The third membrane was made of elastic polymer (thermoplastic elastomer) with a small hole in it. The material is chemical resistance to alcohol since a small amount of alcohol is in the contents of refrigerant. It has a high elongation and adequate tensile strength to keep water inside the capsule, especially at a low temperature. It is adhesive so the hole is closed when there is no volumetric expansion.

#### **3. Propagation through ion exchange membrane**

58 Supercooling

0 10 20 30 0

Pore radius *r*, nm

Cation exchange membranes and anion exchange membranes with various thicknesses were

The second membrane was made of porous elastic polymer material made by foaming. It is elastic and each pore is ideally independent to each other with cracks between them as

The third membrane was made of elastic polymer (thermoplastic elastomer) with a small hole in it. The material is chemical resistance to alcohol since a small amount of alcohol is in the contents of refrigerant. It has a high elongation and adequate tensile strength to keep water inside the capsule, especially at a low temperature. It is adhesive so the hole is closed

shown in Fig. 3. So it is only semi-permeable when the membrane is expanded.

pore crack

Fig. 2. Depression of the melting temperature due to a narrow space

Fig. 3. Image of the cross section of porous polymer membrane

10

**2.2 Porous elastic polymer membrane**

**2.3 Styrene elastomer membrane**

when there is no volumetric expansion.

20

Depression of melting point

used.

30

40

50

Δ*T'*, K

#### **3.1 Experimental apparatus and experimental method**

The type of ion exchange membrane tested in the research is shown in Table 2. Two types of cation exchange membranes and three types of anion exchange membranes were used. The purpose of the experiment was to check whether ice propagates through ion exchange membrane. The apparatus is shown in Fig. 4. The membrane was placed between two cells, namely the upper cell and the lower cell. Several kinds of concentration of NaCl solutions were prepared for both cells. The apparatus was kept under a constant temperature and artificially the nucleation was started in the upper cell under the supercooling state by installing the ice particle from the top of the needle as shown in Fig. 4. The ice gradually grew inside the needle and a single crystal ice appeared at the tip of the needle. Temperature of the solution in the upper cell was measured directly by inserting the thermocouple, and the temperature of the solution in the lower cell was measured indirectly from the outer surface of the cell to avoid natural nucleation due to the existence of the thermocouple. It was confirmed that there is almost no change in concentration during a series of experiments.

Table 2. Type of ion exchange membrane

Fig. 4. Experimental apparatus

#### **3.2 Results and discussions**

Fig. 5 shows the results obtained. The abscissa shows the time after the ice touching the upper surface of the membrane. The ordinate shows the probability of propagation of ice to the lower cell. There were three degrees of supercooling tested, namely, *T*=3 K, 5 K and 7 K. As it can be observed from the figure, the only CMV membrane propagated ice. This is a cation exchange membrane with thin thickness. So, it can be said that a thin cation exchange membrane is better.

Suppressing Method of Supercooling State in Cool Box Using Membrane 61

Ion exchange membrane Rubber sheet

Fig. 6. Example of capsule having an ion exchange membrane on one side and a rubber

In order to solve the problem of complexity, an elastic porous membrane was selected. It is a porous polymer material made by foaming. Similar experiments to the one using the ion exchange membrane were carried out using the membrane. The results are shown in Figs. 7 & 8. Fig. 7 is for 10 wt% NaCl solution having the melting temperature of -6.56 oC and Fig. 8 is for 15 wt% NaCl solution having the melting temperature of -10.89 oC. It can be seen that in the case of 10 wt%, when water in the upper cell solidified, the temperature in the upper cell jumped to the melting temperature. However, since the solution in the lower cell was already in the supercooling state, ice propagated immediately with low degree of supercooling. In the case of 15 wt%, when water in the upper cell solidified, the temperature in the lower cell was above the melting temperature, so no propagation occurred at this stage. After the temperature reached down below the melting temperature for the solution,

0 2000 4000 6000

Influence of the cooling rate on propagation of the ice was examined. Under two kinds of cooling conditions, temperature of the solution in the lower cell may differ to each other at the time of propagation. It is due to the existence of thermal resistance of the capsule. So the

Fig. 7. Experimental result of propagation using porous polymer membrane with 10 wt%

melting temp. for 10wt%

C

Time *t*, s

*T*

**4. Porous polymer material made by foaming**

the ice propagated immediately with a low degree of supercooling.

NaCl -6.6 o

 NaCl water brine

Temperature

NaCl solution

*T*, o

C

0

sheet on the other side

Fig. 5. Time wise variation of probability of propagation using various ion exchanger membranes

It may have a problem of durability.Since there is a volumetric expansion due to solidification, the capsule needs to have a function to absorb the volumetric change. Fig. 6 shows some of the example of having rubber material on the other side of the capsule. It has a demerit since the device becomes a little bit complicated.

0

0

0.2

0.4

Probability of freezing

0 100 200 300 400 500 600

time *t*, s

DSV membrane

It may have a problem of durability.Since there is a volumetric expansion due to solidification, the capsule needs to have a function to absorb the volumetric change. Fig. 6 shows some of the example of having rubber material on the other side of the capsule. It has

Fig. 5. Time wise variation of probability of propagation using various ion exchanger

*P*

0.6

0.8

1

AMV

*T* = 3K *T* = 5K *T* = 7K

0.2

0.4

Probability of freezing

*P*

0.6

0.8

1

CMT

0 100 200 300 400 500 600

0 100 200 300 400 500 600

time *t*, s

time *t*, s

*T* = 3K *T* = 5K *T* = 7K

> *T* = 3K *T* = 5K *T* = 7K

*T* = 3K *T* = 5K *T* = 7K

CMV membrane CMT membrane

*T* = 3K *T* = 5K *T* = 7K

AMT membrane AMV membrane

0 100 200 300 400 500 600

time *t*, s

0 100 200 300 400 500 600

0

a demerit since the device becomes a little bit complicated.

0.2

0.4

Probability of freezing

0.6

0.8

1

DSV

time *t*, s

*P*

0

0

membranes

0.2

0.4

Probability of freezing

*P*

0.6

0.8

1

AMT

0.2

0.4

Probability of freezing

*P*

0.6

0.8

1

CMV

Fig. 6. Example of capsule having an ion exchange membrane on one side and a rubber sheet on the other side

#### **4. Porous polymer material made by foaming**

In order to solve the problem of complexity, an elastic porous membrane was selected. It is a porous polymer material made by foaming. Similar experiments to the one using the ion exchange membrane were carried out using the membrane. The results are shown in Figs. 7 & 8. Fig. 7 is for 10 wt% NaCl solution having the melting temperature of -6.56 oC and Fig. 8 is for 15 wt% NaCl solution having the melting temperature of -10.89 oC. It can be seen that in the case of 10 wt%, when water in the upper cell solidified, the temperature in the upper cell jumped to the melting temperature. However, since the solution in the lower cell was already in the supercooling state, ice propagated immediately with low degree of supercooling. In the case of 15 wt%, when water in the upper cell solidified, the temperature in the lower cell was above the melting temperature, so no propagation occurred at this stage. After the temperature reached down below the melting temperature for the solution, the ice propagated immediately with a low degree of supercooling.

Fig. 7. Experimental result of propagation using porous polymer membrane with 10 wt% NaCl solution

Influence of the cooling rate on propagation of the ice was examined. Under two kinds of cooling conditions, temperature of the solution in the lower cell may differ to each other at the time of propagation. It is due to the existence of thermal resistance of the capsule. So the

Suppressing Method of Supercooling State in Cool Box Using Membrane 63

location of the ice appearance in the capsule. Especially when liquid in the upper cell was a solution, the probability of propagation became low when the ice appeared at the location far away from the membrane. The reason seems to be that the temperature in the upper cell rises above the melting temperature for the lower cell and also concentration in the upper cell near the membrane becomes higher due to the elimination of solute from the ice during the solidification. Hence the melting temperature in the upper cell becomes lower which

Capsules were made using porous polymer membrane. The material can easily be melted by heating, so as shown in Fig. 10, one sheet of membrane was put on top of the other having a spacing material between them. The reason for the spacing material to put between them was to avoid water in the capsule to be vanished away from the capsule due to the osmotic pressure. The circumference was sealed by heating. Air in the capsule was removed from the pipe and water was installed instead. The thermocouple was inserted from the pipe as

100 ml of 15 wt% of NaCl solution was put inside the beaker and the capsule was installed in the solution. The beaker was cooled down with a constant cooling rate and the temperature at ice propagation was measured. Two types of membranes were selected. One membrane had a thickness of 2.2 mm, the average pore size of 7 m and the porosity of 66 %. The other membrane had a thickness of 3 mm, the average pore size of 7 m and porosity

The results are shown in Fig. 11. The figure shows the frequency distribution against the degree of supercooling at propagation. It can be seen that in both cases, propagation occurred at around 1 K of the degree of supercooling. So it can be said that the capsule is

The instant of propagation was observed. The typical example of the propagation was shown in Fig. 12. The camera was set at the side of the capsule in order to observe the ice appearance on the membrane. It can be seen that ice grew slowly from the membrane.

leads to a decrease of the degree of supercooling in the upper cell.

well. The diameter of the membrane was 25 mm.

Fig. 10. Capsule made of porous polymer membrane

of 75 %.

effective.

phenomenon was confirmed by the following method. The water in the upper cell was completely frozen and the whole apparatus was kept at the melting temperature of the solution in the lower cell. Then, experiments were started with constant cooling rates. The results are shown in Fig. 9. In both cases, there were ice propagations in 80 % of the experiments and the natural freezing in the lower cell occurred in 20 %. Fig. 9 shows that there is no difference in the degree of supercooling at propagation by changing the cooling rate. Hence it was found that the temperature of the membrane is the important factor for propagation of ice.

Fig. 8. Experimental result of propagation using porous polymer membrane with 15 wt% NaCl solution

Fig. 9. Probability of propagation under two different cooling rates

By varying the location of ice appearance in the upper cell, it was found that the time taken for propagation of the ice and the probability of propagation are strongly influenced by the

phenomenon was confirmed by the following method. The water in the upper cell was completely frozen and the whole apparatus was kept at the melting temperature of the solution in the lower cell. Then, experiments were started with constant cooling rates. The results are shown in Fig. 9. In both cases, there were ice propagations in 80 % of the experiments and the natural freezing in the lower cell occurred in 20 %. Fig. 9 shows that there is no difference in the degree of supercooling at propagation by changing the cooling rate. Hence it was found that the temperature of the membrane is the important factor for

> 15wt% NaCl water

melting point (15wt% NaCl)

 0.25 K/min. 0.50 K/min.

*T*

0 1000 2000 3000

Time *t,* s

0 2 4 6

By varying the location of ice appearance in the upper cell, it was found that the time taken for propagation of the ice and the probability of propagation are strongly influenced by the

Degree of supercooling at freezing *T*, K

Fig. 8. Experimental result of propagation using porous polymer membrane with 15 wt%

melting temp. for 15wt%

C

NaCl -10.9 o

propagation of ice.

NaCl solution

0

Fig. 9. Probability of propagation under two different cooling rates

0.1

0.2

0.3

Probability of freezing

*P*

0.4

0.5

0.6

Temperature *T,*

℃

0

location of the ice appearance in the capsule. Especially when liquid in the upper cell was a solution, the probability of propagation became low when the ice appeared at the location far away from the membrane. The reason seems to be that the temperature in the upper cell rises above the melting temperature for the lower cell and also concentration in the upper cell near the membrane becomes higher due to the elimination of solute from the ice during the solidification. Hence the melting temperature in the upper cell becomes lower which leads to a decrease of the degree of supercooling in the upper cell.

Capsules were made using porous polymer membrane. The material can easily be melted by heating, so as shown in Fig. 10, one sheet of membrane was put on top of the other having a spacing material between them. The reason for the spacing material to put between them was to avoid water in the capsule to be vanished away from the capsule due to the osmotic pressure. The circumference was sealed by heating. Air in the capsule was removed from the pipe and water was installed instead. The thermocouple was inserted from the pipe as well. The diameter of the membrane was 25 mm.

Fig. 10. Capsule made of porous polymer membrane

100 ml of 15 wt% of NaCl solution was put inside the beaker and the capsule was installed in the solution. The beaker was cooled down with a constant cooling rate and the temperature at ice propagation was measured. Two types of membranes were selected. One membrane had a thickness of 2.2 mm, the average pore size of 7 m and the porosity of 66 %. The other membrane had a thickness of 3 mm, the average pore size of 7 m and porosity of 75 %.

