Before proceeding to the discussion of the results of the experiment on the determination of the growth rate of lithium disilicate crystals in glasses no. 1–no. 3, let us consider the main notions used when determining the crystal growth. The crystal growth occurs as a result of accidental attachments and detachments of structural units from the surface of a supercritical nucleus (the supercritical nucleus is a nucleus with the size exceeding a critical size). The growth of the super-critical nucleus is thermodynamically favorable because this is accompanied by a decrease in the free energy of the system ΔΦ. The linear growth rate of the nucleus is determined by the equation

$$\left(\mathsf{U}\mathsf{I} = \frac{dr}{dt} = \mathsf{I}\left(\beta\_{+} - \beta\_{-}\right)\right),\tag{7}$$

Nucleation and Crystal Growth in Phase Separated Glasses in the Lithium Silicate System 37

*k Т* ,

*T T*

1 / *M*

The quantity *kTmelt*/*HA* in formula (12) usually does not exceed 1/10. Therefore, the temperature of the maximum growth rate is close to the melting temperature, and it is very difficult to experimentally determine this temperature. This inference is very important for

There is a large number of works on the rate of crystal nucleation (see Table 1) in lithium silicate glasses, whereas investigations of the concentration dependences of the growth rate are absent. There exist works in which the results are presented only for specific compositions of glasses in this system [108, 99]. The photographs of characteristic lithium disilicate crystals grown at a development temperature of 600 °C in glasses no. 1–no. 3 are displayed in Fig. 5. In order to determine the growth rate *U*, the dependences *Rmax*(*t*) were constructed for a series of samples held for different times *t* at a specified temperature. It was revealed that these dependences exhibit a linear behavior, which allows us to determine the crystal growth rate from their slope. In the case of spherical particles, their growth rate *U*  is determined from the expression *U* = *dRmax*/*dt*', where *t*' is the time of holding of the glass at the development temperature. In the case of ellipsoids of revolution, the growth rate *U* is determined from the change in their average radius defined as the half sum of the major and minor radii of the ellipsoid. The crystal growth rates were studied at temperatures of 570, 590, 600, 640, 685, 718, 720, and 730 °C. The growth rates *U*(*T*) of lithium disilicate crystals in glasses no. 1–no. 4 as a function of the temperature are given in Table 4. Figure 7 shows the temperature dependences of the growth rate *U* for glasses no. 1–no. 3. The data presented in Fig. 7c allow us to compare the growth rates *U* determined in our work with those obtained in [109]. As can be seen from Fig. 7c, Burgner et al. [109] performed investigations at higher temperatures. We more thoroughly studied the growth rate at low temperatures. In the temperature range 575–600 °C, the results of both studies are in good agreement. It can be seen from Fig. 7 that the growth rate *U*(*T*) increases with an increase in the temperature. In the temperature range 640–718 °C, the growth rate increases linearly. It is difficult to judge the accurate position of the maximum of the growth rate *U*(*T*), because the error in the measurement of the size at temperatures above 720 °C can increase in an uncontrollable manner due to the fact that the time of heat treatment of the sample (1–2 min) is comparable to the time of its heating to a specified development temperature and we cannot state with assurance that crystals grew accurately at this temperature. Therefore, although the temperature dependence of the growth rate should have a maximum, we cannot argue that its true position is recorded. It is quite probable that the maximum of the growth rate is located at higher temperatures. The maxima of the nucleation rates are located at lower temperatures than the maxima of the growth rates. Theoretically, they can either be located at a large

*kT H*

*melt*

*melt A*

(12)

*A melt*

*U*

**6. Experimental results on the crystal growth and their discussion**

**6.1 Crystal growth rate: Dependence on the temperature**

we find that

experimenters.

*H T Т*

1 0 *Um*

*melt Т Т*

where *l* is the thickness of the monomolecular layer growing on the crystal for some time interval, equal to the linear size of the structural unit), and β+ and β– are the probabilities of attachment and detachment of one structural unit from the crystal surface per unit time:

$$\mathcal{J}\_{+} = \frac{1}{\tau\_{0}} \exp\left(-\frac{\Phi\_{A}}{kT}\right), \ \mathcal{J}\_{-} = \mathcal{J}\_{+} \exp\left[\frac{\left(\frac{\partial \Delta \Phi}{\partial a}\right)}{kT}\right] \tag{8}$$

Here, τ0 is the time of the order of the period of vibrations of atoms in a chemical bond, whose switching provides the transition of the structural unit to the ordered state of the crystal structure of the nucleus; ΔΦ is the change in the free energy of the system; and *a* is the linear size of the structural unit. Substitution of these expressions into formula (7) gives

$$\mathcal{U} = \frac{1}{\tau\_0} \exp\left(-\frac{\Phi\_A}{kT}\right) \left| 1 - \exp\left[-\frac{\Delta\phi}{kT} \left(1 - \frac{r}{r}\right) \right] \right| \tag{9}$$

where is the difference between the free energies of the glass and nucleus per unit volume of the nucleus and *r*\* is the radius of the critical nucleus. It follows from this relationship that the growth rate of the critical nucleus is equal to zero. Since the critical nucleus is in an unstable equilibrium state, both the increase and decrease in its size is accompanied by a decrease in the difference and is equally probable. With an increase in the size of the supercritical nucleus, the growth rate increases and becomes constant when the nucleus radius is larger than the critical radius ( \* *r r* ).

Now, we consider the temperature dependence of the stationary growth rate. Under the assumption that ≈ *q*(1 – *T*/*Tmelt*), where *q* is the specific heat capacity, we have

$$\mathcal{U}(T) \approx \exp\left(-\frac{\Phi\_A}{kT}\right) \left| 1 - \exp\left[-\frac{q\left(1 - T \left/ \, \boldsymbol{T}\_{\rm melt} \right)}{kT}\right] \right| \right.\tag{10}$$

According to expression (10), the growth rate is equal to zero at *T* → 0 and *T* → *Tmelt*. At some temperature (0 << *Tmelt*), the growth rate is maximum. This temperature can be evaluated. We assume that exp[–*q*(1 – *T*/*Tmelt*)/*kT*] ≈ 1 – *q*(1 – *T*/*Tmelt*)/*kT* and substitute into expression (10). After the differentiation with respect to the temperature, we obtain

$$\frac{d\mathcal{U}I}{dT} \approx \frac{1}{T^2} \exp\left(-\frac{\Phi\_A}{kT}\right) \left[\frac{H\_A}{k}\left(\frac{T\_{m\text{left}}}{T} - \mathbf{1}\right) - T\_{m\text{left}}\right] \tag{11}$$

From the extremum condition

$$\left[\frac{H\_A}{k}\left(\frac{T\_{melt}}{T} - 1\right) - T\_{melt}\right] \Big|\_{T = T\_{LI\_m}} = 0 \quad .$$

we find that

36 Crystallization and Materials Science of Modern Artificial and Natural Crystals

is accompanied by a decrease in the free energy of the system ΔΦ. The linear growth rate of

*dt*

where *l* is the thickness of the monomolecular layer growing on the crystal for some time interval, equal to the linear size of the structural unit), and β+ and β– are the probabilities of attachment and detachment of one structural unit from the crystal surface per unit time:

, exp

Here, τ0 is the time of the order of the period of vibrations of atoms in a chemical bond, whose switching provides the transition of the structural unit to the ordered state of the crystal structure of the nucleus; ΔΦ is the change in the free energy of the system; and *a* is the linear size of the structural unit. Substitution of these expressions into formula (7)

1 exp 1 exp 1 *A r U*

volume of the nucleus and *r*\* is the radius of the critical nucleus. It follows from this relationship that the growth rate of the critical nucleus is equal to zero. Since the critical nucleus is in an unstable equilibrium state, both the increase and decrease in its size is

in the size of the supercritical nucleus, the growth rate increases and becomes constant when

Now, we consider the temperature dependence of the stationary growth rate. Under the

1 / ( ) exp 1 exp *A melt q TT U T*

According to expression (10), the growth rate is equal to zero at *T* → 0 and *T* → *Tmelt*. At some temperature (0 << *Tmelt*), the growth rate is maximum. This temperature can be evaluated. We assume that exp[–*q*(1 – *T*/*Tmelt*)/*kT*] ≈ 1 – *q*(1 – *T*/*Tmelt*)/*kT* and substitute into

exp 1 *A A melt*

*dU H T Т*

expression (10). After the differentiation with respect to the temperature, we obtain

*dT T kT k Т*

≈ *q*(1 – *T*/*Tmelt*), where *q* is the specific heat capacity, we have

*kT kT*

*kT kT r*

is the difference between the free energies of the glass and nucleus per unit

  , (7)

(8)

, (9)

(10)

*a kT*

\*

*melt*

(11)

and is equally probable. With an increase

the nucleus is determined by the equation

( *dr U l*

0 1

0

the nucleus radius is larger than the critical radius ( \* *r r* ).

2 1

accompanied by a decrease in the difference

From the extremum condition

gives

where

assumption that

exp *A kT*

$$T\_{L\_M} = \frac{T\_{melt}}{1 + kT\_{melt} \, / \, H\_A} \tag{12}$$

The quantity *kTmelt*/*HA* in formula (12) usually does not exceed 1/10. Therefore, the temperature of the maximum growth rate is close to the melting temperature, and it is very difficult to experimentally determine this temperature. This inference is very important for experimenters.

#### **6. Experimental results on the crystal growth and their discussion**

#### **6.1 Crystal growth rate: Dependence on the temperature**

There is a large number of works on the rate of crystal nucleation (see Table 1) in lithium silicate glasses, whereas investigations of the concentration dependences of the growth rate are absent. There exist works in which the results are presented only for specific compositions of glasses in this system [108, 99]. The photographs of characteristic lithium disilicate crystals grown at a development temperature of 600 °C in glasses no. 1–no. 3 are displayed in Fig. 5. In order to determine the growth rate *U*, the dependences *Rmax*(*t*) were constructed for a series of samples held for different times *t* at a specified temperature. It was revealed that these dependences exhibit a linear behavior, which allows us to determine the crystal growth rate from their slope. In the case of spherical particles, their growth rate *U*  is determined from the expression *U* = *dRmax*/*dt*', where *t*' is the time of holding of the glass at the development temperature. In the case of ellipsoids of revolution, the growth rate *U* is determined from the change in their average radius defined as the half sum of the major and minor radii of the ellipsoid. The crystal growth rates were studied at temperatures of 570, 590, 600, 640, 685, 718, 720, and 730 °C. The growth rates *U*(*T*) of lithium disilicate crystals in glasses no. 1–no. 4 as a function of the temperature are given in Table 4. Figure 7 shows the temperature dependences of the growth rate *U* for glasses no. 1–no. 3. The data presented in Fig. 7c allow us to compare the growth rates *U* determined in our work with those obtained in [109]. As can be seen from Fig. 7c, Burgner et al. [109] performed investigations at higher temperatures. We more thoroughly studied the growth rate at low temperatures. In the temperature range 575–600 °C, the results of both studies are in good agreement. It can be seen from Fig. 7 that the growth rate *U*(*T*) increases with an increase in the temperature. In the temperature range 640–718 °C, the growth rate increases linearly. It is difficult to judge the accurate position of the maximum of the growth rate *U*(*T*), because the error in the measurement of the size at temperatures above 720 °C can increase in an uncontrollable manner due to the fact that the time of heat treatment of the sample (1–2 min) is comparable to the time of its heating to a specified development temperature and we cannot state with assurance that crystals grew accurately at this temperature. Therefore, although the temperature dependence of the growth rate should have a maximum, we cannot argue that its true position is recorded. It is quite probable that the maximum of the growth rate is located at higher temperatures. The maxima of the nucleation rates are located at lower temperatures than the maxima of the growth rates. Theoretically, they can either be located at a large

Nucleation and Crystal Growth in Phase Separated Glasses in the Lithium Silicate System 39

retardation of the growth process is caused simultaneously by nucleation, growth, and

1 0.045 0.125 0.25 0.68 1.25 - 1.72 1.92 2.0 2 0.036 0.125 0.31 0.44 1.12 - 1.66 1.58 - 3 0.042 0.130 0.40 0.42 1.14 1.40 1.70 1.85 - 4 0.225 0.625 1.25 3.4 6.25 8.6 - 10.0 - Table 4. Growth rates *U*(*T*) (μm/min) of lithium disilicate crystals in glasses no. 1–no. 4 as a

Growth temperature, °C 570 590 600 640 685 700 718 720 730 Growth rate *U*(*T*), μm/min

t, h 0 20 40 60 80 100

t, h

0 20 40 60 80 100

t, h

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5

0,0

0,2

0,4

U,

*c d* Fig. 8. Dependences of the growth rate *U* on the time of heat treatment at 450 °C for glasses (*a*) no. 1, (*b*) no. 2, (*c*) no. 3 and (*d*) no. 4. Samples were developed at the temperature of

m/min

0,6

0,8

1,0

**U, m / min**

*a b*

dissolution of particles.

function of the temperature

0 20 40 60 80 100

0,0 0,5 1,0 1,5 2,0 2,5 3,0

t, h

Glass no.

