99% of micro-domains have the angle between

*c*-axis and control axis within ± 53 degree inside the powder compact with Lotgering factor of 0.62 (slip-casted under 2

*B*

_{min}).

### 4.4 Distribution of the crystal orientation in Nd:FAP ceramics

*c*-axis and control axis, we evaluated change in branching ratio in XRD patterns during sintering process. From Eqs. (4) and (7), the ratio between

*ρ*

_{s}(

*hkl*) and

*ρ*

_{r}(

*hkl*) is given by

*ρ*

_{s}(

*hkl*) and

*ρ*

_{r}(

*hkl*) has a Gaussian dependence on

*θ*, this ratio in sintered anisotropic laser ceramics has an exponential dependence as shown in Fig. 8. While various causes can be expected about this difference: the influence due to low-angle grain boundary, the size distribution in micro-domains, and so on. However, the orientation distribution changes certainly before and after sintering process, which strongly suggest the existence of the improvement of orientation control due to preferential grain growth.

*θ*in Nd:FAP ceramics-1 is smaller than Nd:FAP ceramics-2. However, some diffraction peaks with a certain

*θ*still remain considerably in Nd:FAP ceramics-1. For example, the diffraction peak of (201) from domains with

*θ*of 19.9 degree did not reduced by sintering. On the contrary, the ratio of branching ratio in Nd:FAP ceramics-2 shows clear dependence on

*θ*, which indicates the fluctuation of

*θ*in Nd:FAP ceramics-2 is well controlled by the kinetics of orientation control discussed in this work. The large deviation of the ratio in Nd:FAP ceramics-1 suggest the problems in the uniformity of the orientation control, such as insufficient pulverization under slurry treatments.

## 5. Summary

## Appendix A: Equation of the motion under magnetic field

*T**can be derived by differential of the magnetostatic energy of*

_{k}*k*-th domain

*U*with angle and is given bywhere

_{k}

*e**is the unit vector for the axis of coordinate-*

_{i}*i*.

*R*due to Brownian motion by slurry molecules. Consequently, the equation of motion for the anisotropic micro-domain is given bywhere

_{k}*I*,

_{k}*η*, and

*t*are the moment of inertia of

*k*-th micro-domain, viscosity of the slurry, and time, respectively. Because

*R*is a random force,

_{k}*θ*cannot be determined as a unique value in a certain time. In other words,

_{k}*θ*and

_{k}*φ*can be estimated stochastically from the distribution function

_{k}*f*(

*θ*,

*φ*,

*V*,

*B*,

*t*) for the orientation of micro-domain with volume

*V*. This fact indicates that in principle the alignment of crystal orientation is not perfect. Therefore, it is important to reduce the deviation of

*θ*in order to improve the optical quality of anisotropic laser ceramics.

*f*(

*θ*,

*φ*,

*V*,

*B*,

*t*) for the orientation of micro-domain with volume

*V*is stabilized to the thermal equilibrium within a finite time

*τ*. In the normal fabrication process where

*R*can be treated as a white noise and the inertial force is enough small to ignore comparing

_{k}*η*, Eq. (18) gives

*τ*as

## Appendix B: Derivation of Eq. (7)

*f*

_{eq}(

*θ*,

*φ*,

*V*,

*B*)

*dΩ*can be given by

## Acknowledgments

## References and links

1. | I. Shoji, S. Kurimura, Y. Sato, T. Taira, A. Ikesue, and K. Yoshida, “Optical properties and laser characteristics of highly Nd |

2. | A. A. Kaminskii, M. S. Akchurin, R. V. Gainutdinov, K. Takaichi, A. Shirakava, H. Yagi, T. Yanagitani, and K. Ueda, “Microhardness and fracture toughness of Y |

3. | S. J. McNaught, H. Komine, S. B. Weiss, R. Simpson, A. M. F. Johnson, J. Machan, C. P. Asman, M. Weber, G. C. Jones, M. M. Valley, A. Jankevics, D. Burchman, M. McClellan, J. Sollee, J. Marmo, and H. Injeyan, “100 kW coherently combined slab MOPAs,” in |

4. | Y. Sato, J. Saikawa, T. Taira, and A. Ikesue, “Characteristics of Nd |

5. | Y. Sato, A. Ikesue, and T. Taira, “Tailored spectral designing of layer-by-layer type composite Nd:Y |