The results are shown in Fig. 11. The figure shows the frequency distribution against the degree of supercooling at propagation. It can be seen that in both cases, propagation occurred at around 1 K of the degree of supercooling. So it can be said that the capsule is effective.

The instant of propagation was observed. The typical example of the propagation was shown in Fig. 12. The camera was set at the side of the capsule in order to observe the ice appearance on the membrane. It can be seen that ice grew slowly from the membrane.

Suppressing Method of Supercooling State in Cool Box Using Membrane 65

(a) 0min (b) 1min

(c) 2min (d) 3min

Fig. 12. Typical example of photos during propagation

(e) 4min (e) 5min

Fig. 11. Frequency distribution of ice propagation through porous polymer membrane

Fig. 13 shows a typical example of the difference with and without using the capsule. It can be seen that the installation of the capsule was effective for propagation of ice. Fig. 14 shows a typical example of the performance under the repeated use of the capsule. It can be seen that the capsule is effective for a repeated use. However, by repeating the experiment and measuring the concentration inside the capsule, it was found that there was an individuality in the quality of the membrane, such as pore size and the distribution of the pore in the membrane. So, it was rather difficult to produce a membrane with a constant quality.

(a) 0min (b) 1min

64 Supercooling

*T*ave=0.42K

0 0.5 1 1.5 0

0 0.5 1 1.5 0

Degree of supercooling at freezing *T*, K

(b)

Fig. 13 shows a typical example of the difference with and without using the capsule. It can be seen that the installation of the capsule was effective for propagation of ice. Fig. 14 shows a typical example of the performance under the repeated use of the capsule. It can be seen that the capsule is effective for a repeated use. However, by repeating the experiment and measuring the concentration inside the capsule, it was found that there was an individuality in the quality of the membrane, such as pore size and the distribution of the pore in the membrane. So, it was rather difficult to produce a

Fig. 11. Frequency distribution of ice propagation through porous polymer membrane

*h* = 2.2 mm *d* = 7 m

= 66 %

Degree of supercooling at freezing *T*, K

(a)

*T*ave=0.96K

0.1

0.1

*h* = 3 mm *d* = 7 m

= 75 %

0.2

Probability of freezing,

membrane with a constant quality.

*P*

0.3

0.4

0.5

0.2

Probability of freezing,

*P*

0.3

0.4

0.5

(c) 2min (d) 3min

Fig. 12. Typical example of photos during propagation

Suppressing Method of Supercooling State in Cool Box Using Membrane 67

The logic of the propagation of freezing using the capsule is explained as follows. There is a difference in melting temperature between the refrigerant in a cool box and water in the capsule. After installing the capsule in the cool box, the cool box is cooled down. Then the water in the capsule starts to freeze first because the melting temperature is higher. When the temperature in the cool box reaches the melting temperature of refrigerant, the water in the capsule is frozen completely. The volume of the capsule is expanded due to phase change of the water contained inside. So, the membrane is expanded and ice in the capsule is uncovered to the refrigerant through pinhole. Then the pinhole becomes a trigger for the

Experiments were carried out by preparing 100 ml of 15 wt% of NaCl solution in a beaker. The capsule was also installed inside the solution. Temperature inside the capsule and the solution near the capsule were measured. The melting temperature of the solution was - 10.89 oC. The solution in the beaker was cooled down with a constant cooling rate of 0.15

2000 4000 6000 8000

Fig. 16. Typical example of temperature record in a case of 15 wt% NaCl with a styrene

15%NaCl solution

brine

Freezing point -10.89 ℃

water

Time *t*,s

Δ*T*

refrigerant to freeze with a low degree of supercooling.

Fig. 15. Capsule using styrene elastomer

elastomer capsule installed

Temperature

*T*,℃

0

Fig. 13. Typical example of freezing of solution with and without capsule (a) without capsule, (b) with capsule

Fig. 14. Typical example of the performance under the repeated use of the capsule

#### **5. Styrene elastomer membrane with pinhole**

The details of the capsule are shown in Fig. 15. The main body of the capsule is made of polypropylene tube. The open part of the two tubes was facing to each other having 2 mm gap between them. Then styrene elastomer membrane with several pinholes was rolled to cover the gap. The membrane can easily be melted by heating, so it can be glued together. The styrene elastomer has a phenomenon that it is elastic at a low temperature. So, the membrane was enlarged in one direction up to 300 % and 6 holes having 0.4 mm diameter were created. It was confirmed that the hole was closed under the condition of no enlargement, and the elasticity was kept at the low temperature.

(a) (b)

capsule, (b) with capsule

K

**5. Styrene elastomer membrane with pinhole**

⊿*T*pro1=1.6

enlargement, and the elasticity was kept at the low temperature.

Temperature *T,*

℃

0

5

Fig. 13. Typical example of freezing of solution with and without capsule (a) without

Water

Brine

15wt% NaCl solution

⊿*T*pro Temperature of freezing

Freezing point -10.89oC

10000 20000 30000 -20

Fig. 14. Typical example of the performance under the repeated use of the capsule

Time *t,*s

⊿*T*pro2=0.95K⊿*T*pro3=1.42K

The details of the capsule are shown in Fig. 15. The main body of the capsule is made of polypropylene tube. The open part of the two tubes was facing to each other having 2 mm gap between them. Then styrene elastomer membrane with several pinholes was rolled to cover the gap. The membrane can easily be melted by heating, so it can be glued together. The styrene elastomer has a phenomenon that it is elastic at a low temperature. So, the membrane was enlarged in one direction up to 300 % and 6 holes having 0.4 mm diameter were created. It was confirmed that the hole was closed under the condition of no The logic of the propagation of freezing using the capsule is explained as follows. There is a difference in melting temperature between the refrigerant in a cool box and water in the capsule. After installing the capsule in the cool box, the cool box is cooled down. Then the water in the capsule starts to freeze first because the melting temperature is higher. When the temperature in the cool box reaches the melting temperature of refrigerant, the water in the capsule is frozen completely. The volume of the capsule is expanded due to phase change of the water contained inside. So, the membrane is expanded and ice in the capsule is uncovered to the refrigerant through pinhole. Then the pinhole becomes a trigger for the refrigerant to freeze with a low degree of supercooling.

Experiments were carried out by preparing 100 ml of 15 wt% of NaCl solution in a beaker. The capsule was also installed inside the solution. Temperature inside the capsule and the solution near the capsule were measured. The melting temperature of the solution was - 10.89 oC. The solution in the beaker was cooled down with a constant cooling rate of 0.15

Fig. 15. Capsule using styrene elastomer

Fig. 16. Typical example of temperature record in a case of 15 wt% NaCl with a styrene elastomer capsule installed

Suppressing Method of Supercooling State in Cool Box Using Membrane 69

One of the typical examples of the time wise variation of the temperature is shown in Fig. 16. Due to the solidification of water inside the capsule, the temperature of the water jumped to 0 oC. Then, the water kept the temperature for a while and it started to drop down again until the temperature of the solution started to jump. It was the instance of

One of the examples of propagation view is shown in Fig. 17. Since the water inside the capsule was frozen completely at this moment, the color inside the capsule looked white and the middle part of the capsule swelled. The original color of the membrane was transparent and it was difficult to observe the propagation on the surface. So, the inner surface of the membrane was painted in black. From the photos it can be observed that the ice propagation started at two points on the membrane and the ice grew slowly. The reason for the slow growth was because of the low degree of supercooling and the high

A method to induce solidification of supercooled refrigerant using membranes was introduced. Three kinds of membranes were selected, namely, ion exchange membrane, porous elastic polymer membrane and styrene elastomer membrane. The following

1. Cation exchange membrane and anion exchange membrane with different thickness were examined. It was found that cation exchange membrane with thin thickness

2. Porous elastic polymer membrane was used to establish the capsule and experiments were carried out. It was found that the membrane was effective for propagating the ice with a low degree of supercooling. Influence of various factors on propagation of ice through membrane, such as difference in the melting temperature between the refrigerant and the liquid material, difference in the cooling rate, difference in the location of the initial ice appearance, difference in the thickness of membrane, were investigated. It was confirmed that the refrigerant with a low melting temperature can be frozen with a small degree of supercooling by inserting a capsule with water or solution having a higher melting temperature in it and a sheet of membrane to separate between them. The propagation time becomes longer when the difference in the melting temperature of two solutions is big. It is because the solution with higher melting temperature solidifies before the solution with the lower melting temperature reaches its solidification temperature. The temperature rises due to solidification and it needs

3. Styrene elastomer membrane which has high elasticity at low temperature was selected and a small pinhole was put through it under the expanding condition in one direction. The shape of the capsule to suit with the characteristics of expansion was prepared and water was put in the capsule. The capsule was set in the refrigerant and the propagation of freezing with a low degree of supercooling was investigated, experimentally. As a result, due to the volume expansion by freezing, the pinhole on the membrane was expanded and it became a trigger for the refrigerant to freeze under the degree of supercooling at around 2 K. Hence, it can be proved that the idea of installing the

propagation of freezing. The degree of supercooling at freezing was around 2 K.

concentration of the solution.

**6. Conclusion**

conclusions were made.

propagated the ice.

time to lowdown the temperature again.

capsule can suppress the supercooling phenomenon.

K/min. The temperatures in the capsule and outside the capsule were measured and the propagation of freezing was checked. Due to the freezing of the water inside the capsule, the middle part of the capsule expanded and the size of the pinhole became bigger.

(a) 0 s

(b) 30 s

(d) 180 s

Fig. 17. Typical example of ice propagation using styrene elastomer capsule

One of the typical examples of the time wise variation of the temperature is shown in Fig. 16. Due to the solidification of water inside the capsule, the temperature of the water jumped to 0 oC. Then, the water kept the temperature for a while and it started to drop down again until the temperature of the solution started to jump. It was the instance of propagation of freezing. The degree of supercooling at freezing was around 2 K.

One of the examples of propagation view is shown in Fig. 17. Since the water inside the capsule was frozen completely at this moment, the color inside the capsule looked white and the middle part of the capsule swelled. The original color of the membrane was transparent and it was difficult to observe the propagation on the surface. So, the inner surface of the membrane was painted in black. From the photos it can be observed that the ice propagation started at two points on the membrane and the ice grew slowly. The reason for the slow growth was because of the low degree of supercooling and the high concentration of the solution.

#### **6. Conclusion**

68 Supercooling

K/min. The temperatures in the capsule and outside the capsule were measured and the propagation of freezing was checked. Due to the freezing of the water inside the capsule, the

(a) 0 s

(b) 30 s

(c) 60 s

(d) 180 s

Fig. 17. Typical example of ice propagation using styrene elastomer capsule

middle part of the capsule expanded and the size of the pinhole became bigger.

A method to induce solidification of supercooled refrigerant using membranes was introduced. Three kinds of membranes were selected, namely, ion exchange membrane, porous elastic polymer membrane and styrene elastomer membrane. The following conclusions were made.

**1. Introduction**

lithography, materials processing and other.

of information and others.