> 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0

> > 0,0 0,5 1,0 1,5 2,0 2,5 3,0

600 °C for 10 min.

**U, m / min**

**U, m / min**

distance from each other or overlap. Since the accurate position of the maximum of the growth rate is not known, we can only to make the inference that, for the compositions under investigation, the positions of the temperature maxima of the nucleation and growth rates are spaced along the temperature scale by an uncertain value no less than 260 °C (720–460 °C).

Fig. 7. Temperature dependences of the growth rate *U* for glasses (*a*) no. 1, (*b*) no. 2, (*с*) no. 3, and (*d*) no. 4 according to the data (*1*) obtained in this work and (*2*) taken from [109].

#### **6.2 Crystal growth rate: Dependence on the time of holding of the glasses at the temperature of 450 °C**

The growth rates of lithium disilicate crystals as a function of the holding time glasses at a constant temperature were investigated at the temperature of 450 °C. The results of the measurements are presented in Fig. 8. It can be seen from this figure that the dependence of the growth rate of lithium disilicate crystals in glasses no. 1, no. 2 and no. 3 on the time of low-temperature heat treatment exhibit an oscillatory behavior with a gradual decay of the oscillation amplitude and a retardation of the growth process. In the framework of the existing theories of nucleation and growth, it was demonstrated for metal alloys [110] that the

distance from each other or overlap. Since the accurate position of the maximum of the growth rate is not known, we can only to make the inference that, for the compositions under investigation, the positions of the temperature maxima of the nucleation and growth rates are spaced along the temperature scale by an uncertain value no less than 260 °C (720–460 °C).

0,0

0

2

4

U,

*c d* Fig. 7. Temperature dependences of the growth rate *U* for glasses (*a*) no. 1, (*b*) no. 2, (*с*) no. 3,

The growth rates of lithium disilicate crystals as a function of the holding time glasses at a constant temperature were investigated at the temperature of 450 °C. The results of the measurements are presented in Fig. 8. It can be seen from this figure that the dependence of the growth rate of lithium disilicate crystals in glasses no. 1, no. 2 and no. 3 on the time of low-temperature heat treatment exhibit an oscillatory behavior with a gradual decay of the oscillation amplitude and a retardation of the growth process. In the framework of the existing theories of nucleation and growth, it was demonstrated for metal alloys [110] that the

and (*d*) no. 4 according to the data (*1*) obtained in this work and (*2*) taken from [109].

**6.2 Crystal growth rate: Dependence on the time of holding of the glasses at the**

m/min

6

8

10

0,4

0,8

U,

*a b*

m/min

1,2

1,6

2,0

560 580 600 620 640 660 680 700 720

450 500 550 600 650 700

**Т, °С**

T, °C

2

1

560 580 600 620 640 660 680 700 720 740

**Т, °С**

560 580 600 620 640 660 680 700 720 740

**T, °C**

0,0

0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 1,8 2,0

**temperature of 450 °C**

**U, m/min**

0,5

1,0

**U,**

**m/min**

1,5

2,0

retardation of the growth process is caused simultaneously by nucleation, growth, and dissolution of particles.

Table 4. Growth rates *U*(*T*) (μm/min) of lithium disilicate crystals in glasses no. 1–no. 4 as a function of the temperature

Fig. 8. Dependences of the growth rate *U* on the time of heat treatment at 450 °C for glasses (*a*) no. 1, (*b*) no. 2, (*c*) no. 3 and (*d*) no. 4. Samples were developed at the temperature of 600 °C for 10 min.

Nucleation and Crystal Growth in Phase Separated Glasses in the Lithium Silicate System 41

Thus, we have investigated the kinetics of phase separation in three lithium silicate glasses containing 23.4, 26.0 and 29.1 mol % Li2O with the compositions lying in the phase separation region in the Li2O–SiO2 system. It has been found that the temperature dependences of the ratio of the number of traces to the maximum number of traces *NS*/*NS*max and the ratio of the radius of the particle trace to the maximum radius *R*/*Rmax* exhibit maxima. The crystal nucleation rate has been studied in glasses with the compositions displaced from the stoichiometric lithium disilicate composition toward an

i. The absolute values of the crystal nucleation rate in glasses with the lower content Li2O than that in the glass with the stoichiometric lithium disilicate composition vary

ii. The position of the temperature maximum of the nucleation rate for the glass with the lower lithium oxide content is shifted on the temperature scale by 12 °C. This relatively small change in the quantities *Ist*(*Tmax*) and *T*max is explained by the fact that the glasses with the displaced composition correspond to the phase separation region in the Li2O– SiO2 system, so that the lithium disilicate in glasses with displaced composition nucleates in phase separated inhomogeneities with compositions that are displaced toward the lithium disilicate composition and depend weakly on the temperature. Variations in the crystal growth rate have been investigated as a function of the temperature and time of heat treatment. It has been shown that the temperature dependences of the crystal growth rate tend to an extremum. It has been found that the crystal growth rate depends substantially on the heat treatment time for phase separated glasses and that the growth rate is constant for the glass with the stoichiometric lithium disilicate composition. The dependence of the crystal growth rate on the heat treatment time in phase separated glasses exhibits a damped oscillatory behavior, which is explained by the depletion of the building material for crystals and the necessity of some time for its supply. Therefore, the performed investigations allow us to draw the following conclusion. Unlike the universally accepted opinion, the phase separation not only does not facilitate the conditions for the formation and growth of crystals but, in some cases (see, for example, the dependences of the crystal growth rate on the time of low-temperature heat treatment), even retards the crystal growth process. The data obtained on the nucleation and growth of crystals help to find optimum compositions and temperature–time conditions for the preparation of many glass-ceramic nanomaterials in lithium silicate system, including photostructured materials.

[1] *Hinz W., Kunth P.* Vitrokeramprodukte mit niedrigem Warmeausdehnungskoeffizienten //

[2] *Vogel W., Gerth K.* Catalyzed Crystallization in Glass, *Proceedings of the Symposium on Nucleation and Crystallization in Glasses and Melts, Columbus, Ohio, United States, 1962*,

[3] *Filipovich V.N.* Initial Stages of Crystallization of Glasses and Formation of Glass Ceramics, in *Stekloobraznoe sostoyanie*, Vyp. 1: *Katalizirovannaya kristallizatsiya stekla*  (Vitreous State, vol. 1: Catalyzed Crystallization in Glass), Moscow: Nauka, 1963, pp.

Silikattechnik. 1960. V. 11. N 12. P. 506–511.

Reser, M.K., Ed., Columbus, 1962, pp. 11–22.

increase in the SiO2 content. The following features have been revealed.

**7. Conclusions**

insignificantly.

**8. References**

9–24 (In Russian).

For glass no. 4 (with the stoichiometric lithium disilicate composition), the crystal growth rate does not depend on the time of low-temperature heat treatment (Fig. 8*c*).

The gradual decrease in the crystal growth rate with an increase in the time of holding of the glass at a temperature of 450 °C for glasses no. 1, no. 2 and no. 3 (Figs. 8*a*, 8*b*, 8*c*) can be explained by the fact that the crystals grow in the glass with the composition lying in the metastable phase separation region. A part of the material is spent for forming the crystal layer, and the growing crystal appears to be surrounded by the region depleted in the building material (the so-called diffusion zone) (Fig. 9).

*b*

Fig. 9. Electron microscopic images of glasses (*a*) no. 1 and (*b*) no. 2 preliminarily heat treated at the temperature of 440 °C for 2 h and developed at 600 °C for 10 min.

The formation of the next crystal layer requires the time it takes for phase separated droplets to come to the crystal growth region. In this case, the phase separated droplets consist predominantly of SiO2. When the droplets come closer to the growing lithium disilicate crystal, they will block its growth, which manifests itself in the dependences (Figs. 8*a*, 8*b*, 8*c*) as a decrease in the crystal growth rate.

#### **7. Conclusions**

40 Crystallization and Materials Science of Modern Artificial and Natural Crystals

For glass no. 4 (with the stoichiometric lithium disilicate composition), the crystal growth

The gradual decrease in the crystal growth rate with an increase in the time of holding of the glass at a temperature of 450 °C for glasses no. 1, no. 2 and no. 3 (Figs. 8*a*, 8*b*, 8*c*) can be explained by the fact that the crystals grow in the glass with the composition lying in the metastable phase separation region. A part of the material is spent for forming the crystal layer, and the growing crystal appears to be surrounded by the region depleted in the

*a*

*b*

The formation of the next crystal layer requires the time it takes for phase separated droplets to come to the crystal growth region. In this case, the phase separated droplets consist predominantly of SiO2. When the droplets come closer to the growing lithium disilicate crystal, they will block its growth, which manifests itself in the dependences (Figs. 8*a*, 8*b*, 8*c*)

Fig. 9. Electron microscopic images of glasses (*a*) no. 1 and (*b*) no. 2 preliminarily heat treated at the temperature of 440 °C for 2 h and developed at 600 °C for 10 min.

as a decrease in the crystal growth rate.

rate does not depend on the time of low-temperature heat treatment (Fig. 8*c*).

building material (the so-called diffusion zone) (Fig. 9).

Thus, we have investigated the kinetics of phase separation in three lithium silicate glasses containing 23.4, 26.0 and 29.1 mol % Li2O with the compositions lying in the phase separation region in the Li2O–SiO2 system. It has been found that the temperature dependences of the ratio of the number of traces to the maximum number of traces *NS*/*NS*max and the ratio of the radius of the particle trace to the maximum radius *R*/*Rmax* exhibit maxima. The crystal nucleation rate has been studied in glasses with the compositions displaced from the stoichiometric lithium disilicate composition toward an increase in the SiO2 content. The following features have been revealed.

Variations in the crystal growth rate have been investigated as a function of the temperature and time of heat treatment. It has been shown that the temperature dependences of the crystal growth rate tend to an extremum. It has been found that the crystal growth rate depends substantially on the heat treatment time for phase separated glasses and that the growth rate is constant for the glass with the stoichiometric lithium disilicate composition. The dependence of the crystal growth rate on the heat treatment time in phase separated glasses exhibits a damped oscillatory behavior, which is explained by the depletion of the building material for crystals and the necessity of some time for its supply. Therefore, the performed investigations allow us to draw the following conclusion. Unlike the universally accepted opinion, the phase separation not only does not facilitate the conditions for the formation and growth of crystals but, in some cases (see, for example, the dependences of the crystal growth rate on the time of low-temperature heat treatment), even retards the crystal growth process. The data obtained on the nucleation and growth of crystals help to find optimum compositions and temperature–time conditions for the preparation of many glass-ceramic nanomaterials in lithium silicate system, including photostructured materials.