6. | Y. Sato, T. Taira, N. Pavel, and V. Lupei, “Laser operation with near quantum-defect slope efficiency in Nd:YVO |

7. | S. A. Payne, L. K. Smith, L. D. Deloach, W. L. Kway, J. B. Tassano, and W. F. Krupke, “Laser, optical, and thermomechanical properties of Yb-doped fluoroapatite,” IEEE J. Quantum Electron. |

8. | T. Taira, “Domain-controlled laser ceramics toward Giant Micro-photonics [Invited],” Opt. Mater. Express |

9. | A. Ikesue, T. Kinoshita, K. Kamata, and K. Yoshida, “Fabrication and optical properties of high performance polycrystalline Nd:YAG ceramics for solid-state lasers,” J. Am. Ceram. Soc. |

10. | W. D. Kingerly, H. K. Brown, and D. R. Uhlmann, |

11. | M. Harada, K. Muramatsu, Y. Iwasaki, S. Kurimura, and T. Taira, “Periodic twinning in crystal quartz for optical quasi-phase matched secondary harmonic conversion,” J. Mater. Res. |

12. | H. Ishizuki and T. Taira, “Half-joule output optical-parametric oscillation by using 10-mm-thick periodically poled Mg-doped congruent LiNbO |

13. | H. Morikawa, Y. Sassa, and S. Asai, “Control of precipitating phase alignment and crystal orientation by imposition of a high magnetic field,” Mater. Trans., JIM |

14. | M. Yamaguchi, S. Ozawa, I. Yamamoto, and T. Kimura, “Characterization of three-dimensional magnetic alignment for magnetically biaxial particles,” Jpn. J. Appl. Phys. |

15. | J. Akiyama, Y. Sato, and T. Taira, “Laser ceramics with rare-earth-doped anisotropic materials,” Opt. Lett. |

16. | J. Akiyama, Y. Sato, and T. Taira, “Laser demonstration of diode-pumped Nd |

17. | F. K. Lotgering, “Topotactical reactions with ferrimagnetic oxides having hexagonal crystal structures-II,” J. Inorg. Nucl. Chem. |

18. | S. Mizuta and K. Koumoto, |

19. | Y. Sato, J. Akiyama, and T. Taira, “Fundamental investigations in orientation control process for anisotropic laser ceramics,” Phys. Status Solidi C (2013), doi:. [CrossRef] |

20. | N. Leroy and E. Bres, “Structure and substitutions in fluorapatite,” Eur. Cell. Mater. |

**OCIS Codes**

(160.1190) Materials : Anisotropic optical materials

(160.3380) Materials : Laser materials

(160.5690) Materials : Rare-earth-doped materials

**ToC Category:**

Laser Materials

**History**

Original Manuscript: March 15, 2013

Revised Manuscript: May 9, 2013

Manuscript Accepted: May 13, 2013

Published: May 17, 2013

**Virtual Issues**

Optical Ceramics (2013) *Optical Materials Express*

**Citation**

Yoichi Sato, Jun Akiyama, and Takunori Taira, "Orientation control of micro-domains in anisotropic laser ceramics," Opt. Mater. Express **3**, 829-841 (2013)