**application**

**5**

*Bulgaria*

**Supercooling at Melt Growth of Single**

**1.1 Single and mixed fluoride crystal compositions – Properties, peculiarities and**

The artificially grown single fluoride crystal compositions of alkali-earth metals (*MF2*, *M = Ca, Sr, Ba*) as well as the congruently melted compounds of the type *Ca1-xSrxF2* or *Ca1-yBayF2* possess a high cubic crystal symmetry structure that determines isotropic optical properties of these materials and correspondingly – their broad usability in various optics devices. Amongst the single fluoride, CaF2 stands out by its unique optical, physical, and mechanical characteristics: light transmission from far ultraviolet (UV) up to middle infrared (MIR), impressive dispersive power and partial dispersion, controllably low stress-induced birefringence, negligible solubility in water and insolubility in many acids, adequate hardness, fairly good thermal conductivity, high resistance to radiation at low luminescence level. The ability of this crystal being a good matrix for rare earth (RE) and actinide completes its superior characteristics and determines exclusively wide application – by variety of optical elements – in UV/MIR and laser optics, holography, dosimetry,

Similarly to CaF2, the matrix of several congruently melted alkali earth (AE) fluoride crystal systems of the type *M1-xM'xF2* (*M ≠ M'*) can contain highly concentrated activator RE ions. The physical-chemical properties of such systems, being put under effective control by adjustment of growing conditions, provide production of crystals with desired characteristics. To gain experience in this direction is as much important as the applicability of the mixed AE fluoride compounds expands as: windows in powerful optical quantum generators for laser thermonuclear synthesis, excellent luminescence coverage, Anti-Stokes luminophors, radiation dosimeters, various devices for preservation and optical processing

Other mixed systems, formed by congruent melting and crystallization of AE and RE fluorides *MF2*–*REF3*, represent a high-potential source of new materials in many fields of science and technology. Peculiarity of such systems is that the introduction in the matrix of large proportion of RE ions may result in formation of hetero-valence, strongly non-

 **and Mixed Fluorite Compounds:**

**Criteria of Interface Stability**

*Institute of Mineralogy and Crystallography*

Jordan T. Mouchovski

*Bulgarian Academy of Sciences*

#### **7. References**

*Bulgaria*

## **Supercooling at Melt Growth of Single and Mixed Fluorite Compounds: Criteria of Interface Stability**

Jordan T. Mouchovski *Institute of Mineralogy and Crystallography Bulgarian Academy of Sciences*

#### **1. Introduction**

70 Supercooling

Hozumi T., Saito A. & Okawa S. (1999). Research on Effect of Ultrasonic Waves on Freezing

Hozumi T., Saito A., Okawa S. & Matsui T. (2002). Freezing Phenomena of Supercooled

Hozumi T., Saito A., Okawa S. & Matsumura T. (2002). Effect of Bubble Nuclei on Freezing of Supercooled Water, *International Journal of Refrigeration*, 25, 2, 243-249 Hozumi T., Saito A., Okawa S. & Eshita Y. (2005). Effects of shapes of electrodes on freezing

Inada T., Zhang X., Yabe A. & Kozawa Y. (2001). Active control of phase change from

Okawa S., Saito A. & T. Harada T. (1997). Experimental Study on the Effect of the Electric

Okawa S., Saito A. & Fukao T. (1999). Freezing of Supercooled Water by Applying the

Okawa S., Saito A. & Minami R. (2001). The Solidification Phenomenon of the Supercooled Water Containing Solid Particles, *International Journal of Refrigeration*, 24, 1, 108-117 Okawa S., Saito A., Kadoma Y. & Kumano H. (2010). Study on a method to induce freezing

Okawa S. & Taniguchi Y. (2010). Fundamental research on freezing of refrigerant by making

Saito A., Okawa S., Tojiki A., Une H. & Tanogashira K. (1992). Fundamental Research on

Shichiri T. & Araki Y. (1986). Nucleation Mechanism of Ice Crystals under Electrical Effect, *J.*

temperature, *International Journal of Heat and Mass Transfer*, 44, 4523-4531 Ishikiriyama K., Todoki M. & Motomura K. (1995). Pore size distribution (PSD)

JSME Date Book Thermophysical Properties of Fluids (1983). 468 (in Japanese)

*Colloid and Interface Science*,171,92-102

(in CD-ROM), (Tokyo, Japan, June 2010).

*of Heat and Mass Transfer*, 35, [10] 2527-2536

*Crystal Growth*, Vol.78, pp.502-508

*Energy Conversion '97*, 347-352

of Supercooled Water, *6th International Symposium on Thermal Engineering and Science*

Water under Impacts of Ultrasonic Wave, *International Journal of Refrigeration*, 25, 7,

of supercooled water in electric freeze control, *International Journal of Refrigeration*,

supercooled water to ice by ultrasonic vibration, 1. Control of phase-change

measurements of Silica gels by means of differential scanning calorimetry, *Journal of*

Field on the Freezing of Supercooled Water, *Proc. Int. Conf. on Fluid and Thermal*

Electric Charge,*5th ASME-JSME Thermal Engineering Joint Conference*, (in CD-

of supercooled solution using a membrane, *International Journal of Refrigeration*, 33,

use of a membrane, *The 5th Asian Conference on Refrigeration and Air-conditioning*, 042,

External Factors Affecting the Freezing of Supercooled Water, *International Journal*

**7. References**

948-953

28, 389-395

ROM)

7, 1459-1464

*for Cold Regions*, 65-72

#### **1.1 Single and mixed fluoride crystal compositions – Properties, peculiarities and application**

The artificially grown single fluoride crystal compositions of alkali-earth metals (*MF2*, *M = Ca, Sr, Ba*) as well as the congruently melted compounds of the type *Ca1-xSrxF2* or *Ca1-yBayF2* possess a high cubic crystal symmetry structure that determines isotropic optical properties of these materials and correspondingly – their broad usability in various optics devices. Amongst the single fluoride, CaF2 stands out by its unique optical, physical, and mechanical characteristics: light transmission from far ultraviolet (UV) up to middle infrared (MIR), impressive dispersive power and partial dispersion, controllably low stress-induced birefringence, negligible solubility in water and insolubility in many acids, adequate hardness, fairly good thermal conductivity, high resistance to radiation at low luminescence level. The ability of this crystal being a good matrix for rare earth (RE) and actinide completes its superior characteristics and determines exclusively wide application – by variety of optical elements – in UV/MIR and laser optics, holography, dosimetry, lithography, materials processing and other.

Similarly to CaF2, the matrix of several congruently melted alkali earth (AE) fluoride crystal systems of the type *M1-xM'xF2* (*M ≠ M'*) can contain highly concentrated activator RE ions. The physical-chemical properties of such systems, being put under effective control by adjustment of growing conditions, provide production of crystals with desired characteristics. To gain experience in this direction is as much important as the applicability of the mixed AE fluoride compounds expands as: windows in powerful optical quantum generators for laser thermonuclear synthesis, excellent luminescence coverage, Anti-Stokes luminophors, radiation dosimeters, various devices for preservation and optical processing of information and others.

Other mixed systems, formed by congruent melting and crystallization of AE and RE fluorides *MF2*–*REF3*, represent a high-potential source of new materials in many fields of science and technology. Peculiarity of such systems is that the introduction in the matrix of large proportion of RE ions may result in formation of hetero-valence, strongly non-

negative *T*-gradient. The phenomenon appears within a very short zone (layer) ahead the crystallization front (CF) as a result of constitutional changes around the moving *IFL/S* or/and specific heat transport throughout the loaded container (crucible) possessing a complex thermal conductivity. No matter what mechanism results the supercooling, it may affect the smoothness and stability of the CF in micro-level, this way accelerating spontaneous growth of protuberating structures on the *IFL/S* resulting in growth of crystals

The nature of CMSc is liable for explanation on the grounds of *conventional crystallization theory* according to which, an extremely fast ionic exchange provides a local quasiequilibrium on the *IFL/S*. This equilibrium can be disrupted owing to inconformity between the CR and the effective diffusion rates of the ions throughout the supercooled melt zone before their incorporation into the structure of growing crystal. This way the crystalchemical reactions nearby the *IFL/S* are diffusion-controlled that is manifested on the phase diagram as difference between liquidus and solidus curves. Under such conditions, the specific change in melt composition ahead the *IFL/S* causes its geometric and/or morphological instability: the polyhedral crystal form stays unstable, sprouting protrusions at its corners and edges where the degree of supersaturation attains the highest level. The process leads to accelerating spontaneous growth of protruding structures on the *IFL/S* and growth of crystal with cellular substructure, which is transformed into a dendroid structure.

For concentrated, two metal component, congruently or incongruently melted, representing mixed fluoride systems of the types MF2—M'F2 and MF2—REF3, the simplest onedimensional model for *IFM/Cr*-instability considers a normal mechanism in Tiller–Chalmers microscopic approximation (Tiller, 1953), where several factors – the released heat and density jump at crystallization, the contribution of surface energy as well as the anisotropy of segregation coefficient are not taken into consideration. Besides, the low heat of melting for these fluoride systems implies the heat-transport to be neglected at the account of a vast mass transport. Nevertheless when growing large sizes crystals, the mass transport may be concurred strongly by a heat-transport owing to significant radiation flow throughout the growing transparent crystal, surpassing increasingly the conductive flow therein (Deshko et al., 1986, 1990). Disregarding this case, if the thermal field inside the furnace may ensure *T*gradient in the melt at the CF, *G*, that is larger than the gradient of the equilibrium solidification temperature *TL*, the derivation of criterion for *IFL/S*-stability starts from the

Further, the material balance at liquid-solid interface is considered relating the CR *Vcryst*, that is, the growing rate, to interdiffusion coefficient (solute diffusivity) *D* by the equation:

*G = (dT/dz)CF > dTL/dl* (1)

*Vcryst(xS – xL) = –D(dx/dl)* (2)

with cellular structure.

**3. Criteria for interface stability**

**3.1 Constitutional supercooling principle**

inequality (Kuznetsov & Fedorov, 2008).

where *l* is the thickness of the short layer zone ahead the CF.

stoichiometric solid solutions of the type *M1-xRExF2+x*, Here *x* ≤ 0.5 and such imposed variations in solid solution composition can be controlled precisely so that to ensure structural micro-homogeneity of the grown crystals with, eventually, some extraordinary physicochemical properties: mechanical (increasing considerably micro-hardness without any cleavage), electrical (amplifying ion conductivity by several orders of magnitude) as well as spectroscopic (strongly improved optical parameters). These type solid solution fluoride crystal systems are used as: structural materials in UV–IR optics; solid electrolytes with high fluorine ion conductivity fast radiation-resistant scintillators, photo-refractive materials, substrates with controlled lattice parameter and low-temperature heat-insulating material; in quantum electronics as host materials and as active elements in solid-state lasers.

#### **1.2 Growing techniques – Interface stability at melt growth**

*Melt growth* of any single or mixed fluoride systems possesses substantial advantages over *growth from solutions* because it may provide conditions for growing homogeneous crystals with two–three orders of magnitude higher crystallization rate (CR). Understandable, the melt growth turned out to be the only method that is appropriate for developing effective techniques for production of crystals with high optical quality (Yushkin, 1983). Specifically, various modifications of Bridgman-Stockbarger (BS) technique are usually employed whereat a container with molten material is moved in vertical two-zone (or multi-zone) furnace, the axial *T*-gradient of which provides a gradual melt/crystal phase transition (Mouchovski, 2007a). Under thermodynamically grounded segregation effect on liquid/solid interface (*IFL/S*), the distribution of components (impurities) in growing crystal body may become inhomogeneous that causes relevant non-uniformity in its optical characteristics. Nevertheless, in many cases when growing mixed fluoride systems, the residual cationic impurities are of trace concentration so that their impact upon general components' distribution is left negligible.

The growth kinetics itself differs depending on the heat and entropy of melting for particular system, thus determining either face-type (atomically smooth surfaces) or normal (atomically rough surfaces) growth mechanism. As being proved for crystals with fluorite type structure, their heat and entropy of melting turn out sufficiently low for melt growing to proceed by "normal growth" mechanism according to Jackson criterion (Chernov, 1984; Jackson, 2004). In this case the fluoride admixture enriches the crystallization zone (CZ), owing to which *IFL/S* loses stability in conformity with the degree of so called *constitutional melt supercooling* (CMSc).

#### **2. Melt supercooling as physical-chemical phenomenon**

Supercooling phenomenon represents a thermodynamic process of lowering the temperature *T* of a liquid or gaseous substance below its freezing point *Tfr* without solid phase transition to occur. Below *Tfr* crystallization can start in any liquid (melt) in the presence of either spontaneously arisen nucleus or artificially introduced seed crystal or nucleus. When lacking any such nucleus the liquid may stay down to the temperature whereat homogeneous nucleation to start.

Supercooling arises inevitably when artificial crystals have been grown by BS method, that is, in a container moving downwards in parallel to gravity force temperature field with negative *T*-gradient. The phenomenon appears within a very short zone (layer) ahead the crystallization front (CF) as a result of constitutional changes around the moving *IFL/S* or/and specific heat transport throughout the loaded container (crucible) possessing a complex thermal conductivity. No matter what mechanism results the supercooling, it may affect the smoothness and stability of the CF in micro-level, this way accelerating spontaneous growth of protuberating structures on the *IFL/S* resulting in growth of crystals with cellular structure.