#### **8. References**

Nucleation and Crystal Growth in Phase Separated Glasses in the Lithium Silicate System 43

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42 Crystallization and Materials Science of Modern Artificial and Natural Crystals

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**3**

*France*

**Elaboration of a Specific Class of**

Guy Dhalenne and Romuald Saint Martin

*Université de Paris Sud 11, Orsay,*

*Institut de Chimie Moléculaire et des Matériaux d'Orsay,*

**Metamaterial: Glass in Single Crystal**

Bertrand Poumellec, Matthieu Lancry, Santhi Ani-Joseph,

Metamaterials are composite materials, which can exhibit some interesting optical properties, which is usually not found in natural materials. Some examples of the properties that can be obtained and controlled in these materials are negative refraction and artificial magnetism [1, 2]. The wave vector k of a wave propagating through such a left-handed substance is anti-parallel to its Poynting-vector S. This remarkable property has far-reaching consequences. Usually these materials have some periodic or quasi-periodic structure. Such photonic metacrystals are composed of periodic assembly of dielectric or metallo-dielectric structures that are designed to affect the propagation of electromagnetic waves. In particular, they should have refraction indices significantly different. The assembly can be unidimensional as in periodic multilayered materials, or two-dimensional with periodically

When the characteristic length scale of constituting elements is much smaller than the radiation wavelength, they are called photonic crystals (PC). This is because these structures avoid diffraction. The electromagnetic wave passing through the PC will feel only the effective parameters of the material such as the effective magnetic permeability, effective permittivity, etc. They therefore can in good approximation be considered as continuous media with some effective electrodynamic properties. For these reasons, metamaterials do not necessarily possess a photonic band gap (PBG), which is a striking difference compared to PCs. Nevertheless it is interesting to look for the possibilities of opening up a PBG, for extensive applications and also for better insight into the optical

There are various methods to fabricate ordered composites like metamaterials. They are collected, for instance, in the book Nanophotonics by Prasad [5] or in Chemistry of Nanomaterials by Rao et al.[1]. They are classified into two categories, those, which are carried out by engineering, and those, which are obtained by self-organization. In the former category, there are various techniques including nanolithography, the deposit of the multilayer films (CVD, Langmuir-Blodget) etc. The most commonly practiced method for preparing these materials are largely based on lithography, which uses a pre-fabricated photo mask as a master from which the final pattern is derived. Self-organization is an

spaced rods or fibers aligned in arrays, or 3D like silica spheres in Opal.

**1. Introduction**

properties of the material [3, 4].

## **Elaboration of a Specific Class of Metamaterial: Glass in Single Crystal**

Bertrand Poumellec, Matthieu Lancry, Santhi Ani-Joseph, Guy Dhalenne and Romuald Saint Martin *Institut de Chimie Moléculaire et des Matériaux d'Orsay, Université de Paris Sud 11, Orsay, France*

#### **1. Introduction**

48 Crystallization and Materials Science of Modern Artificial and Natural Crystals

[101] Sycheva G.A, Sigaev V.N. Crystallization kinetics in Ca, Sr and Ba borophosphate systems Proceedings of Second Intern. Conf. on borate glasses, crystals and melts. Edited by A.C.Wright, S.H.Feller and A.C.Hannon. The Society of Glass Technology,

[102] Sycheva G.A. Formation of the Bubble Structure in the 26Li2O ⋅ 74SiO2 Glass // Fiz. Khim. Stekla. 2009. V. 35. N 3. P. 342–351 [Glass Phys. Chem. (Engl. transl.), 2009, vol.

[103] Sycheva G.A. Influence of the Presence of Bubbles on the Parameters of Crystal Nucleation in the 26Li2O ⋅ 74SiO2 Glass // Fiz. Khim. Stekla. 2009. V. 35. N 6. P.

[105] Holler W.S., Blackburn D.H., Simmons J.H. Miscibility Gaps in Alkali-Silicate Binaries—Data and Thermodynamic Interpretation // Am. Ceram. Soc. Bull. 1974. V.

[106] Andreev N.S., Mazurin O.V., Porai\_Koshits E.A., Roskova G.P., Filipovich V.N. Yavleniya likvatsii v steklakh, Leningrad: Nauka, 1974. Translated under the title

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[108] Braetsch V., Frischta G.H. Crystallization and Phase Separation in the System Li2O–

[109] Burgner L.L., Weinberg M.C., Symmons J.H. Early Stage Crystallization Kinetics of Lithium Disilicate and Sodium Silicate Glasses, Final Report of US Department of Energy Grant DE\_FG03ER45500, Tucson, Arizona, United States: The University of

[110] Ardell A.S. The Effect of Volume Fraction on Particle Coarsening: The Theoretical

SiO2 // Journal of Non-Crystalline Solids. 1987. V. 95–96. P. 457–462.

Phase Separation in Glasses, Amsterdam: North-Holland, 1984.

Applications), Varshal, B.G., Ed., Leningrad: Nauka, 1991.

Consideration // Acta Metall. 1972. V. 20. N 1. P. 61–71.

798–811 [Glass Phys. Chem. (Engl. transl.), 2009, vol. 35, no. 6, pp. 602–612]. [104] Mel'nichenko T.D., Rizak V.M., Mel'nichenko T.N., Fedelesh V.I. Parameters of the Fluctuation Free Volume Theory for Glasses in the Ge–As–Se System // Fiz. Khim. Stekla. 2004. V. 30. N 5. P. 553–564 [Glass Phys. Chem. (Engl. transl.), 2004, vol. 30,

Sheffild, 1997. 554s. P. 254-260.

35, no. 3, pp. 267–273].

no. 5, pp. 406–414].

57. N 3. P. 120–126.

Arizona, p. 85721;

Metamaterials are composite materials, which can exhibit some interesting optical properties, which is usually not found in natural materials. Some examples of the properties that can be obtained and controlled in these materials are negative refraction and artificial magnetism [1, 2]. The wave vector k of a wave propagating through such a left-handed substance is anti-parallel to its Poynting-vector S. This remarkable property has far-reaching consequences. Usually these materials have some periodic or quasi-periodic structure. Such photonic metacrystals are composed of periodic assembly of dielectric or metallo-dielectric structures that are designed to affect the propagation of electromagnetic waves. In particular, they should have refraction indices significantly different. The assembly can be unidimensional as in periodic multilayered materials, or two-dimensional with periodically spaced rods or fibers aligned in arrays, or 3D like silica spheres in Opal.

When the characteristic length scale of constituting elements is much smaller than the radiation wavelength, they are called photonic crystals (PC). This is because these structures avoid diffraction. The electromagnetic wave passing through the PC will feel only the effective parameters of the material such as the effective magnetic permeability, effective permittivity, etc. They therefore can in good approximation be considered as continuous media with some effective electrodynamic properties. For these reasons, metamaterials do not necessarily possess a photonic band gap (PBG), which is a striking difference compared to PCs. Nevertheless it is interesting to look for the possibilities of opening up a PBG, for extensive applications and also for better insight into the optical properties of the material [3, 4].

There are various methods to fabricate ordered composites like metamaterials. They are collected, for instance, in the book Nanophotonics by Prasad [5] or in Chemistry of Nanomaterials by Rao et al.[1]. They are classified into two categories, those, which are carried out by engineering, and those, which are obtained by self-organization. In the former category, there are various techniques including nanolithography, the deposit of the multilayer films (CVD, Langmuir-Blodget) etc. The most commonly practiced method for preparing these materials are largely based on lithography, which uses a pre-fabricated photo mask as a master from which the final pattern is derived. Self-organization is an

Elaboration of a Specific Class of Metamaterial: Glass in Single Crystal 51

obtain a self-organized composite. Moreover, as we increase the atomic fraction of Si in the starting composition, the nature and quality of the crystal varies. In fact, we could not confirm if we really have a eutectic or not in our case. This is because, a detailed chemical analysis by the electron microprobe method [6], both on the crystalline matrix and on the fibers, confirmed the presence of Si at these two locations. If it were a eutectic, more chances would be that Si

Phase diagram from Figure 3(a) shows that the cooling of CuGeO3 from liquid phase is congruent. So, there is no phase separation in the pure compound. With silicon dioxide substitution a priori to GeO2, the pseudo ternary diagram Figure 3(b) leads to expect that this property is conserved until 50% but our observations contradict this result. On the other hand, the phase diagram in Fig. 3 shows the existence of a multiphase domain limited by the single phase line CuGe1-xSixO3. The idea is thus to exploit this domain for inducing the second phase precipitation more strongly. The compositions of the starting material were CuGexSiyO1+2(x+y) with x between 0.89 and 1.2 and for y between 0 and 0.3 and 1x+y1.4, 0y/x0.33 (see Figure 4). They break up from the liquid state into a monocrystalline matrix,

*Crystal growth technique:* The technique of crystalline growth known as 'floating zone method' (see Figure 1), that we used here, makes it possible to obtain monocrystals of a few cm long without stacking faults. By the way, the advantage of this technique is the possibility to grow high-purity single crystals since the melt has no contact with a crucible during the experiment. Homogeneous and dense feed rods were prepared using >99.99% pure powders of CuO, GeO2 and SiO2 as described in reference [7]. Required amounts of these powders were accurately weighted and mixed thoroughly. This mixture was then pressed into cylindrical rods of about 4mm in diameter and 8 cm in length under isostatic pressure of about 2500 bar. The feed rods, were then sintered in air at 1050°C for 24h. All composites (varying in the composition x and y) were grown from a pure CuGeO3 monocrystalline seed aligned along its 'a' axis that defined the growth axis (see Figure 5). Growth was carried out with the 'a' axis of CuGeO3 along the growth direction. The essential idea of this crystal growth technique is to establish a liquid zone between the single crystal seed and the polycrystalline feed rod. The vertical high-temperature gradients

would not replace Ge in the crystalline matrix, instead stand alone as silica fibers.

Fig. 2. CuGeO3 crystallographic stacking.

incorporating some precipitates.

unconventional way of manufacturing photonic crystals by purely chemical method and it would open up new fields of application avoiding the problems related to engraving. Especially, in self organization, no one has described a way of synthesis starting from high temperatures, although it has some advantages like low cost, large volume and weak OH content (absorbing in the near-IR range). This last advantage can be made profitable to obtain materials compatible with the existing optical fiber networks.

In the past, researchers [6-12] showed the possibility of obtaining such ordered structures starting from eutectic compositions. Eutectics correspond to particular points in the phase diagram between the liquid phase and a solid phase, the composition of the liquid is fixed but it decomposes into two solid phases of different compositions, on cooling. A microstructure appears, but very often it is disordered. Nevertheless for some sets of elaboration parameters, ordered microstructures settle. Those can be defined by the kinetics of the growing environment, speed of cooling and also by some intrinsic parameters like diffusion, chemical kinetics of phase formation etc.

In this paper, we report the use of this method for synthesizing a new category of composites: equidistant glass fibers in a single crystal matrix. This is carried out at high temperature, by a technique called 'floating zone'/'melting zone' associated with an image furnace [7, 10]. The apparatus is described in Figure 1. A floating zone is established between a single crystal (the seed) and a polycrystalline rod (the feed rod) by using the radiative heat provided by two halogen lamps in an image furnace. These ways of development are essentially adapted to the development of large volume of single crystal (see Fig. 1).

Fig. 1. Scheme of the floating zone apparatus using a double image furnace.