http://www.opticsinfobase.org/ome/abstract.cfm?URI=ome-3-6-829

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### References

- I. Shoji, S. Kurimura, Y. Sato, T. Taira, A. Ikesue, and K. Yoshida, “Optical properties and laser characteristics of highly Nd3+-doped Y3Al5O12 ceramics,” Appl. Phys. Lett.77(7), 939–941 (2000). [CrossRef]
- A. A. Kaminskii, M. S. Akchurin, R. V. Gainutdinov, K. Takaichi, A. Shirakava, H. Yagi, T. Yanagitani, and K. Ueda, “Microhardness and fracture toughness of Y2O3- and Y3Al5O12-based nanocrystalline laser ceramics,” Crystallogr. Rep.50(5), 869–873 (2005). [CrossRef]
- S. J. McNaught, H. Komine, S. B. Weiss, R. Simpson, A. M. F. Johnson, J. Machan, C. P. Asman, M. Weber, G. C. Jones, M. M. Valley, A. Jankevics, D. Burchman, M. McClellan, J. Sollee, J. Marmo, and H. Injeyan, “100 kW coherently combined slab MOPAs,” in Conference on Lasers and Electro-Optics/International Quantum Electronics Conference, OSA Technical Digest (CD) (Optical Society of America, 2009), paper CThA1. [CrossRef]
- Y. Sato, J. Saikawa, T. Taira, and A. Ikesue, “Characteristics of Nd3+-doped Y3ScAl4O12 ceramic laser,” Opt. Mater.29(10), 1277–1282 (2007). [CrossRef]
- Y. Sato, A. Ikesue, and T. Taira, “Tailored spectral designing of layer-by-layer type composite Nd:Y3ScAl4O12/Nd:Y3Al5O12 ceramics,” IEEE J. Sel. Top. Quantum Electron.13(3), 838–843 (2007). [CrossRef]
- Y. Sato, T. Taira, N. Pavel, and V. Lupei, “Laser operation with near quantum-defect slope efficiency in Nd:YVO4 under direct pumping into emitting level,” Appl. Phys. Lett.82(6), 844–846 (2003). [CrossRef]
- S. A. Payne, L. K. Smith, L. D. Deloach, W. L. Kway, J. B. Tassano, and W. F. Krupke, “Laser, optical, and thermomechanical properties of Yb-doped fluoroapatite,” IEEE J. Quantum Electron.30(1), 170–179 (1994). [CrossRef]
- T. Taira, “Domain-controlled laser ceramics toward Giant Micro-photonics [Invited],” Opt. Mater. Express1(5), 1040–1050 (2011). [CrossRef]
- A. Ikesue, T. Kinoshita, K. Kamata, and K. Yoshida, “Fabrication and optical properties of high performance polycrystalline Nd:YAG ceramics for solid-state lasers,” J. Am. Ceram. Soc.78(4), 1033–1040 (1995). [CrossRef]
- W. D. Kingerly, H. K. Brown, and D. R. Uhlmann, Introduction to Ceramics, 2nd ed. (John Wiley & Sons, 1975), Chap. 5.
- M. Harada, K. Muramatsu, Y. Iwasaki, S. Kurimura, and T. Taira, “Periodic twinning in crystal quartz for optical quasi-phase matched secondary harmonic conversion,” J. Mater. Res.19(04), 969–972 (2004). [CrossRef]
- H. Ishizuki and T. Taira, “Half-joule output optical-parametric oscillation by using 10-mm-thick periodically poled Mg-doped congruent LiNbO3.,” Opt. Express20(18), 20002–20010 (2012). [CrossRef] [PubMed]
- H. Morikawa, Y. Sassa, and S. Asai, “Control of precipitating phase alignment and crystal orientation by imposition of a high magnetic field,” Mater. Trans., JIM39(8), 814–818 (1998).
- M. Yamaguchi, S. Ozawa, I. Yamamoto, and T. Kimura, “Characterization of three-dimensional magnetic alignment for magnetically biaxial particles,” Jpn. J. Appl. Phys.52, 013003 (2013). [CrossRef]
- J. Akiyama, Y. Sato, and T. Taira, “Laser ceramics with rare-earth-doped anisotropic materials,” Opt. Lett.35(21), 3598–3600 (2010). [CrossRef] [PubMed]
- J. Akiyama, Y. Sato, and T. Taira, “Laser demonstration of diode-pumped Nd3+-doped fluorapatite anisotropic ceramics,” Appl. Phys. Express4(2), 022703 (2011). [CrossRef]
- F. K. Lotgering, “Topotactical reactions with ferrimagnetic oxides having hexagonal crystal structures-II,” J. Inorg. Nucl. Chem.9(2), 113–123 (1959). [CrossRef]
- S. Mizuta and K. Koumoto, Materials Science for Ceramics (University of Tokyo Press, Tokyo, 1996), p. 237.
- Y. Sato, J. Akiyama, and T. Taira, “Fundamental investigations in orientation control process for anisotropic laser ceramics,” Phys. Status Solidi C (2013), doi:. [CrossRef]
- N. Leroy and E. Bres, “Structure and substitutions in fluorapatite,” Eur. Cell. Mater.2, 36–48 (2001). [PubMed]

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