The nature of CMSc is liable for explanation on the grounds of *conventional crystallization theory* according to which, an extremely fast ionic exchange provides a local quasiequilibrium on the *IFL/S*. This equilibrium can be disrupted owing to inconformity between the CR and the effective diffusion rates of the ions throughout the supercooled melt zone before their incorporation into the structure of growing crystal. This way the crystalchemical reactions nearby the *IFL/S* are diffusion-controlled that is manifested on the phase diagram as difference between liquidus and solidus curves. Under such conditions, the specific change in melt composition ahead the *IFL/S* causes its geometric and/or morphological instability: the polyhedral crystal form stays unstable, sprouting protrusions at its corners and edges where the degree of supersaturation attains the highest level. The process leads to accelerating spontaneous growth of protruding structures on the *IFL/S* and growth of crystal with cellular substructure, which is transformed into a dendroid structure.

#### **3. Criteria for interface stability**

72 Supercooling

stoichiometric solid solutions of the type *M1-xRExF2+x*, Here *x* ≤ 0.5 and such imposed variations in solid solution composition can be controlled precisely so that to ensure structural micro-homogeneity of the grown crystals with, eventually, some extraordinary physicochemical properties: mechanical (increasing considerably micro-hardness without any cleavage), electrical (amplifying ion conductivity by several orders of magnitude) as well as spectroscopic (strongly improved optical parameters). These type solid solution fluoride crystal systems are used as: structural materials in UV–IR optics; solid electrolytes with high fluorine ion conductivity fast radiation-resistant scintillators, photo-refractive materials, substrates with controlled lattice parameter and low-temperature heat-insulating material; in

*Melt growth* of any single or mixed fluoride systems possesses substantial advantages over *growth from solutions* because it may provide conditions for growing homogeneous crystals with two–three orders of magnitude higher crystallization rate (CR). Understandable, the melt growth turned out to be the only method that is appropriate for developing effective techniques for production of crystals with high optical quality (Yushkin, 1983). Specifically, various modifications of Bridgman-Stockbarger (BS) technique are usually employed whereat a container with molten material is moved in vertical two-zone (or multi-zone) furnace, the axial *T*-gradient of which provides a gradual melt/crystal phase transition (Mouchovski, 2007a). Under thermodynamically grounded segregation effect on liquid/solid interface (*IFL/S*), the distribution of components (impurities) in growing crystal body may become inhomogeneous that causes relevant non-uniformity in its optical characteristics. Nevertheless, in many cases when growing mixed fluoride systems, the residual cationic impurities are of trace concentration so that their impact upon general

The growth kinetics itself differs depending on the heat and entropy of melting for particular system, thus determining either face-type (atomically smooth surfaces) or normal (atomically rough surfaces) growth mechanism. As being proved for crystals with fluorite type structure, their heat and entropy of melting turn out sufficiently low for melt growing to proceed by "normal growth" mechanism according to Jackson criterion (Chernov, 1984; Jackson, 2004). In this case the fluoride admixture enriches the crystallization zone (CZ), owing to which *IFL/S* loses stability in conformity with the degree of so called *constitutional*

Supercooling phenomenon represents a thermodynamic process of lowering the temperature *T* of a liquid or gaseous substance below its freezing point *Tfr* without solid phase transition to occur. Below *Tfr* crystallization can start in any liquid (melt) in the presence of either spontaneously arisen nucleus or artificially introduced seed crystal or nucleus. When lacking any such nucleus the liquid may stay down to the temperature

Supercooling arises inevitably when artificial crystals have been grown by BS method, that is, in a container moving downwards in parallel to gravity force temperature field with

quantum electronics as host materials and as active elements in solid-state lasers.

**1.2 Growing techniques – Interface stability at melt growth**

**2. Melt supercooling as physical-chemical phenomenon**

components' distribution is left negligible.

whereat homogeneous nucleation to start.

*melt supercooling* (CMSc).

#### **3.1 Constitutional supercooling principle**

For concentrated, two metal component, congruently or incongruently melted, representing mixed fluoride systems of the types MF2—M'F2 and MF2—REF3, the simplest onedimensional model for *IFM/Cr*-instability considers a normal mechanism in Tiller–Chalmers microscopic approximation (Tiller, 1953), where several factors – the released heat and density jump at crystallization, the contribution of surface energy as well as the anisotropy of segregation coefficient are not taken into consideration. Besides, the low heat of melting for these fluoride systems implies the heat-transport to be neglected at the account of a vast mass transport. Nevertheless when growing large sizes crystals, the mass transport may be concurred strongly by a heat-transport owing to significant radiation flow throughout the growing transparent crystal, surpassing increasingly the conductive flow therein (Deshko et al., 1986, 1990). Disregarding this case, if the thermal field inside the furnace may ensure *T*gradient in the melt at the CF, *G*, that is larger than the gradient of the equilibrium solidification temperature *TL*, the derivation of criterion for *IFL/S*-stability starts from the inequality (Kuznetsov & Fedorov, 2008).

$$\mathbf{G} = (\mathbf{d}\mathbf{T}/\mathbf{dz})\_{\rm CF} \rhd \mathbf{T}\_1 \mathbf{/dl} \tag{1}$$

where *l* is the thickness of the short layer zone ahead the CF.

Further, the material balance at liquid-solid interface is considered relating the CR *Vcryst*, that is, the growing rate, to interdiffusion coefficient (solute diffusivity) *D* by the equation:

$$V\_{cryst}(\mathbf{x}\_S - \mathbf{x}\_L) = -\mathbf{D} \, (d\mathbf{x}/d\mathbf{l}) \tag{2}$$

which, in case of slightly soluble into the lattice impurities (*ko*<< 1), can be approximate by:

The physical explanation of criterion (10) is grounded on so called *segregation mechanism*, under which impact are all kind of impurities penetrated in trace concentrations within the layer ahead the moving CF. These impurities are usually several metal cations and oxygen containing anions, any concentration gradients of which arisen can not be equalized at a lack of sufficiently intensive convection into the melt bulk and relatively low diffusion rates. Hence, kinetic limitations determine the real crystallization conditions liable for impurities' segregation at the CF. Mathematical this phenomenon is convenient being expressed by introducing in (5) an extra multiplayer, *keff*, the so called effective coefficient of segregation

where the index *∆ = Vcryst δ/D* has a meaning of "*reduced CR*" and *δ* is the thickness of the

Evidently the parameter *∆* decreases with lowering *Vcryst* or *δ* and with an increase of *D* whereat the denominator in (12) enhances, tending to 1 at *∆* → 0. Correspondingly *keff* goes down, tending to the equilibrium segregation coefficient *ko.* The other limiting case (*∆* → ∞) leads to *ko* → 1, that is, the impurity's content in the crystal stays equal to that in

The position of *keff* as regards *ko* depends on whether the equilibrium coefficient stays lower or higher than unity: if *ko > 1*, then the inequalities 1 *< keff < ko* are being fulfilled. Opposite to that, if *ko < 1* (as in the cases for mostly RE in fluorite matrix (Nassau, 1961), then *ko < keff < 1.* The thickness of the diffusion boundary layer *δ*, assessed approximately of the order of 0.1 cm, can be substantially reduced in case of stirring the melt whereat it approaches a limiting value of 0.001 cm, conditioned by liquid to solid adhesion. Thus, providing sufficiently intensive stirring of the melt, one may anticipate considerable enhancement of *IFL/S*-stability. Without such stirring, owing to low diffusion rate in liquid phase for the heterogeneous impurities, they either enrich (at *keff* < 1) or impoverish (at *keff* > 1) the layer ahead the *IFL/S*. This way attained enhancement/lowering of impurities' concentration causes a relevant local lowering of melt temperature in the layer, followed by rejection of the impurities from crystallizing surface and their embedment in crystal lattice. At that the equilibrium freezing temperature in the melt adjacent to CF remains above the real temperature established. As a consequence, numbers of nucleolus centres appear simultaneously in supercooled layer that breaks the morphological *IFL/S* stability. In conformity with that the normal growth turns out replaced by high rate poly-crystalline (dendroidal) solidification with formation of cellular sub-structure. Such trapped impurities appear as well light-scattering centres lowering the

light transmission within corresponding spectral range.

(Sekerka, 1968):

the melt bulk.

diffusion boundary layer.

Here:

*G D/Vcryst > – (RT2/Lmolt)(ko – 1)(1 – ko)xo(b)/ko* = *(RT2/Lmolt)xo(b)(1 – ko)2/ko* (9)

*G D/Vcryst > (RT2/Lmolt)xo(b)[(1/ko) – 1]* (10)

*G/Vcryst > – (m xo/D)[(1 – ko)/ko.]keff* (11)

*keff = ko/[ko.+ (1- ko)exp(-∆)]* (12)

Here *xS* and *xL* are the molar concentrations of solid and liquid phases respectively and *(dx/dl)* – the concentration gradient at the CF.

Eq. (2) unifies the two types mass transport, the jump-like forcing-out (when the equilibrium coefficient of segregation *ko* < 1) and absorption of the second component and its diffusion into the melt bulk or from the bulk (when *ko* > 1).

Using the mathematical expression:

$$\text{d'T\_L/dl} = (\text{dT\_L/dx})(\text{dx/dl}) = m(\text{dx/dl}) \tag{3}$$

where the slope *m* of the liquidus curve (tangent of the slope's angle for the liquidus curve) is being introduced, from (1) and (2) it follows:

$$\mathbf{G} \, \mathbf{D} / V\_{\text{cryt}} \succeq \mathbf{m} \, \Delta \mathbf{x} \tag{4}$$

Here *Δx* = *xS – xL* expresses the apparent discontinuity in component's concentration at the CF.

The inequality (4) is an extension to region of the high concentrations and non-stationary processes of Tiller–Chalmers stability criterion (Chalmers, 1968):

$$\mathbf{G} / V\_{\rm cryt} \succ -m (1 - k\_o) \mathbf{x}\_{v \langle b \rangle} \not{k}\_o \, D \tag{5}$$

where: *xo(b)* is the admixture molar concentration of the second (impurity) component in the melt prior to solidification, equal to its mean concentration in the melt bulk (sufficiently far from the CF), while *ko* is defined by the ratio of solid *xoS* to liquid *xoL* phase concentrations for this component, specified at the CF.

This criterion, derived under suggestions of stationary crystallization caused by CMSc, small concentration for the second component and constant values for *ko* and *m*, has been proved being valid for concentrated melts as well (Kaminskii et al., 2008; Sobolev et al., 2003).

The solution of the van't Hoff equation:

$$\text{dllnk}\_o \text{/dT} = \text{L}\_{mol} \text{/RT} \text{2} \tag{6}$$

for the case of coexisting phases in binary systems with low content of the second component gives the expression:

$$
\Delta m = \Delta T / \Delta \mathbf{x}\_o = (RT^2 / \mathcal{L}\_{mult}) (k\_o - 1) \tag{7}
$$

where *R* is the universal gas constant and *Lmolt* is the enthalpy of melting (latent heat of fusion) for matrix material (the first component).