#### **2. Synthesis route**

The system that we investigated is CuO-SiO2-GeO2 around the composition CuGexSiyO1+2(x+y) with 1x+y1.4 and 0y/x0.4. CuGeO3 is an inorganic crystal that was studied in our laboratory for other properties [6, 10]. This crystal is orthorhombic with a space group Pbmm and the crystalline parameters: a = 4.79Å, b = 8.49 Å, c = 2.94 Å. It can dissolve Ni/Mn on its network of octahedral Cu and Si on its tetrahedral Ge network. Even though it can dissolve Si into its atomic tetrahedral chain, there is a limit (typ.: a fraction of a few %) in the atomic fraction of Si which can thus replace Ge, beyond which it will not produce a single phase on solidification. We discovered this while looking for methods to

unconventional way of manufacturing photonic crystals by purely chemical method and it would open up new fields of application avoiding the problems related to engraving. Especially, in self organization, no one has described a way of synthesis starting from high temperatures, although it has some advantages like low cost, large volume and weak OH content (absorbing in the near-IR range). This last advantage can be made profitable to

In the past, researchers [6-12] showed the possibility of obtaining such ordered structures starting from eutectic compositions. Eutectics correspond to particular points in the phase diagram between the liquid phase and a solid phase, the composition of the liquid is fixed but it decomposes into two solid phases of different compositions, on cooling. A microstructure appears, but very often it is disordered. Nevertheless for some sets of elaboration parameters, ordered microstructures settle. Those can be defined by the kinetics of the growing environment, speed of cooling and also by some intrinsic parameters like

In this paper, we report the use of this method for synthesizing a new category of composites: equidistant glass fibers in a single crystal matrix. This is carried out at high temperature, by a technique called 'floating zone'/'melting zone' associated with an image furnace [7, 10]. The apparatus is described in Figure 1. A floating zone is established between a single crystal (the seed) and a polycrystalline rod (the feed rod) by using the radiative heat provided by two halogen lamps in an image furnace. These ways of development are essentially adapted to the

obtain materials compatible with the existing optical fiber networks.

diffusion, chemical kinetics of phase formation etc.

development of large volume of single crystal (see Fig. 1).

**2. Synthesis route**

Fig. 1. Scheme of the floating zone apparatus using a double image furnace.

The system that we investigated is CuO-SiO2-GeO2 around the composition CuGexSiyO1+2(x+y) with 1x+y1.4 and 0y/x0.4. CuGeO3 is an inorganic crystal that was studied in our laboratory for other properties [6, 10]. This crystal is orthorhombic with a space group Pbmm and the crystalline parameters: a = 4.79Å, b = 8.49 Å, c = 2.94 Å. It can dissolve Ni/Mn on its network of octahedral Cu and Si on its tetrahedral Ge network. Even though it can dissolve Si into its atomic tetrahedral chain, there is a limit (typ.: a fraction of a few %) in the atomic fraction of Si which can thus replace Ge, beyond which it will not produce a single phase on solidification. We discovered this while looking for methods to obtain a self-organized composite. Moreover, as we increase the atomic fraction of Si in the starting composition, the nature and quality of the crystal varies. In fact, we could not confirm if we really have a eutectic or not in our case. This is because, a detailed chemical analysis by the electron microprobe method [6], both on the crystalline matrix and on the fibers, confirmed the presence of Si at these two locations. If it were a eutectic, more chances would be that Si would not replace Ge in the crystalline matrix, instead stand alone as silica fibers.

#### Fig. 2. CuGeO3 crystallographic stacking.

Phase diagram from Figure 3(a) shows that the cooling of CuGeO3 from liquid phase is congruent. So, there is no phase separation in the pure compound. With silicon dioxide substitution a priori to GeO2, the pseudo ternary diagram Figure 3(b) leads to expect that this property is conserved until 50% but our observations contradict this result. On the other hand, the phase diagram in Fig. 3 shows the existence of a multiphase domain limited by the single phase line CuGe1-xSixO3. The idea is thus to exploit this domain for inducing the second phase precipitation more strongly. The compositions of the starting material were CuGexSiyO1+2(x+y) with x between 0.89 and 1.2 and for y between 0 and 0.3 and 1x+y1.4, 0y/x0.33 (see Figure 4). They break up from the liquid state into a monocrystalline matrix, incorporating some precipitates.

*Crystal growth technique:* The technique of crystalline growth known as 'floating zone method' (see Figure 1), that we used here, makes it possible to obtain monocrystals of a few cm long without stacking faults. By the way, the advantage of this technique is the possibility to grow high-purity single crystals since the melt has no contact with a crucible during the experiment. Homogeneous and dense feed rods were prepared using >99.99% pure powders of CuO, GeO2 and SiO2 as described in reference [7]. Required amounts of these powders were accurately weighted and mixed thoroughly. This mixture was then pressed into cylindrical rods of about 4mm in diameter and 8 cm in length under isostatic pressure of about 2500 bar. The feed rods, were then sintered in air at 1050°C for 24h. All composites (varying in the composition x and y) were grown from a pure CuGeO3 monocrystalline seed aligned along its 'a' axis that defined the growth axis (see Figure 5). Growth was carried out with the 'a' axis of CuGeO3 along the growth direction. The essential idea of this crystal growth technique is to establish a liquid zone between the single crystal seed and the polycrystalline feed rod. The vertical high-temperature gradients

Elaboration of a Specific Class of Metamaterial: Glass in Single Crystal 53

Fig. 5. Enlarged photograph showing the typical size of a crystal having the composition CuGe0.915Si0.085O3. Growth rate 0.75-1.3 mm/h. 'a' axis is the growth axis. On the right hand,

The growth rate has an important influence on the final structure of the crystal that evolves from the system. By carrying out the experiments at different rates, we conclude that, at this particular composition it is not possible to obtain ordered fibers if the rate exceeds 4 mm/h. However, there is a trade-off between the periodicity and the size and spacing of these fibers, as we increase the growth rate. Even though the size and spacing becomes smaller with increasing rate, it becomes more disordered. Therefore, we fixed this value at 1.3 mm/h and it was possible to obtain single crystals of about 14-cm long. The optimum growth rate varies with the composition, stoichiometry, etc. In this case also, an increase in the Si content beyond a limit (10%) results in more chaotic distribution of the fibers even if it is advantageous in obtaining smaller fiber diameter. It also restricts us in the range of feasible growth rates. Therefore, it is the compromise between the Si content and the growth rate that acts as the limitation to the minimum fiber size and spacing obtainable by this method. By working at slightly over stoichiometric composition, we are able to reduce this size up to 0.6 m with spacing as small as 1.5 m. What we observed was that, beyond a limit, the increase in Si will inhibit the ordered formation of silica fibers in the matrix.

The crystals are deep blue in color (see Figure 6), translucent, and grow with an elliptical cross section, with the b and c axes of the structure being the minor (0.30 cm) and major (0.75 cm) axes of the ellipse, respectively. They can be easily cleaved along the (100) plane. Characterization of the samples was done by optical microscopy, SEM (secondary electrons and backscattered electrons), X-ray diffraction (XRD), EBSD (electrons backscattered

The obtained composites were cleaved and thin slices were observed by optical microscopy using polarized light. Precipitates are easily seen when they are larger than the micron. This indicates that the refraction index difference is large enough. Upon rotation of the sample about the 'a' axis, the brightness of the matrix varies whereas that of the rods does not change, indicating that the matrix is not isotropic but the rods are. The optical microscopy always showed two phases and not 3 for some composition as it is shown in Fig. 3b. It shows also the regularity of the precipitates, the heterogeneity of the distribution in some

diffraction), electron microprobe analysis, Raman micro-spectroscopy.

we can see the seed that defined the growth axis.

**3. Characterization**

regions.

**3.1 Optical microscopy and SEM**

required to stabilize the molten zone are ensured by focusing the image of two halogen lamps with two ellipsoidal mirrors. The molten zone temperature is precisely controlled by adjusting the lamp electric current magnitude. The crystal growth was carried out in an enclosed quartz tube where 2-atm oxygen pressure was established in order to prevent vaporization of CuO from the molten zone as much as possible. The growth rate (0.6 to 9 mm/h) is the rate at which the ensemble moves down from the furnace and this has a major influence on the characteristics of the grown crystal. The slow relative displacement of the heat source with respect to the liquid zone makes it possible to force the solidification of the fluid on the correctly directed monocrystal and thus the solidified material presents a continuous crystallographic character. This method yields single crystals of Cu(Ge1-xSix)O3 as well as two phase composite in the vicinity of the monocrystalline domain.

Fig. 3. Phase diagrams for CuGeO3.

Fig. 4. Location of the compositions we have explored in the phase diagram of Figure 3

Fig. 5. Enlarged photograph showing the typical size of a crystal having the composition CuGe0.915Si0.085O3. Growth rate 0.75-1.3 mm/h. 'a' axis is the growth axis. On the right hand, we can see the seed that defined the growth axis.

The growth rate has an important influence on the final structure of the crystal that evolves from the system. By carrying out the experiments at different rates, we conclude that, at this particular composition it is not possible to obtain ordered fibers if the rate exceeds 4 mm/h. However, there is a trade-off between the periodicity and the size and spacing of these fibers, as we increase the growth rate. Even though the size and spacing becomes smaller with increasing rate, it becomes more disordered. Therefore, we fixed this value at 1.3 mm/h and it was possible to obtain single crystals of about 14-cm long. The optimum growth rate varies with the composition, stoichiometry, etc. In this case also, an increase in the Si content beyond a limit (10%) results in more chaotic distribution of the fibers even if it is advantageous in obtaining smaller fiber diameter. It also restricts us in the range of feasible growth rates. Therefore, it is the compromise between the Si content and the growth rate that acts as the limitation to the minimum fiber size and spacing obtainable by this method. By working at slightly over stoichiometric composition, we are able to reduce this size up to 0.6 m with spacing as small as 1.5 m. What we observed was that, beyond a limit, the increase in Si will inhibit the ordered formation of silica fibers in the matrix.

#### **3. Characterization**

52 Crystallization and Materials Science of Modern Artificial and Natural Crystals

required to stabilize the molten zone are ensured by focusing the image of two halogen lamps with two ellipsoidal mirrors. The molten zone temperature is precisely controlled by adjusting the lamp electric current magnitude. The crystal growth was carried out in an enclosed quartz tube where 2-atm oxygen pressure was established in order to prevent vaporization of CuO from the molten zone as much as possible. The growth rate (0.6 to 9 mm/h) is the rate at which the ensemble moves down from the furnace and this has a major influence on the characteristics of the grown crystal. The slow relative displacement of the heat source with respect to the liquid zone makes it possible to force the solidification of the fluid on the correctly directed monocrystal and thus the solidified material presents a continuous crystallographic character. This method yields single crystals of Cu(Ge1-xSix)O3 as well as two

Fig. 4. Location of the compositions we have explored in the phase diagram of Figure 3

(b): Pseudo ternary diagram of CuO-SiO2-GeO2 at 600°C and 1kb. Lines are only indicative. [13]

phase composite in the vicinity of the monocrystalline domain.

(a): CuO-GeO2 binary phase diagram [12].

Fig. 3. Phase diagrams for CuGeO3.

The crystals are deep blue in color (see Figure 6), translucent, and grow with an elliptical cross section, with the b and c axes of the structure being the minor (0.30 cm) and major (0.75 cm) axes of the ellipse, respectively. They can be easily cleaved along the (100) plane. Characterization of the samples was done by optical microscopy, SEM (secondary electrons and backscattered electrons), X-ray diffraction (XRD), EBSD (electrons backscattered diffraction), electron microprobe analysis, Raman micro-spectroscopy.

#### **3.1 Optical microscopy and SEM**

The obtained composites were cleaved and thin slices were observed by optical microscopy using polarized light. Precipitates are easily seen when they are larger than the micron. This indicates that the refraction index difference is large enough. Upon rotation of the sample about the 'a' axis, the brightness of the matrix varies whereas that of the rods does not change, indicating that the matrix is not isotropic but the rods are. The optical microscopy always showed two phases and not 3 for some composition as it is shown in Fig. 3b. It shows also the regularity of the precipitates, the heterogeneity of the distribution in some regions.

Elaboration of a Specific Class of Metamaterial: Glass in Single Crystal 55

respectively for 10 minutes each. This was then observed using the optical microscope and the result is given in Figure 9. The photographs show that the fibers are quite uniform and grow along a great length of the crystal. It was observed that the length of the fiber as

photograph from the sample with initial composition CuGe0.915Si0.085O3 is around 4 m and

Towards the sample center

Sample surface

Fig. 7. Example of the structure of the samples: (a) typical image of the central region of the samples. Fibers are ordered on a short distance. (b) When moving out of the center, precipitate distance and size increase (c) when we are going closer to the surface the precipitates disappear. (d) In the intermediate region corresponding to c, the fibers appear

(c) Cu1.000Ge0.915Si0.085O3.000 (UPS416) (d) Intermediate region of

750 m

(b) Cu1.0Ge0.9Si0.1O3.0, 1mm/h (UPS417a2)

Cu1.000Ge0.915Si0.085O3.000 (UPS416) 0.75-1.3 mm/h, size 2.5 microns,

distance 11.5 microns

of the fibers in this

revealed from the photograph is almost 750 µm. The diameter

is more or less uniform throughout the length.