The application of later equation makes it possible to determine *m* from the liquidus curve at *x* → 0 (initial slope of the curve, *minit*) and, correspondingly, *kolim* – the limiting segregation coefficient of the admixture when its concentration tends to zero. Thence:

$$k\_{alim} = (\mathcal{L}\_{mil} \mathcal{J} \mathcal{R} T^2) m\_{init} + 1 \tag{8}$$

Substituting (7) for *m* in (5) the criterion for interface stability takes the form:

$$\mathbf{T} \cdot \mathbf{D} / \mathbf{V}\_{\mathrm{crypt}} \succeq -(\mathbf{RT}^2 / \mathbf{L}\_{\mathrm{mul}}) (\mathbf{k}\_o - 1)(\mathbf{1} - \mathbf{k}\_o) \mathbf{x}\_{\circ (\mathbf{b})} \Downarrow \mathbf{k}\_o = (\mathbf{RT}^2 / \mathbf{L}\_{\mathrm{mul}}) \mathbf{x}\_{\circ (\mathbf{b})} (\mathbf{1} - \mathbf{k}\_o)^2 / \mathbf{k}\_o \tag{9}$$

which, in case of slightly soluble into the lattice impurities (*ko*<< 1), can be approximate by:

$$\text{G D}/V\_{\text{cryst}} \succ (\text{RT} \, \text{\$\forall L}\_{\text{mult}}) \propto\_{o(b)} [(1/k\_o) - 1] \tag{10}$$

The physical explanation of criterion (10) is grounded on so called *segregation mechanism*, under which impact are all kind of impurities penetrated in trace concentrations within the layer ahead the moving CF. These impurities are usually several metal cations and oxygen containing anions, any concentration gradients of which arisen can not be equalized at a lack of sufficiently intensive convection into the melt bulk and relatively low diffusion rates. Hence, kinetic limitations determine the real crystallization conditions liable for impurities' segregation at the CF. Mathematical this phenomenon is convenient being expressed by introducing in (5) an extra multiplayer, *keff*, the so called effective coefficient of segregation (Sekerka, 1968):

$$\text{G/V}\_{\text{cryst}} \succ - (\text{m x}\_{\text{o}} \text{/D}) [(1 - k\_{\text{o}}) \&\_{\text{o}}] \text{k}\_{\text{cyl}} \tag{11}$$

Here:

74 Supercooling

Here *xS* and *xL* are the molar concentrations of solid and liquid phases respectively and

Eq. (2) unifies the two types mass transport, the jump-like forcing-out (when the equilibrium coefficient of segregation *ko* < 1) and absorption of the second component and its diffusion

where the slope *m* of the liquidus curve (tangent of the slope's angle for the liquidus curve)

Here *Δx* = *xS – xL* expresses the apparent discontinuity in component's concentration at the

The inequality (4) is an extension to region of the high concentrations and non-stationary

where: *xo(b)* is the admixture molar concentration of the second (impurity) component in the melt prior to solidification, equal to its mean concentration in the melt bulk (sufficiently far from the CF), while *ko* is defined by the ratio of solid *xoS* to liquid *xoL* phase concentrations

This criterion, derived under suggestions of stationary crystallization caused by CMSc, small concentration for the second component and constant values for *ko* and *m*, has been proved being valid for concentrated melts as well (Kaminskii et al., 2008; Sobolev et al.,

for the case of coexisting phases in binary systems with low content of the second

where *R* is the universal gas constant and *Lmolt* is the enthalpy of melting (latent heat of

The application of later equation makes it possible to determine *m* from the liquidus curve at *x* → 0 (initial slope of the curve, *minit*) and, correspondingly, *kolim* – the limiting segregation

coefficient of the admixture when its concentration tends to zero. Thence:

Substituting (7) for *m* in (5) the criterion for interface stability takes the form:

*dTL*/*dl = (dTL*/*dx)(dx/dl) = m(dx/dl)* (3)

*G D/Vcryst > m Δx* (4)

*G/Vcryst > – m(1 – ko)xo(b)/ko D* (5)

*dlnko/dT = Lmolt/RT2* (6)

*m* = *ΔT/Δxo = (RT2/Lmolt)(ko – 1)* (7)

*kolim = (Lmolt/RT2)minit + 1* (8)

*(dx/dl)* – the concentration gradient at the CF.

into the melt bulk or from the bulk (when *ko* > 1).

is being introduced, from (1) and (2) it follows:

for this component, specified at the CF.

The solution of the van't Hoff equation:

fusion) for matrix material (the first component).

component gives the expression:

processes of Tiller–Chalmers stability criterion (Chalmers, 1968):

Using the mathematical expression:

CF.

2003).

$$k\_{cfl} = k\_o / [k\_o + (1 - k\_o) \exp(-\Delta)] \tag{12}$$

where the index *∆ = Vcryst δ/D* has a meaning of "*reduced CR*" and *δ* is the thickness of the diffusion boundary layer.

Evidently the parameter *∆* decreases with lowering *Vcryst* or *δ* and with an increase of *D* whereat the denominator in (12) enhances, tending to 1 at *∆* → 0. Correspondingly *keff* goes down, tending to the equilibrium segregation coefficient *ko.* The other limiting case (*∆* → ∞) leads to *ko* → 1, that is, the impurity's content in the crystal stays equal to that in the melt bulk.

The position of *keff* as regards *ko* depends on whether the equilibrium coefficient stays lower or higher than unity: if *ko > 1*, then the inequalities 1 *< keff < ko* are being fulfilled. Opposite to that, if *ko < 1* (as in the cases for mostly RE in fluorite matrix (Nassau, 1961), then *ko < keff < 1.*

The thickness of the diffusion boundary layer *δ*, assessed approximately of the order of 0.1 cm, can be substantially reduced in case of stirring the melt whereat it approaches a limiting value of 0.001 cm, conditioned by liquid to solid adhesion. Thus, providing sufficiently intensive stirring of the melt, one may anticipate considerable enhancement of *IFL/S*-stability. Without such stirring, owing to low diffusion rate in liquid phase for the heterogeneous impurities, they either enrich (at *keff* < 1) or impoverish (at *keff* > 1) the layer ahead the *IFL/S*. This way attained enhancement/lowering of impurities' concentration causes a relevant local lowering of melt temperature in the layer, followed by rejection of the impurities from crystallizing surface and their embedment in crystal lattice. At that the equilibrium freezing temperature in the melt adjacent to CF remains above the real temperature established. As a consequence, numbers of nucleolus centres appear simultaneously in supercooled layer that breaks the morphological *IFL/S* stability. In conformity with that the normal growth turns out replaced by high rate poly-crystalline (dendroidal) solidification with formation of cellular sub-structure. Such trapped impurities appear as well light-scattering centres lowering the light transmission within corresponding spectral range.

1968). Besides, this theory provides a description of time evolution of the perturbed interface, following the alterations in thermal and concentration fields. It seams this theory should replace that of CMSc principle despite giving a much more complicated expression for perturbation *IFL/S* stability criterion (Sekerka, 1965). One may assess the reasonability of such replacement after writing the perturbation *IFL/S* stability criterion in a form usable for

Here, the latent heat of fusion *Lmolt* refers to unit volume of the solid crystal phase; *KL* and *KS* present the thermal conductivities of the liquid and solid phase, respectively; *φ* is function of stability that depends in a complex way on *ko, m*, *xo(b)*, *D*, *Vcryst*, the absolute melting point of the pure solvent (first component), and *Γ*-function, that expresses melt-crystal surface free

Since *Γ* ≤ 0.001, *φ* is usually between 0.8 and 0.9, approaching 1 for negligibly small surface free energy. Such relatively small deviations of *φ* from unity illustrate a tendency of the surface free energy to stabilize slightly the *IFL/S*. Here some comments are useful: the divergence between the results, obtained by criterions (5) and (14), becomes greatest for small concentrations, *xo(b)*, of the second (impurities') component. Thence, for the first, highly concentrated component (the matrix), it exists a certain limiting concentration for the second component below which, even at very fast crystallization rates, a cellular structure could not be formed so that the established planar CF would be absolutely stable. Nevertheless, with increasing the concentration of the second component, the contribution of the surface energy lessens rapidly and may be neglected. It should be notice as well that it is possible the corrections due to surface energy stabilizing effect to become significant around the extreme points upon the melting curve for several mixed

The ratio *[(KS + KL)/2KL]* in (14) may affect tangibly the *IFL/S*-stability only if *KS* is appreciably higher than *KL.* Opposite to that, when thermal conductivities of the two phases are close to each other, this factor stays insignificant for assisting the criterion (14) being fulfilled. However in case the growing medium is optically transparent (as appear mostly single and/or mixed fluoride compounds), not conductivity but radiation should determine the heat transport throughout the load. Besides, the melts of these compounds turn out semi-transparent. Despite the thermal processes for such liquid-solid systems have been described by a system of integral-differential equations solved at specific boundary conditions, the developed on this grounds non-linier stability theory leads inevitably to considerable mathematical complications and difficulties for correct interpretation of the results. Both later could be overcome by using a simple model where thermal conductivities were replaced by their effective analogs, representing a sum of thermal and radiative conductivities for corresponding phases (Mouchovski, 2006). This model has been checked being a good approximation in case of CaF2 single crystal growth at usually imposed BS thermal conditions (Mouchovski, 2007a) where the effective conductivities for both phases were estimated being approximately 20 times higher than

Analyzing further the inequality (14) it is seen the second term on its left-hand side, *(Lmolt/2KL)*, should have stabilizing effect upon interface stability; this effect is as much

*(G/Vcryst) + (Lmolt/2KL) > [–m(1 – ko)xo(b)/ko D][(KS + KL)/2KL] φ* (14)

comparative analysis to inequality (5):

energy.

fluoride solid solutions.

their constituting thermal conductivities.

The approximation (10), giving deviation below 1% for *ko* ≤ 0.01, is especially convenient to demonstrate clearly how it could be attained specific conditions, guaranteeing the interface stability, varying appropriately definite substantial and/or apparatus factors so that the influence of the second (impurity) component to be neglected.

#### **3.2 Function of stability and interdiffusion coefficient**

The right-hand side of inequality (4) is denoted by *F(x)* as function of stability (P.I. Fedorov & P.P. Fedorov, 1982; Turkina et al, 1986; Van-der-Vaal's & Konstamm, 1936), possessing dimensionality of temperature. The function is nonnegative *F(x)* ≥ 0, vanishing at points with pure components and at the extremes (points of congruent melting of the solid solutions) where the liquidus and solidus have gotten tangential points. In the range of small impurity's concentrations *F(x)* approximates a straight line (Djurinskij & Bandurkin, 1979). To a first approximation *F(x)* can be expressed by the *T*-difference between liquidus and solidus curves.

The function of stability reveals definite physical meaning as regards the real crystallization process: there is no concentration supercooling and the CF should keep a planar and stable form for any cases when the value of parametric combination *G D/Vcryst* lies higher at a given *xs* than the corresponding point pertaining to *F(x)*-curve*.* In this combination the furnace design and power supply distribution are decisive for determining the steepness for *G*. After establishment of this gradient, *Vcryst* is under control either by altering the speed of crucible (SC) withdrawal or by cooling slowly the immobile load by Stober growing technique so that both cases the growth to proceed within constitutional regions of interface stability. Such approach requires an eventual compositional dependence for interdiffusion coefficient *D* to be searched by using an expression, following from (4):

$$D = m\,\Delta\text{x}(V\_{cryst}/G) = F(\text{x})\_{crit}(V\_{cryst}/G) \tag{13}$$

where *F*(*x*)*crit* is the critical stability function upon the transition from stable plane CF to unstable one, the values of which function are calculated with sufficient precision from corresponding compositional phase diagram for particular fluoride system. Here special attention should be paid to number of points, for which dataset precision is *in situ* high: the melting temperatures for the end members of studied solid solution, the extremes (maximums or minimums) on the melting curve, and eutectic equilibriums.

#### **3.3 Perturbation theory of interface stability**

The inequalities (5), respectively (9), appear appropriate criterion for quantitative analysis about the influence of any impurities on melt supercooling arising ahead the CF in accordance with their amount and solubility into the structure of growing crystal. Processing the experimental data on hand, concerning *IFL/S* stability of many single crystal compositions and metal alloys, Sekerka (1968) outlined mostly of them being in satisfactory agreement with inequalities (5) or (9). However abiding this criterion, based on purely thermodynamic considerations, failed in several other cases due to its shortcomings. Indeed, the constitutional supercooling principle refers only to liquid phase, thus ignoring the latent heat of fusion and the heat flow throughout the growing crystal. Unlike it, the *perturbation theory of interface stability* has been developed being based upon the dynamics of the whole melt-crystal system (Mullins & Sekerka, 1963, 1964; Sekerka, 1965, 1967, as cited in Sekerka,

The approximation (10), giving deviation below 1% for *ko* ≤ 0.01, is especially convenient to demonstrate clearly how it could be attained specific conditions, guaranteeing the interface stability, varying appropriately definite substantial and/or apparatus factors so that the

The right-hand side of inequality (4) is denoted by *F(x)* as function of stability (P.I. Fedorov & P.P. Fedorov, 1982; Turkina et al, 1986; Van-der-Vaal's & Konstamm, 1936), possessing dimensionality of temperature. The function is nonnegative *F(x)* ≥ 0, vanishing at points with pure components and at the extremes (points of congruent melting of the solid solutions) where the liquidus and solidus have gotten tangential points. In the range of small impurity's concentrations *F(x)* approximates a straight line (Djurinskij & Bandurkin, 1979). To a first approximation *F(x)* can be expressed by the *T*-difference between liquidus

The function of stability reveals definite physical meaning as regards the real crystallization process: there is no concentration supercooling and the CF should keep a planar and stable form for any cases when the value of parametric combination *G D/Vcryst* lies higher at a given *xs* than the corresponding point pertaining to *F(x)*-curve*.* In this combination the furnace design and power supply distribution are decisive for determining the steepness for *G*. After establishment of this gradient, *Vcryst* is under control either by altering the speed of crucible (SC) withdrawal or by cooling slowly the immobile load by Stober growing technique so that both cases the growth to proceed within constitutional regions of interface stability. Such approach requires an eventual compositional dependence for interdiffusion coefficient

where *F*(*x*)*crit* is the critical stability function upon the transition from stable plane CF to unstable one, the values of which function are calculated with sufficient precision from corresponding compositional phase diagram for particular fluoride system. Here special attention should be paid to number of points, for which dataset precision is *in situ* high: the melting temperatures for the end members of studied solid solution, the extremes

The inequalities (5), respectively (9), appear appropriate criterion for quantitative analysis about the influence of any impurities on melt supercooling arising ahead the CF in accordance with their amount and solubility into the structure of growing crystal. Processing the experimental data on hand, concerning *IFL/S* stability of many single crystal compositions and metal alloys, Sekerka (1968) outlined mostly of them being in satisfactory agreement with inequalities (5) or (9). However abiding this criterion, based on purely thermodynamic considerations, failed in several other cases due to its shortcomings. Indeed, the constitutional supercooling principle refers only to liquid phase, thus ignoring the latent heat of fusion and the heat flow throughout the growing crystal. Unlike it, the *perturbation theory of interface stability* has been developed being based upon the dynamics of the whole melt-crystal system (Mullins & Sekerka, 1963, 1964; Sekerka, 1965, 1967, as cited in Sekerka,

(maximums or minimums) on the melting curve, and eutectic equilibriums.

*D* = *m Δx*(*Vcryst*/*G*) *= F*(*x*)*crit*(*Vcryst*/*G*) (13)

influence of the second (impurity) component to be neglected.

**3.2 Function of stability and interdiffusion coefficient**

*D* to be searched by using an expression, following from (4):

**3.3 Perturbation theory of interface stability**

and solidus curves.

1968). Besides, this theory provides a description of time evolution of the perturbed interface, following the alterations in thermal and concentration fields. It seams this theory should replace that of CMSc principle despite giving a much more complicated expression for perturbation *IFL/S* stability criterion (Sekerka, 1965). One may assess the reasonability of such replacement after writing the perturbation *IFL/S* stability criterion in a form usable for comparative analysis to inequality (5):

$$(\text{G/V}\_{\text{crypt}}) + (\text{L}\_{\text{myl}}\% \text{K}\_{\text{L}}) \geqslant [-\text{m}(1 - k\_o)\text{x}\_{o\text{(b}}\% \text{k}\_o \text{ D})] (\text{K}\_S + \text{K}\_{\text{L}}) \prime 2\text{K}\_{\text{L}}) \tag{14}$$

Here, the latent heat of fusion *Lmolt* refers to unit volume of the solid crystal phase; *KL* and *KS* present the thermal conductivities of the liquid and solid phase, respectively; *φ* is function of stability that depends in a complex way on *ko, m*, *xo(b)*, *D*, *Vcryst*, the absolute melting point of the pure solvent (first component), and *Γ*-function, that expresses melt-crystal surface free energy.

Since *Γ* ≤ 0.001, *φ* is usually between 0.8 and 0.9, approaching 1 for negligibly small surface free energy. Such relatively small deviations of *φ* from unity illustrate a tendency of the surface free energy to stabilize slightly the *IFL/S*. Here some comments are useful: the divergence between the results, obtained by criterions (5) and (14), becomes greatest for small concentrations, *xo(b)*, of the second (impurities') component. Thence, for the first, highly concentrated component (the matrix), it exists a certain limiting concentration for the second component below which, even at very fast crystallization rates, a cellular structure could not be formed so that the established planar CF would be absolutely stable. Nevertheless, with increasing the concentration of the second component, the contribution of the surface energy lessens rapidly and may be neglected. It should be notice as well that it is possible the corrections due to surface energy stabilizing effect to become significant around the extreme points upon the melting curve for several mixed fluoride solid solutions.

The ratio *[(KS + KL)/2KL]* in (14) may affect tangibly the *IFL/S*-stability only if *KS* is appreciably higher than *KL.* Opposite to that, when thermal conductivities of the two phases are close to each other, this factor stays insignificant for assisting the criterion (14) being fulfilled. However in case the growing medium is optically transparent (as appear mostly single and/or mixed fluoride compounds), not conductivity but radiation should determine the heat transport throughout the load. Besides, the melts of these compounds turn out semi-transparent. Despite the thermal processes for such liquid-solid systems have been described by a system of integral-differential equations solved at specific boundary conditions, the developed on this grounds non-linier stability theory leads inevitably to considerable mathematical complications and difficulties for correct interpretation of the results. Both later could be overcome by using a simple model where thermal conductivities were replaced by their effective analogs, representing a sum of thermal and radiative conductivities for corresponding phases (Mouchovski, 2006). This model has been checked being a good approximation in case of CaF2 single crystal growth at usually imposed BS thermal conditions (Mouchovski, 2007a) where the effective conductivities for both phases were estimated being approximately 20 times higher than their constituting thermal conductivities.

Analyzing further the inequality (14) it is seen the second term on its left-hand side, *(Lmolt/2KL)*, should have stabilizing effect upon interface stability; this effect is as much

retain the *α*2-phase and it is decomposed. As a consequence of these thermal events, the resultant composite material consists at room temperature of two fluorite phases, *α*1 and *β*. Surprisingly, it posses a unique ionic conductivity that is respectively 25 and 330 times higher than those of the parent BaF2 and CaF2 – phenomenon that was explained in terms of the electrical properties of the material's interfaces (Maier, 1995). Materials with such properties appear of great interest for solid-state ionic opening up new opportunities for

Nevertheless, the mixed AEM fluoride crystals have been studied mostly for their promising lasing properties after doping/co-doping by appropriate trivalent Ln-ions with eventual charge-compensation by univalent alkali ions. For Ln3+-based lasers these properties depend strongly on both the symmetry and strength of the crystal field so they may be purposely

As denoted by Fedovov at al. (1988) the phase diagram and resulting function of stability for any MF2*–*REF3 system is in accordance with the type and sizes of RE ionic radius that, in turn, reflects on relevant interdiffusion coefficient. The authors established a regular dependence of *F*(*x*)*crit* on RE ionic radius, with a clear minimum for M = Ba and Sr, corresponding to transition from *ko* > 1 for the larger cations in the beginning of the Lnseries to *ko* < 1 for the smaller cations pertaining to the yttrium sub-series. In case of M = Ca any regularity was not found due, probably, to fine partitioning of the Ln-series on several sub-series (Djurinskij & Bandurkin, 1979). Further calculations led the authors to interdiffusion coefficient for La (the RE with largest ionic radius), *DLa* into fluorite matrix being independent on partition of *LaF3* second component when its concentration exceeded 0.16 mol.%. At the same time *DRE* was shown to keep a linear dependence on RE-radius,

Extending broadly the investigation of MF2*–*REF3 systems, Kuznetzov and Fedorov (2008) established empirically a clear maximum in solidus compositional dependence of *ko* for SrF2–CeF3 system, expressing it by power function equation, the coefficients of which appear complex functions on data extreme. The relevant *F*(*x*)*crit*, showing a slight maximum (10 K) at *xS* = 0.1 mol.% and nearly zero minimum at *xS* = 0.25, goes up hyperbolically attaining so high values at *xS >* 0.5 that excluded, in practical the formation of solid solution crystals. The same specificity for *F*(*x*)*crit* dependence on *xS* were found out for Nd, Pr and La whereas for the rest REs the authors did not reported of any maximum at low RE content. They announced similar results in case M = Ca where *F*(*x*)*crit* started to increase hyperbolically without any intermediate maximum, being less pronounced for Ce whereas for La *F*(*x*)*crit* tended to linearity as for *xS* < 0.5 its values stayed firmly below 25 K. In case M = Ba, again for Ce, Nd, Pr and La it was established the specific for M = Ca maximum – minimum – hyperbolical increasing succession of *F*(*x*)*crit* dependence where the maximum variations were larger when being displaced towards the lower *xS*. For the rest RE significant alterations in curves' course were obtained that suggests formation of variety of complex structures consisting in large Ba ions and much smaller, different in sizes RE ions. To the same suggestion leads the comparison of *F*(*x*)*crit* vs. RE radius dependence for Ca and Ba respectively; one can see a clear minimum for M = Ca, displaced to larger radiuses for higher concentrations whereas the minimum turns out considerably broadened when M =

engineering medium temperature fluoride ion conductors (Sorokin et al., 2008).

decreasing approximately twice from La to Yb in the range of 10*–*5 cm2/s.

controlled by appropriate modification of the host material.

**4.2 Mixed alkali earth – Rare earth fluoride systems**

pronounced as the latent heat of fusion is larger and/or the thermal conductivity of the melt is lower. For crystals with semi-transparent melts where *KLeff ≈ KLrad >> KLconc* the real contribution of *(Lmolt/2KLeff)* should be relevantly small.

#### **3.4 Normal growth criterion**

Any analysis of supercooling effect, causing transition from mono-crystalline to cellular growth for solid solutions with fluorite structure, will be correct only if normal growth proceeds on a rough surface. It is considered this mechanism being involved when the value of normalized latent heat of fusion *Lmolt/kTm.p.* (*k* – is the Boltzmann constant and *Tm.p.* – the m.p. absolute temperature) stays below 2 (Jackson, 1958) or below 3.5 (Alfintzev et al., 1980). As shown in case this criterion failed, the growth proceeds by formation of two-dimensional nucleation. At that the high anisotropy of the process prerequisites both, the segregation coefficient and the surface energy for considered impurity, to depend strongly on growth direction. In this case besides cells' formation, a laminar distribution of impurity can occur caused by capturing of melt's layer adjacent to the growth surface.

#### **4. Function of stability and properties for mixed fluoride systems**

#### **4.1 Mixed alkali earth fluoride systems**

The alkali earth metal single fluorides MF2 may form tri- or four-component solid solutions of type Ca1-xSrxF2, Ca1-yBayF2 or Ca1-x-yBaxSryF2, which retain the cubic symmetry of the fluorite lattice. However, complete solubility of the starting single fluorides has proved to be possible only for the first system, the compositional phase diagram of which manifests a clear minimum for 0.4 < *x* < 0.5 (Chern'evskaya & Anan'eva, 1966; Klimm et al., 2008; Weller, 1965) while the system's properties appear intermediate between those of CaF2 and SrF2 end members. Correspondingly the compositional dependence for stability function starts from and ends to zero K, being insignificant at the extreme minimum (Klimm et al., 2008). This crystal system has been recently an object of thorough investigation within a large compositional interval for specific case of simultaneous growth of batch of parallel boules with different *x* where for CaF2 was utilized fluorspar, concentrated to above 99.6 wt.% (Mouchovski, 2007b; Mouchovski et al., 2009a, b; Mouchovski, 2011).

Completely different behaviour at room temperature possesses CaF2–BaF2 system owing to a large difference in ionic radius between Ca2+ and Ba2+ and, accordingly, in lattice parameter between CaF2 and BaF2. As a result, after a primary crystallization of a continuous series of Ba1-xCaxF2 fluorite solid solutions with a minimum in the liquidus curve at *x* < 0.5, during the cooling there occurs a high-*T* (at 900 oC) solid-state decomposition into a two-phase mixture of dilute CaF2 and BaF2-based solid solutions (Chernevskaya, & Anan'eva, 1966; Wrubel et al., 2006; Zhigarnovskii & Ippolitov, 1969). The phase diagram of this system has been recently reviewed by Fedorov et al. (2005) who showed it does not contain a continuous solid solution. Thus, on cooling the 70CaF2–30BaF2 melt (mol%), a Ca1 yBayF2 solid solution (*α*-phase) solidifies as a primary phase, which at (1039 ± 5) oC reacts with the melt to form a second fluorite α-phase, Ba1-xCaxF2. This *α*-solid solution decomposes on further lowering the temperature into two phases: *α*1 and *α*2. When the temperature reaches (870 ± 5) oC, it comes up to a eutectoid equilibrium between three fluorite phases: *α*1, *α*2, and *β*. This equilibrium is broken at lower temperature since quenching at a rate of 250 oC/s cannot retain the *α*2-phase and it is decomposed. As a consequence of these thermal events, the resultant composite material consists at room temperature of two fluorite phases, *α*1 and *β*. Surprisingly, it posses a unique ionic conductivity that is respectively 25 and 330 times higher than those of the parent BaF2 and CaF2 – phenomenon that was explained in terms of the electrical properties of the material's interfaces (Maier, 1995). Materials with such properties appear of great interest for solid-state ionic opening up new opportunities for engineering medium temperature fluoride ion conductors (Sorokin et al., 2008).