(a) Central region of Cu1.0Ge0.9Si0.1O3.0, 1mm/h (UPS417b) Precipitate distance is 4.1 microns, their size is 2.5 microns

tilted.

1.5 mm

Fig. 6. Global view of a sample cleaved perpendicularly to its growth axis, the « a » axis. The sample composition was CuGe0.915Si0.085O3 and it was grown at the rate 0.75-1.3 mm/h.

As we can see in Figure 6, the samples show concentric heterogeneities. There is a central region (brighter) containing precipitates. There is an external region clear from precipitates. There are regions with ordered precipitates of the same size e.g. the central region (appearing brighter in Fig. 6), and region rather disordered with various sizes and distances e.g. intermediate region. The outside part near the surface of the crystal never shows precipitates. Close to the borderline between intermediate region and external one, precipitates are aligned along equally spaced lines. The lines are more distant than in the central region.

Figure 7 shows precipitates in the different region of the sample. In the central region, there is only a short range order. When moving to the outside, the precipitates` distribution varies. The distance and size increases. When reaching the surface, they disappear. In this region fibers are no more perpendicular but they are all tilted at the same angle. This is probably due to the curvature of the solid-liquid interface characteristic of the floating zone technique. Then, further from the center, the arrangement is more chaotic as we can see in Figure 7b. Figure 8 show the regularity of the precipitates in the central zone that is about 1 mm in diameter, in four of the numerous crystals that we elaborated. The smallest size and distance that we have obtained in the phase domain investigated is 0.6 and 1.4 m respectively (see Figure 8d). Further observations by means of optical microscope in transmission and by successive cleaving perpendicular to 'a' axis showed that, in the central zone, these precipitates were long, columnar and perpendicular to the cleavage surface thus 'a' axis is along the crystal rods. They are therefore analogous to fibers.

One of the best ordered samples showing nearly periodically distributed fibers, was selected for the observation along a longitudinal section containing 'a' axis, to analyze whether these fibers run uniformly along the entire length of crystal without crossing over. A smallcleaved piece of sample was cut into two pieces. It was then fixed in epoxy resin and polished using SiC discs of grain sizes 600 and 1200 mesh for 10 minutes each and then using felt discs and diamond powder of granule sizes 6 µm, 3 µm, 1 µm and 0.25 µm

Fig. 6. Global view of a sample cleaved perpendicularly to its growth axis, the « a » axis. The sample composition was CuGe0.915Si0.085O3 and it was grown at the rate 0.75-1.3 mm/h.

As we can see in Figure 6, the samples show concentric heterogeneities. There is a central region (brighter) containing precipitates. There is an external region clear from precipitates. There are regions with ordered precipitates of the same size e.g. the central region (appearing brighter in Fig. 6), and region rather disordered with various sizes and distances e.g. intermediate region. The outside part near the surface of the crystal never shows precipitates. Close to the borderline between intermediate region and external one, precipitates are aligned along equally spaced lines. The lines are more distant than in the

Figure 7 shows precipitates in the different region of the sample. In the central region, there is only a short range order. When moving to the outside, the precipitates` distribution varies. The distance and size increases. When reaching the surface, they disappear. In this region fibers are no more perpendicular but they are all tilted at the same angle. This is probably due to the curvature of the solid-liquid interface characteristic of the floating zone technique. Then, further from the center, the arrangement is more chaotic as we can see in Figure 7b. Figure 8 show the regularity of the precipitates in the central zone that is about 1 mm in diameter, in four of the numerous crystals that we elaborated. The smallest size and distance that we have obtained in the phase domain investigated is 0.6 and 1.4 m respectively (see Figure 8d). Further observations by means of optical microscope in transmission and by successive cleaving perpendicular to 'a' axis showed that, in the central zone, these precipitates were long, columnar and perpendicular to the cleavage surface thus 'a' axis is along the crystal

One of the best ordered samples showing nearly periodically distributed fibers, was selected for the observation along a longitudinal section containing 'a' axis, to analyze whether these fibers run uniformly along the entire length of crystal without crossing over. A smallcleaved piece of sample was cut into two pieces. It was then fixed in epoxy resin and polished using SiC discs of grain sizes 600 and 1200 mesh for 10 minutes each and then using felt discs and diamond powder of granule sizes 6 µm, 3 µm, 1 µm and 0.25 µm

central region.

rods. They are therefore analogous to fibers.

respectively for 10 minutes each. This was then observed using the optical microscope and the result is given in Figure 9. The photographs show that the fibers are quite uniform and grow along a great length of the crystal. It was observed that the length of the fiber as revealed from the photograph is almost 750 µm. The diameter of the fibers in this photograph from the sample with initial composition CuGe0.915Si0.085O3 is around 4 m and is more or less uniform throughout the length.

(a) Central region of Cu1.0Ge0.9Si0.1O3.0, 1mm/h (UPS417b) Precipitate distance is 4.1 microns, their size is 2.5 microns

(b) Cu1.0Ge0.9Si0.1O3.0, 1mm/h (UPS417a2)

(c) Cu1.000Ge0.915Si0.085O3.000 (UPS416) (d) Intermediate region of

Cu1.000Ge0.915Si0.085O3.000 (UPS416) 0.75-1.3 mm/h, size 2.5 microns, distance 11.5 microns

Fig. 7. Example of the structure of the samples: (a) typical image of the central region of the samples. Fibers are ordered on a short distance. (b) When moving out of the center, precipitate distance and size increase (c) when we are going closer to the surface the precipitates disappear. (d) In the intermediate region corresponding to c, the fibers appear tilted.

Elaboration of a Specific Class of Metamaterial: Glass in Single Crystal 57

Fig. 9. Three images of the longitudinal section of the crystal were taken and carefully joined together to see the extension of the fibers running without distortion. The maximum length is around 750 microns and the fiber diameter is around 4 microns. Of course, as it is not possible to be exactly parallel to fiber axis, they are often cut during polishing. Sample Cu0.20Ge0.183Si0.017O0.60 (UPS416) growth rate 0.75-1.3 mm/h, fiber size 2.5m, fiber distance

Secondary electrons in SEM have a large field depth that is convenient for imaging on a large scale the surface of the sample that shows some topographies. However, it is necessary

Fig. 10. Secondary electron image of the surface of a cleaved sample tilted under 70° upward. The fibers appearing here elliptic due to the tilt are actually circular.

As we can see, the fibers seem to protrude the surface of the sample. We checked using a phase shift interferometer whether it is actually the case. This observation can be made on both faces of the cleavage. Sometimes, fibers are broken below the surface level but commonly, they protrude. This means that a stress relaxation occurs and that they were in compression before cleaving. This can be due to smaller thermal expansion coefficient for

In Fig. 11, the surface of the sample has not been perfectly cleaved and allows us to view the layer structure of the single crystal along an edge. It also reveals the channels where

to tilt the sample sharply. Some images are displayed in Figure 10.

11.5 microns has been used here.

**3.2 SEM-SE on tilted samples**

the fiber than for the matrix.

(a) Sample Cu1.000Ge0.915Si0.085O3.000 grown at 1 mm/h. The average distance between precipitates is 10 µm and the precipitate size is 4 µm.

(b) Sample with average distance between precipitates is 5 µm and the precipitate size is 3 µm.

(c) Sample CuGe0.9Si0.2O3.2 grown at 2 mm/h (UPS430), fiber distance is 2.50,3 microns and diameter is 1.20,13 micron (filling factor 0.18).

(d) Sample CuGe1,1Si0.3O3.8 grown at 2 mm/h (UPS436), fiber distance is 1.40,3 microns and diameter is 0.650,13 micron (filling factor 0.17).

Fig. 8. Variation of the central region of the samples when elaboration parameters are varied (SEM-backscattered electrons).

Fig. 9. Three images of the longitudinal section of the crystal were taken and carefully joined together to see the extension of the fibers running without distortion. The maximum length is around 750 microns and the fiber diameter is around 4 microns. Of course, as it is not possible to be exactly parallel to fiber axis, they are often cut during polishing. Sample Cu0.20Ge0.183Si0.017O0.60 (UPS416) growth rate 0.75-1.3 mm/h, fiber size 2.5m, fiber distance 11.5 microns has been used here.

#### **3.2 SEM-SE on tilted samples**

56 Crystallization and Materials Science of Modern Artificial and Natural Crystals

is 3 µm.

(b) Sample with average distance between precipitates is 5 µm and the precipitate size

(d) Sample CuGe1,1Si0.3O3.8 grown at 2 mm/h (UPS436), fiber distance is 1.40,3 microns and diameter is 0.650,13 micron

(filling factor 0.17).

Fig. 8. Variation of the central region of the samples when elaboration parameters are varied

(a) Sample Cu1.000Ge0.915Si0.085O3.000 grown at 1 mm/h. The average distance between precipitates is 10 µm and the

(c) Sample CuGe0.9Si0.2O3.2 grown at 2 mm/h (UPS430), fiber distance is 2.50,3 microns and diameter is 1.20,13 micron

(SEM-backscattered electrons).

(filling factor 0.18).

precipitate size is 4 µm.

Secondary electrons in SEM have a large field depth that is convenient for imaging on a large scale the surface of the sample that shows some topographies. However, it is necessary to tilt the sample sharply. Some images are displayed in Figure 10.

Fig. 10. Secondary electron image of the surface of a cleaved sample tilted under 70° upward. The fibers appearing here elliptic due to the tilt are actually circular.

As we can see, the fibers seem to protrude the surface of the sample. We checked using a phase shift interferometer whether it is actually the case. This observation can be made on both faces of the cleavage. Sometimes, fibers are broken below the surface level but commonly, they protrude. This means that a stress relaxation occurs and that they were in compression before cleaving. This can be due to smaller thermal expansion coefficient for the fiber than for the matrix.

In Fig. 11, the surface of the sample has not been perfectly cleaved and allows us to view the layer structure of the single crystal along an edge. It also reveals the channels where

Elaboration of a Specific Class of Metamaterial: Glass in Single Crystal 59

Fig. 13. X-ray diffraction pattern of a typical sample of CuGexSiyO1+2(x+y). All the peaks in

EBSD has been used to show that all fibers do not diffract the electron beam. The electron beam is incident on the sample under at an angle 70°. Electrons diffracted by crystal planes form Kossel cones. They are intercepted on a plane where they organized into lines called Kikuchi lines. Their indexation can lead to finding of crystal orientation if the space group and unit cell parameters are known. In our case, just the existence of diffraction is searched for. The resolution of the SEM-FEG used is good enough to analyze only one fiber. The accelerating voltage was 5kV and the working distance 10 mm. As we can see in Fig. 14 precipitates appear black. They do not diffract the backscattered electrons. We can thus

Fig. 14. Electron Backscattered Diffraction images. (a) EBSD image rebuilt from quality index of the Kikuchi diagrams (black = no diffraction). Precipitates appear actually black. The matrix has uniform brightness. (b) Orientation image, the matrix with uniform green

color is a single crystal actually. (c) Combination of the two previous images.

**3.4 SEM-EBSD (complementary observations, glass fibers are vitreous)**

this spectrum can be labeled from CuGeO3 space group.

conclude that the precipitates are vitreous.

the fibers were but in this case they disappeared during the cleaving. However, in the second image, a fiber remains showing that the crystallization around the fiber is not disturbed and the interface is very well defined. A transition region is not appearing. However, in Figure 12, we see better that a ring is appearing around the fiber compatible with a compression effect.