Nevertheless, the mixed AEM fluoride crystals have been studied mostly for their promising lasing properties after doping/co-doping by appropriate trivalent Ln-ions with eventual charge-compensation by univalent alkali ions. For Ln3+-based lasers these properties depend strongly on both the symmetry and strength of the crystal field so they may be purposely controlled by appropriate modification of the host material.

#### **4.2 Mixed alkali earth – Rare earth fluoride systems**

78 Supercooling

pronounced as the latent heat of fusion is larger and/or the thermal conductivity of the melt is lower. For crystals with semi-transparent melts where *KLeff ≈ KLrad >> KLconc* the real

Any analysis of supercooling effect, causing transition from mono-crystalline to cellular growth for solid solutions with fluorite structure, will be correct only if normal growth proceeds on a rough surface. It is considered this mechanism being involved when the value of normalized latent heat of fusion *Lmolt/kTm.p.* (*k* – is the Boltzmann constant and *Tm.p.* – the m.p. absolute temperature) stays below 2 (Jackson, 1958) or below 3.5 (Alfintzev et al., 1980). As shown in case this criterion failed, the growth proceeds by formation of two-dimensional nucleation. At that the high anisotropy of the process prerequisites both, the segregation coefficient and the surface energy for considered impurity, to depend strongly on growth direction. In this case besides cells' formation, a laminar distribution of impurity can occur

The alkali earth metal single fluorides MF2 may form tri- or four-component solid solutions of type Ca1-xSrxF2, Ca1-yBayF2 or Ca1-x-yBaxSryF2, which retain the cubic symmetry of the fluorite lattice. However, complete solubility of the starting single fluorides has proved to be possible only for the first system, the compositional phase diagram of which manifests a clear minimum for 0.4 < *x* < 0.5 (Chern'evskaya & Anan'eva, 1966; Klimm et al., 2008; Weller, 1965) while the system's properties appear intermediate between those of CaF2 and SrF2 end members. Correspondingly the compositional dependence for stability function starts from and ends to zero K, being insignificant at the extreme minimum (Klimm et al., 2008). This crystal system has been recently an object of thorough investigation within a large compositional interval for specific case of simultaneous growth of batch of parallel boules with different *x* where for CaF2 was utilized fluorspar, concentrated to above 99.6

Completely different behaviour at room temperature possesses CaF2–BaF2 system owing to a large difference in ionic radius between Ca2+ and Ba2+ and, accordingly, in lattice parameter between CaF2 and BaF2. As a result, after a primary crystallization of a continuous series of Ba1-xCaxF2 fluorite solid solutions with a minimum in the liquidus curve at *x* < 0.5, during the cooling there occurs a high-*T* (at 900 oC) solid-state decomposition into a two-phase mixture of dilute CaF2 and BaF2-based solid solutions (Chernevskaya, & Anan'eva, 1966; Wrubel et al., 2006; Zhigarnovskii & Ippolitov, 1969). The phase diagram of this system has been recently reviewed by Fedorov et al. (2005) who showed it does not contain a continuous solid solution. Thus, on cooling the 70CaF2–30BaF2 melt (mol%), a Ca1 yBayF2 solid solution (*α*-phase) solidifies as a primary phase, which at (1039 ± 5) oC reacts with the melt to form a second fluorite α-phase, Ba1-xCaxF2. This *α*-solid solution decomposes on further lowering the temperature into two phases: *α*1 and *α*2. When the temperature reaches (870 ± 5) oC, it comes up to a eutectoid equilibrium between three fluorite phases: *α*1, *α*2, and *β*. This equilibrium is broken at lower temperature since quenching at a rate of 250 oC/s cannot

contribution of *(Lmolt/2KLeff)* should be relevantly small.

caused by capturing of melt's layer adjacent to the growth surface.

**4. Function of stability and properties for mixed fluoride systems**

wt.% (Mouchovski, 2007b; Mouchovski et al., 2009a, b; Mouchovski, 2011).

**3.4 Normal growth criterion**

**4.1 Mixed alkali earth fluoride systems**

As denoted by Fedovov at al. (1988) the phase diagram and resulting function of stability for any MF2*–*REF3 system is in accordance with the type and sizes of RE ionic radius that, in turn, reflects on relevant interdiffusion coefficient. The authors established a regular dependence of *F*(*x*)*crit* on RE ionic radius, with a clear minimum for M = Ba and Sr, corresponding to transition from *ko* > 1 for the larger cations in the beginning of the Lnseries to *ko* < 1 for the smaller cations pertaining to the yttrium sub-series. In case of M = Ca any regularity was not found due, probably, to fine partitioning of the Ln-series on several sub-series (Djurinskij & Bandurkin, 1979). Further calculations led the authors to interdiffusion coefficient for La (the RE with largest ionic radius), *DLa* into fluorite matrix being independent on partition of *LaF3* second component when its concentration exceeded 0.16 mol.%. At the same time *DRE* was shown to keep a linear dependence on RE-radius, decreasing approximately twice from La to Yb in the range of 10*–*5 cm2/s.

Extending broadly the investigation of MF2*–*REF3 systems, Kuznetzov and Fedorov (2008) established empirically a clear maximum in solidus compositional dependence of *ko* for SrF2–CeF3 system, expressing it by power function equation, the coefficients of which appear complex functions on data extreme. The relevant *F*(*x*)*crit*, showing a slight maximum (10 K) at *xS* = 0.1 mol.% and nearly zero minimum at *xS* = 0.25, goes up hyperbolically attaining so high values at *xS >* 0.5 that excluded, in practical the formation of solid solution crystals. The same specificity for *F*(*x*)*crit* dependence on *xS* were found out for Nd, Pr and La whereas for the rest REs the authors did not reported of any maximum at low RE content. They announced similar results in case M = Ca where *F*(*x*)*crit* started to increase hyperbolically without any intermediate maximum, being less pronounced for Ce whereas for La *F*(*x*)*crit* tended to linearity as for *xS* < 0.5 its values stayed firmly below 25 K. In case M = Ba, again for Ce, Nd, Pr and La it was established the specific for M = Ca maximum – minimum – hyperbolical increasing succession of *F*(*x*)*crit* dependence where the maximum variations were larger when being displaced towards the lower *xS*. For the rest RE significant alterations in curves' course were obtained that suggests formation of variety of complex structures consisting in large Ba ions and much smaller, different in sizes RE ions. To the same suggestion leads the comparison of *F*(*x*)*crit* vs. RE radius dependence for Ca and Ba respectively; one can see a clear minimum for M = Ca, displaced to larger radiuses for higher concentrations whereas the minimum turns out considerably broadened when M =

compounds predetermines often the region of permissible alterations for the imposed

The present survey deals with different approaches for providing an efficient control upon morphological and geometric stability of the moving CF, keeping its shape near to planar during the growth of particular single or mixed fluoride systems. The accent is being put on single CaF2 and Ca1-xSrxF2, where both cases natural fluorite (fluorspar) is being used as starting material. The influence of several different impurities (eutectic particles of nonsoluble compositions) or ionized molecules (oxygen-containing anions or cations embedded into the lattice) upon interface stability is analyzed semi-quantitatively for single CaF2 by using simple supercooling criterion. Here the aim is how to be implemented an efficient control on trace levels of residual impurities into starting fluorspar and its melt as well as on

The same supercooling criterion is being utilized for investigating the stability function in case of simultaneously growth of Ca1-xSrxF2 boules with different composition. Here the exact knowledge of liquidus–solidus phase diagram is rather important in particular, when for CaF2 end member is being chosen two types fluorspar, differing each other with RE impurities' content (total amount below 100 ppm). Thus specified phase diagram (with expected decrease of the initial negative slope and displacement of the minimum towards the lower Sr-content *x* (Mouchovski, 2007a) should alter relevantly the effective segregation coefficient, which equilibrium value appears a structurally sensitive characteristic of the mixing processes. Here aim is determination of the compositional dependence for *ko* that will offer scope for hypothesizing about possible mechanisms for incorporation of the

The phase diagram character influences decisively as well the interdiffusion coefficient, so

The optical characteristics of thus grown crystals – as most sensitive properties of the medium – are measured, aimed they being related to corresponding values for stability

As general goal of the survey appears determination of appropriate interval for crucible speed towards the cold furnace zone, within which a normal growth to proceed at minimal supercooling effect in conjunction with the established real temperature distribution into the load. This means to be implemented control upon the technological factors so that: *i*) to be attained controllably sufficiently steep axial (vertical) *T*-gradient into the furnace unit (FU), keeping at that a minimal radial *T*-gradient into the load whereat a planar or slightly convex *IFL/S* to be maintained; *ii*) to be set up an appropriate SC, that to be very close to the real CR

Complex growing methodology is applied to provide minimum contamination into CZ, and

It is implemented efficient deep preliminary purification (PP) of the starting fluorspar for substantial reduction of mostly metal cations (except those pertaining to Ln-group) and some metal oxides resulted from decomposition of accessory minerals accompanying the

that the determination of its compositional dependence is another aim of the study.

other contaminants that enter eventually the CZ during crystallization itself.

second component – Sr ions with CaF2 matrix (first component).

function for comparative analysis to be implemented.

effective control on the thermal condition into the FU.

along the mostly crystal length.

**6. Methodology**

thermal conditions and the speed of crucible withdrawal.

Ba. Accordingly to these results one may anticipate specific behavior of the key physicalchemical properties for studied mixed fluoride systems grounded on their structural peculiarities. Here of particular importance appears the thermal conductivity *T*-dependence mostly used for assessing the eventual application of particular MF2*–*REF3 systems for creation of highly effective solid-state lasers, with RE3+ as activator.

Nowadays it is apprehended M1-xYb*x*F2+*x* systems with *x* < 0.5 being with priority as comparing corresponding M1-xNd*x*F2+*x* systems since the usage of Yb3+ instead Nd3+ as ionactivator in laser materials leads to several approved advantages: *i*) simplicity of the structures of the laser levels for ytterbium; *ii*) smaller difference between the wavelength of pumping and generation of pulses; *iii*) lowered heat losses; *iv*) wider bands of the emission spectrum that is especially convenient for generation of short pulses and creation of tunable lasers with larger lifetime of the upper excited level. Here again the usage of CaF2 for the matrix seams most promising and combination CaF2–YbF3 leads to formation of heterovalent fluoride solid solution Ca1-*x*Yb*x*F2+*x* at *x* ≤ 0.41 (Ito et al., 2004; Lucca et al., 2004; Sobolev, & Fedorov, 1978). The thermal conductivity *T*-dependence for this system has been investigated thoroughly by Popov and co-workers (2008a) that established specific alteration in conjunction with considered compositional interval. At the lowest compositions, *x*≤0.001, a decrease of thermal conductivity with increasing *T* was observed within 50-300 K, that is, up to room temperature. Such behavior is typical for crystalline materials and corresponds to a decrease in the mean free path of phonons with temperature. For *x* within 0.005–0.03, smooth maximums were noted in the curves that turned out displaced downward into the range of high *T* with increasing *x*. Importantly, in the low *T*-interval there occurs a transition from behavior typical of single crystals (with a maximum in *T*-dependence) to behavior characteristic for glasses, with a monotonic decrease in thermal conductivity with *T*decrease. As firmly considered such transition to glasslike structures is connected with formation, accumulation, and agglomeration of clusters of oppositely charged defects, which leads to disturbance of short-range order with retention of long-range order [Fedorov, 1991, 2000; Kazanskii et al., 2005; Sobolev et al., 2003). Within the next compositional interval, *x* = 0.09–0.25, the thermal conductivity, was found out to grow up monotonically with *T*, which is typical behavior for any disordered systems but is completely inadmissible for crystals of simple stoichiometric compounds. Similar glasslike behavior of any concentrated heterovalent solid solutions M1–xRExF2+*x* of REF3 into fluoritetype compounds MF2 (M = Ca, Sr, Ba, Cd, Pb) was noted repeatedly (Fedorov, 2000) for the thermal conductivity as well (Popov et al., 2008b, c). Here unifying result appeared the mean free path for phonons (within 0.09–0.25 compositional interval) was found only weakly *T*-dependent, becoming on the order of unit-cell dimensions (Popov et al., 2006). This fact suggests "growth-in" of clusters into the fluorite-type lattice and appearance of nanoheterogeneity in the relevant solid solutions.