Fig. 11. Secondary electron image of surface of a sample not perfectly cleaved and tilted 70° upward. The ellipse in the photograph on the left shows a fiber end.

Fig. 12. This picture shows that around the fiber, a contrast is detected that seems to show a compression effect in agreement with the topography seen in SEM (sample UPS416).

#### **3.3 X-ray diffraction**

We have performed X-ray diffraction experiment (see fig.13) in order to identify additional lines among those of the well known cuprogermanate. Surprisingly, we detected the presence of only one crystalline phase, in contrast with the two phases observed by optical microscopy or SEM-EBS. This leads to think that the fibers are not crystallized.

the fibers were but in this case they disappeared during the cleaving. However, in the second image, a fiber remains showing that the crystallization around the fiber is not disturbed and the interface is very well defined. A transition region is not appearing. However, in Figure 12, we see better that a ring is appearing around the fiber compatible

Fig. 11. Secondary electron image of surface of a sample not perfectly cleaved and tilted 70°

Fig. 12. This picture shows that around the fiber, a contrast is detected that seems to show a compression effect in agreement with the topography seen in SEM (sample UPS416).

We have performed X-ray diffraction experiment (see fig.13) in order to identify additional lines among those of the well known cuprogermanate. Surprisingly, we detected the presence of only one crystalline phase, in contrast with the two phases observed by optical

microscopy or SEM-EBS. This leads to think that the fibers are not crystallized.

upward. The ellipse in the photograph on the left shows a fiber end.

100 m

with a compression effect.

**3.3 X-ray diffraction**

Fig. 13. X-ray diffraction pattern of a typical sample of CuGexSiyO1+2(x+y). All the peaks in this spectrum can be labeled from CuGeO3 space group.

#### **3.4 SEM-EBSD (complementary observations, glass fibers are vitreous)**

EBSD has been used to show that all fibers do not diffract the electron beam. The electron beam is incident on the sample under at an angle 70°. Electrons diffracted by crystal planes form Kossel cones. They are intercepted on a plane where they organized into lines called Kikuchi lines. Their indexation can lead to finding of crystal orientation if the space group and unit cell parameters are known. In our case, just the existence of diffraction is searched for. The resolution of the SEM-FEG used is good enough to analyze only one fiber. The accelerating voltage was 5kV and the working distance 10 mm. As we can see in Fig. 14 precipitates appear black. They do not diffract the backscattered electrons. We can thus conclude that the precipitates are vitreous.

Fig. 14. Electron Backscattered Diffraction images. (a) EBSD image rebuilt from quality index of the Kikuchi diagrams (black = no diffraction). Precipitates appear actually black. The matrix has uniform brightness. (b) Orientation image, the matrix with uniform green color is a single crystal actually. (c) Combination of the two previous images.

Elaboration of a Specific Class of Metamaterial: Glass in Single Crystal 61

matrix, which is perfectly crystalline, is made of rather fine structures corresponding to the solid state vibration modes allowed in Raman scattering. Concerning the spectrum collected from the fiber region, even though the lateral resolution is larger than the fiber diameter, it can be seen however that it is identical to the spectrum of the matrix, except for an additional band around 430 cm-1. To clearly express the difference between the two spectra, we performed a weighted difference between the spectrum of the precipitate and that of the matrix so as to eliminate, as much as possible, the lines from the matrix. One can thus clearly see the additional band in Figure 15, which has its maximum at 433 cm-1. Its width is 107 cm-1, which is significantly larger than the width of the crystal lines. It is thus possible to conclude that, in the fibers, there is no long-range order, in contrast to the crystals. Taking into account the results of the chemical analysis of the samples, it can be concluded that the fibers have a glass structure with an average composition intermediate between GeO2 and SiO2. We find the Raman spectra for this type of glass in the literature recently [14, 15], Raman spectra for SiO2 and GeO2 glasses are also well known. They present both a wide and intense band, respectively at 440 cm-1 and 416 cm-1. Their respective widths are 234 cm-1 and 94 cm-1 [16]. The values that we measured are thus intermediate between these values in agreement with [15] and we can thus conclude from the Raman measurements that the

**3.6 Chemical composition by EPMA (it reveals that compensation occurs on the metal**

In order to understand the phase separation, we performed chemical analyses. They were done by electron microprobe analysis (EPMA). Cu, Ge, Si and O, were analyzed independently. In this way, one could exclude measurements presenting a sum deviating from 100 wt %. A composition profiling was carried out in the zones containing precipitates. Monocrystal of CuGeO3 and a silica blade were the references of measurement. Correct measurements were then converted into atomic percentage. The error bars were obtained by statistical average. For a global view of the content variations, we have recorded chemical profiles in dynamic mode. We can see large variations in Si and Cu profiles. It means that the profile is crossing a fiber. However, as we have no chance to cross exactly a fiber in the center, contrarily to static mode, the variation is never extremum. Of course, precise positioning has been achieved in the static mode. We made profiles on different samples with SiO2 content

from 8.5 to 30 mol %, only three measurements are shown below for demonstration.

The external area, without precipitate, was measured separately in static mode, it showed the composition Cu0.203Ge0.191Si0.014O0.593. The ratio of the number of oxygen to metal (Cu, Ge, and Si) is a useful quantity for analyzing the differences in chemical composition at various regions. It is around 1.43 in this area instead of 1.5 in the perfect crystal. The external matter is thus slightly under-stoichiometric. One can also see, by comparing the chemical formula

At the central area, the matter surrounding the fibers has the chemical composition: Cu0.205Ge0.197Si0.015O0.583 and here the ratio is 1.4. Thus this matter is also under-stoichiometric and again Si depressed (0.015 instead of 0.017). Under-stoichiometry undoubtedly corresponds to a natural reduction of Cu at high temperature during the elaboration

**3.6.1 In UPS 416, the nominal composition is Cu(Ge0.915, Si0.085)O3**

process. The single crystal therefore should include oxygen vacancies.

at this place with the nominal one, that it presents a loss in Si.

probed precipitate is indeed vitreous.

**=Cu0.20Ge0.183Si0.017O0.60**

**lattice)**

#### **3.5 Raman micro-spectroscopy analysis**

Raman micro-spectroscopy can be used to identify a compound by its spectroscopic signature. It is based on the inelastic scattering of usually visible electromagnetic waves (energy exchange between the optical wave and the material vibration waves. Raman spectroscopy was performed in confocal mode with a pinhole of 300 microns and an objective of magnification 100x. The source of excitation was the 514 nm emission from an Ar+ laser. To record the Raman spectra, the beam was focused on the surface of the material. Under these conditions, the lateral resolution is of the order of the micron whereas the depth resolution is about 30 microns. The acquisition time for the spectrum was fixed at 60s. By taking measurements between 200 and 4000 cm-1, it was observed that, for our materials, Raman features were located only between 200 and 1500 cm-1. The Raman spectra were recorded on the reference sample CuGeO3 and on the sample doped with 10 % Si/(Si+Ge). Figure 15 shows the spectra collected between fibers and on a fiber. The spectrum of the

Fig. 15. Raman scattering spectra recorded at the surface of a cleaved sample with 10 % Si/(Si+Ge). One of the spectra has been recorded in the matrix surrounding the precipitate and the other one on the fiber itself. The third spectrum has been computed by weighted difference of the two others in order to eliminate the matrix spectrum as much as possible (UPS422). The second picture is an enhancement of the first one.

#### UPS422

Raman micro-spectroscopy can be used to identify a compound by its spectroscopic signature. It is based on the inelastic scattering of usually visible electromagnetic waves (energy exchange between the optical wave and the material vibration waves. Raman spectroscopy was performed in confocal mode with a pinhole of 300 microns and an objective of magnification 100x. The source of excitation was the 514 nm emission from an Ar+ laser. To record the Raman spectra, the beam was focused on the surface of the material. Under these conditions, the lateral resolution is of the order of the micron whereas the depth resolution is about 30 microns. The acquisition time for the spectrum was fixed at 60s. By taking measurements between 200 and 4000 cm-1, it was observed that, for our materials, Raman features were located only between 200 and 1500 cm-1. The Raman spectra were recorded on the reference sample CuGeO3 and on the sample doped with 10 % Si/(Si+Ge). Figure 15 shows the spectra collected between fibers and on a fiber. The spectrum of the

UPS422

100 300 500<PHONE_NUMBER> 1300 1500 wavenumber (cm-1)

100 300 500<PHONE_NUMBER> 1300 1500 wavenumber (cm-1)

Fig. 15. Raman scattering spectra recorded at the surface of a cleaved sample with 10 % Si/(Si+Ge). One of the spectra has been recorded in the matrix surrounding the precipitate and the other one on the fiber itself. The third spectrum has been computed by weighted difference of the two others in order to eliminate the matrix spectrum as much as possible

(UPS422). The second picture is an enhancement of the first one.

UPS422

matrix z=0 precipitate z=0 difference

matrix z=0 precipitate z=0 difference

**3.5 Raman micro-spectroscopy analysis**

Raman intensity (a.u.)

Raman intensity (a.u.)

matrix, which is perfectly crystalline, is made of rather fine structures corresponding to the solid state vibration modes allowed in Raman scattering. Concerning the spectrum collected from the fiber region, even though the lateral resolution is larger than the fiber diameter, it can be seen however that it is identical to the spectrum of the matrix, except for an additional band around 430 cm-1. To clearly express the difference between the two spectra, we performed a weighted difference between the spectrum of the precipitate and that of the matrix so as to eliminate, as much as possible, the lines from the matrix. One can thus clearly see the additional band in Figure 15, which has its maximum at 433 cm-1. Its width is 107 cm-1, which is significantly larger than the width of the crystal lines. It is thus possible to conclude that, in the fibers, there is no long-range order, in contrast to the crystals. Taking into account the results of the chemical analysis of the samples, it can be concluded that the fibers have a glass structure with an average composition intermediate between GeO2 and SiO2. We find the Raman spectra for this type of glass in the literature recently [14, 15], Raman spectra for SiO2 and GeO2 glasses are also well known. They present both a wide and intense band, respectively at 440 cm-1 and 416 cm-1. Their respective widths are 234 cm-1 and 94 cm-1 [16]. The values that we measured are thus intermediate between these values in agreement with [15] and we can thus conclude from the Raman measurements that the probed precipitate is indeed vitreous.

#### **3.6 Chemical composition by EPMA (it reveals that compensation occurs on the metal lattice)**

In order to understand the phase separation, we performed chemical analyses. They were done by electron microprobe analysis (EPMA). Cu, Ge, Si and O, were analyzed independently. In this way, one could exclude measurements presenting a sum deviating from 100 wt %. A composition profiling was carried out in the zones containing precipitates. Monocrystal of CuGeO3 and a silica blade were the references of measurement. Correct measurements were then converted into atomic percentage. The error bars were obtained by statistical average. For a global view of the content variations, we have recorded chemical profiles in dynamic mode. We can see large variations in Si and Cu profiles. It means that the profile is crossing a fiber. However, as we have no chance to cross exactly a fiber in the center, contrarily to static mode, the variation is never extremum. Of course, precise positioning has been achieved in the static mode. We made profiles on different samples with SiO2 content from 8.5 to 30 mol %, only three measurements are shown below for demonstration.

#### **3.6.1 In UPS 416, the nominal composition is Cu(Ge0.915, Si0.085)O3 =Cu0.20Ge0.183Si0.017O0.60**

The external area, without precipitate, was measured separately in static mode, it showed the composition Cu0.203Ge0.191Si0.014O0.593. The ratio of the number of oxygen to metal (Cu, Ge, and Si) is a useful quantity for analyzing the differences in chemical composition at various regions. It is around 1.43 in this area instead of 1.5 in the perfect crystal. The external matter is thus slightly under-stoichiometric. One can also see, by comparing the chemical formula at this place with the nominal one, that it presents a loss in Si.

At the central area, the matter surrounding the fibers has the chemical composition: Cu0.205Ge0.197Si0.015O0.583 and here the ratio is 1.4. Thus this matter is also under-stoichiometric and again Si depressed (0.015 instead of 0.017). Under-stoichiometry undoubtedly corresponds to a natural reduction of Cu at high temperature during the elaboration process. The single crystal therefore should include oxygen vacancies.

Elaboration of a Specific Class of Metamaterial: Glass in Single Crystal 63

oxygen to metal ration of 1.48. The other one shows the composition: Cu0.2060.005Ge0.1850.004 Si0.0170.001O0.5920.006 with oxygen to metal ratio of 1.45. In the central region, the matrix composition is Cu0.203Ge0.187Si0.018O0.592. The precipitates have the following composition: Cu0.055Ge0.197Si0.104O0.644 with an oxygen-to-metal ratio of 1.91. We observe thus similar

Fig. 17. Chemical profile of typical sample with the composition Cu0.20Ge0.180Si0.020O0.60. The region with precipitates is on the right hand side, the one without is on the left. Note the constancy of Ge atomic content from matrix and fibers. Note also the Cu variation, which is

**=Cu0.20Ge0.160Si0.040O0.60. This sample contains precipitates but they are disordered**  In the central region, the matrix appears with composition: Cu0.2100.005Ge0.1770.003 Si0.0280.001O0.5800.004 with oxygen to metal ratio of 1.4 and with an depletion in Si compared to the nominal composition. The average composition of the precipitates is Cu0.1250.015 Ge0.1600.004Si0.110.01O0.620.01 with oxygen to metal ratio of 1.57. Compared to the previous sample, here, the nominal composition contains more Si and therefore the precipitates contain more silicon too but the exchange is not in proportion. For starting compounds with 8.5 and 10% mol SiO2/(SiO2+GeO2), the fiber contains about 27% of Si/(Si+Ge+Cu), 59% of Ge/(Si+Ge+Cu) and 14% Cu/(Si+Ge+Cu). In this sample with 20% mol SiO2/(SiO2+GeO2) in the nominal compound, the fiber is made with 28% Si/(Si+Ge+Cu) and 41% Ge/(Si+Ge+Cu) but 31% Cu/(Si+Ge+Cu) i.e. more than twice the Cu content of the previous samples. Therefore, the oxygen to metal ratio does not reach 2 and even it is understoichiometric in oxygen. Nevertheless, the Ge content remains the same in all parts of the sample. However, the Cu/Si balance is now degraded (see Cu+2Si curve in Figure 18). What we observe also, is that the oxygen content does not show variation in proportion for

roughly the double of the Si one (Cu+2.Si curve).

balancing. We can suspect an electric charge of the fiber.

**3.6.3 In UPS 423, the nominal composition is Cu(Ge0.800, Si0.200)O3**

contrasts between matrix and precipitates similar to the previous sample.

The fibers show an average composition of Cu0.05Ge0.20Si0.09O0.66 with a ratio of oxygen to metal of 1.94. This is clearly a different chemical formula than the one for the matrix. This leads us to consider a formal composition of type MeO2; like SiO2 or GeO2. In addition, we can also observe a clear enrichment of the net content in Si and a reduction in the quantity of Cu on comparison with the nominal composition. On the other hand, the sum of the oxygen quantities for saturating the metal content should yield 1.91 instead of 1.94. It can be thus considered that the compound is over-stoichiometric in oxygen. It should also be noted that the quantity of Si in the precipitates is higher than it can be obtained by a simple expulsion of CuO from the starting composition as can be seen by comparing the nominal composition rewritten in the form of CuO+(Ge0.915Si0.085)O2 with that of the precipitates (Cu0.15Ge0.61Si0.27)O2. The chemical profile of Cu, Ge, Si and O made across the center (containing precipitates) and the surrounding area (clear from precipitate) is displayed in Fig. 16. It shows that O and Ge are almost constant along the profile. In contrast Si and Cu vary when a precipitate is crossed. We suspected a balance between only Si and Cu variation between the phases. The curve Cu+2Si actually confirms this observation. This curve does not show large variation anymore and suggests a valence charge balance between Cu and Si atoms.

Fig. 16. Chemical profile of typical sample with the composition Cu0.20Ge0.183Si0.017O0.60. The region with precipitates is on the right hand side, the one without is on the left. Note the constancy of Ge atomic content from matrix and fibers. Note also the Cu variation, which is roughly the double of the Si one (Cu+2.Si curve).

#### **3.6.2 In UPS 417, the nominal composition is Cu(Ge0.900, Si0.100)O3**

#### **=Cu0.20Ge0.180Si0.020O0.60**

In this case also, several regions are analyzed (Fig. 17). The outermost region is without any precipitate and going towards the center; there is a region with disordered precipitates; a clear region and a small central region with ordered precipitates. The clear area exhibits a Si rich content with 2.5 to 3 at% and another area with a content poor in Si close to the biphased area. The average composition is Cu0.1890.007Ge0.1870.002Si0.0270.004O0.5970.004 with

The fibers show an average composition of Cu0.05Ge0.20Si0.09O0.66 with a ratio of oxygen to metal of 1.94. This is clearly a different chemical formula than the one for the matrix. This leads us to consider a formal composition of type MeO2; like SiO2 or GeO2. In addition, we can also observe a clear enrichment of the net content in Si and a reduction in the quantity of Cu on comparison with the nominal composition. On the other hand, the sum of the oxygen quantities for saturating the metal content should yield 1.91 instead of 1.94. It can be thus considered that the compound is over-stoichiometric in oxygen. It should also be noted that the quantity of Si in the precipitates is higher than it can be obtained by a simple expulsion of CuO from the starting composition as can be seen by comparing the nominal composition rewritten in the form of CuO+(Ge0.915Si0.085)O2 with that of the precipitates (Cu0.15Ge0.61Si0.27)O2. The chemical profile of Cu, Ge, Si and O made across the center (containing precipitates) and the surrounding area (clear from precipitate) is displayed in Fig. 16. It shows that O and Ge are almost constant along the profile. In contrast Si and Cu vary when a precipitate is crossed. We suspected a balance between only Si and Cu variation between the phases. The curve Cu+2Si actually confirms this observation. This curve does not show large variation

anymore and suggests a valence charge balance between Cu and Si atoms.

Fig. 16. Chemical profile of typical sample with the composition Cu0.20Ge0.183Si0.017O0.60. The region with precipitates is on the right hand side, the one without is on the left. Note the constancy of Ge atomic content from matrix and fibers. Note also the Cu variation, which is

In this case also, several regions are analyzed (Fig. 17). The outermost region is without any precipitate and going towards the center; there is a region with disordered precipitates; a clear region and a small central region with ordered precipitates. The clear area exhibits a Si rich content with 2.5 to 3 at% and another area with a content poor in Si close to the biphased area. The average composition is Cu0.1890.007Ge0.1870.002Si0.0270.004O0.5970.004 with

roughly the double of the Si one (Cu+2.Si curve).

**=Cu0.20Ge0.180Si0.020O0.60**

**3.6.2 In UPS 417, the nominal composition is Cu(Ge0.900, Si0.100)O3**

oxygen to metal ration of 1.48. The other one shows the composition: Cu0.2060.005Ge0.1850.004 Si0.0170.001O0.5920.006 with oxygen to metal ratio of 1.45. In the central region, the matrix composition is Cu0.203Ge0.187Si0.018O0.592. The precipitates have the following composition: Cu0.055Ge0.197Si0.104O0.644 with an oxygen-to-metal ratio of 1.91. We observe thus similar contrasts between matrix and precipitates similar to the previous sample.

Fig. 17. Chemical profile of typical sample with the composition Cu0.20Ge0.180Si0.020O0.60. The region with precipitates is on the right hand side, the one without is on the left. Note the constancy of Ge atomic content from matrix and fibers. Note also the Cu variation, which is roughly the double of the Si one (Cu+2.Si curve).

#### **3.6.3 In UPS 423, the nominal composition is Cu(Ge0.800, Si0.200)O3 =Cu0.20Ge0.160Si0.040O0.60. This sample contains precipitates but they are disordered**

In the central region, the matrix appears with composition: Cu0.2100.005Ge0.1770.003 Si0.0280.001O0.5800.004 with oxygen to metal ratio of 1.4 and with an depletion in Si compared to the nominal composition. The average composition of the precipitates is Cu0.1250.015 Ge0.1600.004Si0.110.01O0.620.01 with oxygen to metal ratio of 1.57. Compared to the previous sample, here, the nominal composition contains more Si and therefore the precipitates contain more silicon too but the exchange is not in proportion. For starting compounds with 8.5 and 10% mol SiO2/(SiO2+GeO2), the fiber contains about 27% of Si/(Si+Ge+Cu), 59% of Ge/(Si+Ge+Cu) and 14% Cu/(Si+Ge+Cu). In this sample with 20% mol SiO2/(SiO2+GeO2) in the nominal compound, the fiber is made with 28% Si/(Si+Ge+Cu) and 41% Ge/(Si+Ge+Cu) but 31% Cu/(Si+Ge+Cu) i.e. more than twice the Cu content of the previous samples. Therefore, the oxygen to metal ratio does not reach 2 and even it is understoichiometric in oxygen. Nevertheless, the Ge content remains the same in all parts of the sample. However, the Cu/Si balance is now degraded (see Cu+2Si curve in Figure 18). What we observe also, is that the oxygen content does not show variation in proportion for balancing. We can suspect an electric charge of the fiber.

Elaboration of a Specific Class of Metamaterial: Glass in Single Crystal 65

For a ternary system like the one we used (CuO-SiO2-GeO2), when the starting composition falls into a 3-phase domain, the composition of the phases is defined by the intersection of a complex line (the connodal) with the domain boundaries. In this case, the molar proportion of each phase is defined by the position of the starting composition from each phase boundaries along the connodal, there is no free parameter. It is different with 2-phases case, the compositions of each phase depends on the starting composition. The phase molar proportion is defined by other constraint (not thermodynamically) e.g. the constancy of the

In our case, we obtain a two phase composite with varying phase compositions when we change the starting composition. This is in principle, in agreement with the phase diagram of Breuer et al. [17]. However, the compositions of our phases are not the expected ones. We should obtain a combination of the CuMeO3 phase and an almost pure GeO2 phase. Instead, the fibers contain about 30% of Si, 40-60% of Ge and 15-30% Cu. We thus deduce that the formation of a vitreous phase allows germanosilicate glass to dissolve CuO and changes significantly the ternary CuO-SiO2-GeO2 phase diagram in which the two-phase domain is

The rod-type shape of the second phase and the regularity of the microstructure can then be explained as characteristic of a eutectic structure resulting from the decomposition, upon solidification, of a liquid into two solid phases. It is the right compromise between the thermodynamical energy and the surface energy. In such a eutectic transformation, taking place in a temperature gradient like the one available in the growth technique used in this study, the resulting aligned microstructures found are generally either lamellar or fibrous, depending on the volume fraction of the minor phase: when this fraction is smaller than about 30% the microstructure is rod-shaped, which is obviously the case here. Above 30% it is lamellar [18, 19]. The space period and also the size of the precipitates changes in the same samples and also the size but not continuously. There are region with precipitates of the same size and rather region rather disordered with various size and distance. When the size is constant, this means that the theory above is

*The last question is why there is a spatially regular distribution?* Some interpretation is given by Flemings [20] for eutectic solidification but can be applied more generally to our case. Decomposition of the liquid into two solid phases by eutectic reaction is limited by diffusion and reaction rate. This leads to a constrained control of the size and period of the minority phase distribution. In order to understand the relevant forces that lead to this situation, we tried a rationalization of our observations on a kinetics basis. We studied the differences in volume chemical content between the two solid phases (Table 1). Knowing the number of molecules per cm-3 of each phase (CuMeO3 for the matter surrounding fibers and germaniasilica glass for fibers themselves), we computed the changes per unit volume of each species in Table 1. The concentration of germanium and oxygen decreases only a little. This is apparently in contradiction with the atomic fraction (second column of Table 1) but change in atomic fraction does not mimic directly the change in the unit volume (third and above columns in Table 1) as we have to take into account the molecule volume. About the difference in Cu and Si, they appear complementary really in a ratio of 1 Si to 2 Cu. Therefore, we can conclude that during solidification of the liquid phase, there is an exchange of Cu and Si, on one hand, and a slight decrease in GeO2 content in the fiber

Ge content whatever the phase.

extended (see Figure 19).

applicable.

compared to the matrix.

Fig. 18. Chemical profile of typical sample with the composition Cu0.20Ge0.160Si0.040O0.60. In this sample precipitates are everywhere across the profile. Note the constancy of Ge atomic content from matrix and fibers. Note also the Cu variation, which is roughly the double of the Si one (Cu+2Si curve).

#### **3.6.4 Discussion and conclusions on the chemical observations**

When precipitates are correctly ordered, we observed an over-stoichiometry in oxygen in fiber and an under-stoichiometry in the matrix. Although all the samples exhibit heterogeneities, the chemical composition of the cuprogermanate phase is rather homogeneous and close to the nominal one. Some small fluctuations in the Si content are nevertheless observed. They are compensated by Cu content variations.

For all the samples studied, which contain varying amounts of Si, we observed that the fibers are vitreous and have a (Si+Ge)O2 type composition containing a small amount of CuO whereas the matrix has a composition close to the nominal one. We also observed that the fibers present a clear enrichment in Si in comparison with the surrounding matrix. More surprising is that this exchange does not disturb the Ge content. Quantitatively, the GeO2 proportion is constant but the SiO2 variation is 1/2 the one of CuO i.e. 2 CuO molecules in matrix structure is replaced by 1 molecules of SiO2 in precipitate when they are ordered. We observe that the oxygen content increases consistently (we remind that oxygen content has been measured independently of the other constituents). The most external region is slightly silicon depressed when the central region is ordered. The constancy of Ge throughout the sample is remarkable. This is not the case for Si and Cu. Even more surprising is the balance between Cu and Si. There should be no constraint on oxygen diffusion in the liquid and so we expected no relation between Cu and Si. Therefore, the existence of such a relation means that oxygen migration is much lower than Cu and Si.

*Now, one has to find an answer to three questions in order to master metamaterial elaboration: why is a second phase formed, why does it appear as micro rods (or fibers) and why are these rods regularly distributed?*

Fig. 18. Chemical profile of typical sample with the composition Cu0.20Ge0.160Si0.040O0.60. In this sample precipitates are everywhere across the profile. Note the constancy of Ge atomic content from matrix and fibers. Note also the Cu variation, which is roughly the double of

When precipitates are correctly ordered, we observed an over-stoichiometry in oxygen in fiber and an under-stoichiometry in the matrix. Although all the samples exhibit heterogeneities, the chemical composition of the cuprogermanate phase is rather homogeneous and close to the nominal one. Some small fluctuations in the Si content are

For all the samples studied, which contain varying amounts of Si, we observed that the fibers are vitreous and have a (Si+Ge)O2 type composition containing a small amount of CuO whereas the matrix has a composition close to the nominal one. We also observed that the fibers present a clear enrichment in Si in comparison with the surrounding matrix. More surprising is that this exchange does not disturb the Ge content. Quantitatively, the GeO2 proportion is constant but the SiO2 variation is 1/2 the one of CuO i.e. 2 CuO molecules in matrix structure is replaced by 1 molecules of SiO2 in precipitate when they are ordered. We observe that the oxygen content increases consistently (we remind that oxygen content has been measured independently of the other constituents). The most external region is slightly silicon depressed when the central region is ordered. The constancy of Ge throughout the sample is remarkable. This is not the case for Si and Cu. Even more surprising is the balance between Cu and Si. There should be no constraint on oxygen diffusion in the liquid and so we expected no relation between Cu and Si. Therefore, the existence of such a relation

*Now, one has to find an answer to three questions in order to master metamaterial elaboration: why is a second phase formed, why does it appear as micro rods (or fibers) and why are these rods regularly*

**3.6.4 Discussion and conclusions on the chemical observations**

nevertheless observed. They are compensated by Cu content variations.

means that oxygen migration is much lower than Cu and Si.

the Si one (Cu+2Si curve).

*distributed?*

For a ternary system like the one we used (CuO-SiO2-GeO2), when the starting composition falls into a 3-phase domain, the composition of the phases is defined by the intersection of a complex line (the connodal) with the domain boundaries. In this case, the molar proportion of each phase is defined by the position of the starting composition from each phase boundaries along the connodal, there is no free parameter. It is different with 2-phases case, the compositions of each phase depends on the starting composition. The phase molar proportion is defined by other constraint (not thermodynamically) e.g. the constancy of the Ge content whatever the phase.

In our case, we obtain a two phase composite with varying phase compositions when we change the starting composition. This is in principle, in agreement with the phase diagram of Breuer et al. [17]. However, the compositions of our phases are not the expected ones. We should obtain a combination of the CuMeO3 phase and an almost pure GeO2 phase. Instead, the fibers contain about 30% of Si, 40-60% of Ge and 15-30% Cu. We thus deduce that the formation of a vitreous phase allows germanosilicate glass to dissolve CuO and changes significantly the ternary CuO-SiO2-GeO2 phase diagram in which the two-phase domain is extended (see Figure 19).

The rod-type shape of the second phase and the regularity of the microstructure can then be explained as characteristic of a eutectic structure resulting from the decomposition, upon solidification, of a liquid into two solid phases. It is the right compromise between the thermodynamical energy and the surface energy. In such a eutectic transformation, taking place in a temperature gradient like the one available in the growth technique used in this study, the resulting aligned microstructures found are generally either lamellar or fibrous, depending on the volume fraction of the minor phase: when this fraction is smaller than about 30% the microstructure is rod-shaped, which is obviously the case here. Above 30% it is lamellar [18, 19]. The space period and also the size of the precipitates changes in the same samples and also the size but not continuously. There are region with precipitates of the same size and rather region rather disordered with various size and distance. When the size is constant, this means that the theory above is applicable.

*The last question is why there is a spatially regular distribution?* Some interpretation is given by Flemings [20] for eutectic solidification but can be applied more generally to our case. Decomposition of the liquid into two solid phases by eutectic reaction is limited by diffusion and reaction rate. This leads to a constrained control of the size and period of the minority phase distribution. In order to understand the relevant forces that lead to this situation, we tried a rationalization of our observations on a kinetics basis. We studied the differences in volume chemical content between the two solid phases (Table 1). Knowing the number of molecules per cm-3 of each phase (CuMeO3 for the matter surrounding fibers and germaniasilica glass for fibers themselves), we computed the changes per unit volume of each species in Table 1. The concentration of germanium and oxygen decreases only a little. This is apparently in contradiction with the atomic fraction (second column of Table 1) but change in atomic fraction does not mimic directly the change in the unit volume (third and above columns in Table 1) as we have to take into account the molecule volume. About the difference in Cu and Si, they appear complementary really in a ratio of 1 Si to 2 Cu. Therefore, we can conclude that during solidification of the liquid phase, there is an exchange of Cu and Si, on one hand, and a slight decrease in GeO2 content in the fiber compared to the matrix.

Elaboration of a Specific Class of Metamaterial: Glass in Single Crystal 67

 Cu Ge Si O Total Matrix filling factor=0.84 1.71 1.64 0.125 4.87 8.34 Fiber filling factor=0.16 0.354 1.43 0.637 4.67 7.08

estimate for the liquid phase 1.49 1.60 0.207 4.84 8.14

composition of the ceramics rod 1.67 1.53 0.14 5.01 8.35 Difference average minus nominal -0.18 0.07 0.07 -0.17 -0.21

Fig. 19. Ternary diagram for CuO-SiO2-GeO2 from [13] (1kb, 600°C) on which we have placed the estimated composition for the liquid phase (red point), the matrix composition (open blue circle) and the fiber composition (solid blue circle). Dashed blue lines are reestimate of the location of the phase boundaries. The green point is the starting composition

Figure 18 shows the location of the previous example in the phase diagram. As we can see, the rod composition in green moved to the red point on losing CuO, and then under cooling, the two phase composition is defined by the line linking the two blue points and the matter conservation. On one hand, the matrix composition surrounding the fiber is defined by the congruent line. On the other hand, since the fibers contain CuO, this means that a phase boundary (dashed blue line) is missing in the diagram. Below this line, there exists a silica glass containing CuO. In addition, we can say, that between the two blue points, it is not a three phases domain but a two phases one. Finally, as we had seen that the liquid phase corresponding to ordered area, moved to glass boundary, we made further experiments with increasing content in glass formers. This gave then results with the

In order to obtain spacing between fibers compatible with the second telecom window (1.5

*2/4d2=f,*

µm), we studied several parameters of growth. For a fixed filling factor *f*: *d2R=a,*

Number of atoms 1022 per cm3

Chemical formula (atomic %)

Sample UPS416 Cu0.20Ge0.183Si0.017O0.60

Average composition in at.cm-3 i.e.

Nominal composition in at.cm-3 i.e.

for samples Cu0.20Ge0.18Si0.02O0.60

smallest size and distances between fibers (See Figure 8d).

Table 3. Computation of the liquid phase composition.

N.B.: There are 1.67 1022 molecules per cm3 for CuGeO3 crystal and 2.36 1022 molecules per cm3 about for 1/3SiO2+2/3GeO2 glass.

Table 1. Atomic content in each phase and their difference.

N.B.: There are 1.67 1022 molecules per cm3 for CuGeO3 crystal and 2.36 1022 molecules per cm3 about for 1/3SiO2+2/3GeO2 glass.

Table 2. Atomic content in each phase and their difference.

We can now arise the question on the transformation of the system under cooling in the phase diagram. As a matter of fact, the observation of a first phase with composition CuMeO3 where Me is a mixture between Ge, Si, and a second phase with composition Me'O2 where Me' is also a metallic mixture, means that the starting liquid did not keep the composition of the rod in particular, CuMeO3 in the case considered in the table. Unfortunately, it was not possible to know the composition of the liquid phase, but we can deduce it by calculation. In this direction, considering the sample with 8.5% at Si, from the optical and SEM observations, we know the filling factor in the central region in 2D is 16%. With this information, and considering that there is invariance along the growth axis, we can achieve the calculation by combining the chemical formula of one phase with the other one. This is done in the Table 3.

Table 3 shows for the central region of the sample that the Ge or Si content have not changed whereas Cu and O show a similar decrease. It is thus clear that the liquid has been losing CuO under heating. We can remind here that from chemical analysis, we found that the external region without precipitates is SiO2 depressed. As the GeO2 content remains almost constant, and because a flux of SiO2 is compensated by a flux of CuO, it yields that CuO migrated from the center to the surface and SiO2 migrated in the reverse direction. This is likely due to thermal gradient. Therefore, for this specific composition, this migration moved the system (corresponding to the ordered region) from the congruent line to the two phase domain as we show in Figure 18.

Matrix composition Cu0.203Ge0.187Si0.018O0.592 1.70 1.56 0.15 4.94 8.35 Fiber composition Cu0.055Ge0.197Si0.104O0.644 0.39 1.39 0.74 4.56 7.08

(matrix to fiber) Cu-0.148Ge0.010Si0.086O0.052 -1.31 - 0.15 0.594 - 0.33 -1.27

N.B.: There are 1.67 1022 molecules per cm3 for CuGeO3 crystal and 2.36 1022 molecules per cm3 about for

Matrix composition Cu0.205Ge0.197Si0.015O0.583 1.71 1.64 0.125 4.87 8.35 Fiber composition Cu0.05Ge0.20Si0.09O0.66 0.354 1.43 0.637 4.67 7.08

(matrix to fiber) Cu-0.148Ge0.010Si0.086O0.052 -1.35 - 0.21 0.61 - 0.20 -1.27

N.B.: There are 1.67 1022 molecules per cm3 for CuGeO3 crystal and 2.36 1022 molecules per cm3 about