#### **5. Purpose and aims: Control upon interface stability and crystal quality**

It can be distinguished two types of factors, influencing the stability of growing interface: substantial factors that refer to peculiarities of growing compounds, determining liquidus– solidus phase diagram, and technological (apparatus) factors related to configuration of the thermal field, established into the load, and the real CR. Nevertheless the technological factors are in subordinated position to the substantial ones since the proportion of starting compounds predetermines often the region of permissible alterations for the imposed thermal conditions and the speed of crucible withdrawal.

The present survey deals with different approaches for providing an efficient control upon morphological and geometric stability of the moving CF, keeping its shape near to planar during the growth of particular single or mixed fluoride systems. The accent is being put on single CaF2 and Ca1-xSrxF2, where both cases natural fluorite (fluorspar) is being used as starting material. The influence of several different impurities (eutectic particles of nonsoluble compositions) or ionized molecules (oxygen-containing anions or cations embedded into the lattice) upon interface stability is analyzed semi-quantitatively for single CaF2 by using simple supercooling criterion. Here the aim is how to be implemented an efficient control on trace levels of residual impurities into starting fluorspar and its melt as well as on other contaminants that enter eventually the CZ during crystallization itself.

The same supercooling criterion is being utilized for investigating the stability function in case of simultaneously growth of Ca1-xSrxF2 boules with different composition. Here the exact knowledge of liquidus–solidus phase diagram is rather important in particular, when for CaF2 end member is being chosen two types fluorspar, differing each other with RE impurities' content (total amount below 100 ppm). Thus specified phase diagram (with expected decrease of the initial negative slope and displacement of the minimum towards the lower Sr-content *x* (Mouchovski, 2007a) should alter relevantly the effective segregation coefficient, which equilibrium value appears a structurally sensitive characteristic of the mixing processes. Here aim is determination of the compositional dependence for *ko* that will offer scope for hypothesizing about possible mechanisms for incorporation of the second component – Sr ions with CaF2 matrix (first component).

The phase diagram character influences decisively as well the interdiffusion coefficient, so that the determination of its compositional dependence is another aim of the study.

The optical characteristics of thus grown crystals – as most sensitive properties of the medium – are measured, aimed they being related to corresponding values for stability function for comparative analysis to be implemented.

As general goal of the survey appears determination of appropriate interval for crucible speed towards the cold furnace zone, within which a normal growth to proceed at minimal supercooling effect in conjunction with the established real temperature distribution into the load. This means to be implemented control upon the technological factors so that: *i*) to be attained controllably sufficiently steep axial (vertical) *T*-gradient into the furnace unit (FU), keeping at that a minimal radial *T*-gradient into the load whereat a planar or slightly convex *IFL/S* to be maintained; *ii*) to be set up an appropriate SC, that to be very close to the real CR along the mostly crystal length.

#### **6. Methodology**

80 Supercooling

Ba. Accordingly to these results one may anticipate specific behavior of the key physicalchemical properties for studied mixed fluoride systems grounded on their structural peculiarities. Here of particular importance appears the thermal conductivity *T*-dependence mostly used for assessing the eventual application of particular MF2*–*REF3 systems for

Nowadays it is apprehended M1-xYb*x*F2+*x* systems with *x* < 0.5 being with priority as comparing corresponding M1-xNd*x*F2+*x* systems since the usage of Yb3+ instead Nd3+ as ionactivator in laser materials leads to several approved advantages: *i*) simplicity of the structures of the laser levels for ytterbium; *ii*) smaller difference between the wavelength of pumping and generation of pulses; *iii*) lowered heat losses; *iv*) wider bands of the emission spectrum that is especially convenient for generation of short pulses and creation of tunable lasers with larger lifetime of the upper excited level. Here again the usage of CaF2 for the matrix seams most promising and combination CaF2–YbF3 leads to formation of heterovalent fluoride solid solution Ca1-*x*Yb*x*F2+*x* at *x* ≤ 0.41 (Ito et al., 2004; Lucca et al., 2004; Sobolev, & Fedorov, 1978). The thermal conductivity *T*-dependence for this system has been investigated thoroughly by Popov and co-workers (2008a) that established specific alteration in conjunction with considered compositional interval. At the lowest compositions, *x*≤0.001, a decrease of thermal conductivity with increasing *T* was observed within 50-300 K, that is, up to room temperature. Such behavior is typical for crystalline materials and corresponds to a decrease in the mean free path of phonons with temperature. For *x* within 0.005–0.03, smooth maximums were noted in the curves that turned out displaced downward into the range of high *T* with increasing *x*. Importantly, in the low *T*-interval there occurs a transition from behavior typical of single crystals (with a maximum in *T*-dependence) to behavior characteristic for glasses, with a monotonic decrease in thermal conductivity with *T*decrease. As firmly considered such transition to glasslike structures is connected with formation, accumulation, and agglomeration of clusters of oppositely charged defects, which leads to disturbance of short-range order with retention of long-range order [Fedorov, 1991, 2000; Kazanskii et al., 2005; Sobolev et al., 2003). Within the next compositional interval, *x* = 0.09–0.25, the thermal conductivity, was found out to grow up monotonically with *T*, which is typical behavior for any disordered systems but is completely inadmissible for crystals of simple stoichiometric compounds. Similar glasslike behavior of any concentrated heterovalent solid solutions M1–xRExF2+*x* of REF3 into fluoritetype compounds MF2 (M = Ca, Sr, Ba, Cd, Pb) was noted repeatedly (Fedorov, 2000) for the thermal conductivity as well (Popov et al., 2008b, c). Here unifying result appeared the mean free path for phonons (within 0.09–0.25 compositional interval) was found only weakly *T*-dependent, becoming on the order of unit-cell dimensions (Popov et al., 2006). This fact suggests "growth-in" of clusters into the fluorite-type lattice and appearance of

creation of highly effective solid-state lasers, with RE3+ as activator.

nanoheterogeneity in the relevant solid solutions.

**5. Purpose and aims: Control upon interface stability and crystal quality**

It can be distinguished two types of factors, influencing the stability of growing interface: substantial factors that refer to peculiarities of growing compounds, determining liquidus– solidus phase diagram, and technological (apparatus) factors related to configuration of the thermal field, established into the load, and the real CR. Nevertheless the technological factors are in subordinated position to the substantial ones since the proportion of starting Complex growing methodology is applied to provide minimum contamination into CZ, and effective control on the thermal condition into the FU.

It is implemented efficient deep preliminary purification (PP) of the starting fluorspar for substantial reduction of mostly metal cations (except those pertaining to Ln-group) and some metal oxides resulted from decomposition of accessory minerals accompanying the

(Mouchovski et al., 2011). The corresponding light extinction losses per unit optical path are calculated from formulas grounded on Lambert-Beer law approximation where *t* is presented as a sum of absorption and scattering parts and window's plane surface

The stress-induced birefringence in the windows is determined by means of

The concentration and distribution of residual impurities along the height of grown boules is determined applying Atomic Absorption Spectroscopy (AAS), Solid Sampling Electrothermal Atomic Absorption Spectrometry (SS-ETAAS) and Neutron Activation Analysis (NAA) – for the RE elements – techniques (Detcheva & Hassler, 2001; Detcheva &

For assessing the CF position, *xCF*, along the FU it is applied originally developed indirect technique, which involved determination of a Quenched Interface (QI) in multicameral crucible loaded by portions of concentrated grained fluorspar (Mouchovski et al., 1996a).

Empirically derived formulas, given elsewhere (Mouchovski, 2007a), relate *xCF* to *x1*-variable (expressing crucible movement from the starting position) and the changes in the set *T*regimes of furnace heaters. The origin of the basic axial coordinate, *z*, lies on FU crosssection separating AdZ from Z2. The height of crystallizing part of the boules is determined by the difference (*xCF* – *xcon*) where *xcon* defines the distance passed by the plane section of the conical cameras' tips. All variables are reduced appropriately to take dimensionless form. The boules' height *hboule* is expressed in *x1\** units as partition from crucible withdrawal for a

*z = zincon – lcrmov*

where *zincon* is the initial position of cameras' tips wherefrom the crystallization begins to propagate at the chosen starting position of the crucible into Z1 and *lcrmov* represents the

The real *T*-gradient along the FU is assessed approximating the calculated axial *T*-gradient obtained after differentiating the axial *T*-profile of the furnace, measured in empty crucible

The real CR is estimated as the set SC is being corrected accordingly the varying shift in *IFL/S*

The liquidus – solidus curves for particularly studied Ca1-xSrxF2 system (Bulgarian fluorspar as starting material) are obtained by DTA measurements accomplished using a Stenton Red Croft STA instrument in argon flow and using graphite crucibles. The measured points lie within the experimental error on curves estimated by using formula, adopting recently specified phase diagram for the system in case both end members are chemical compounds

*x1\* = (zincon – z)/lcrmov* (15)

 *x1\** (16)

reflectivity is being excluded.

given run by formulas:

and correspondingly:

(Mouchovski et al., 1996).

position during crucible movement.

[see Fig. 3 in Mouchovski et al., 2011]:

polariscope/polarimeter PKS-250.

Havezov, 1994, 2001, 2005, as cited in Mouchovski, 2007).

distance whereto the crucible moves downward into the FU.

basic fluorite. The PP includes consecutive chemical treatment of grained fluorspar portions into HCl and HF acids followed by high-*T* treatment in the presence of melt scavenger (up to 2 wt.% PbF2 or/and ZnF2 additives) so that the purity of the produced polycrystalline/sintered precursors attains level of 99.6 wt.% (Mouchovski et al., 1999; Mouchovski, 2007a).

Crystal growth by BS technique, where the constant speed of crucible withdrawal is stepwise changed between 0.2 and 0.6 cm per hour, is accomplished into two types' multicameral crucibles, the specific construction of which assist for restricting considerably penetration into CZ of undesired oxygen/water vapour ionised molecule from the vacuum chamber environment. Specifically, the sizes of the openings (channels) into inner cameras' lids are adjusted of order of the mean free path of gaseous ions so that their back movement inside the cameras being under control by Knudsen diffusion (Mouchovski, 2007a).

Both types' multicameral crucibles consist of central nest surrounded by several axisymmetric peripheral cameras (Mouchovski et al., 1996). The first type crucible ("Tube support") contains 9 cameras-inserts (with diameter of 2.48 cm), placed in parallel and close to each other in peripheral tubular compartment. The free spaces around and above the inserts are, essentially, additional mass-transport resistance as regards all gaseous species penetrating from vacuum environment. Opposite to that, the 8 cameras (with diameter of 2.56 cm) of the other type crucible ("Revolver support") are surrounded my solid graphite mass, the relatively large thermal conductivity of which ensures the radial heat flux throughout the load (graphite walls surrounding crystallizing molten portions of fluorspar and several connected free spaces) being correspondingly released.

The thermal field in such complex load follows the set *T*-programs for furnace power supply of tailor-made Bridgman-Stockbarger Growth System (BSGS). Its FU consists of close package graphite screens wherein upper and lower heaters differentiate hot (Z1) and cold (Z2) zones, respectively, separated by diaphragm (SD), differentiating relatively long adiabatic zone (AdZ) wherein the radial heat-exchange should expect being insignificant. Supplementary, system of molybdenum shields, part of which is related to moving crucible, has been introduced into the FU. The altering configuration of this system allows much more precise regulation of the thermal field into the FU and into the load, respectively (Mouchovski, 2007a). The details are discussed in sub-section 7.2.

Products of crystal growth are batch of simultaneously grown boules wherefrom are prepared pairs of parallel optical windows, taken from one and the same sections of the cylindrical body of the boules. The windows are finished according to requirements for laser grade CaF2 (Mouchovski et al., 2011) that reduces considerably the reflectivity radiation losses from the parallel planes surfaces within and below UV region. Approximately equal distance (0.6–1 cm) between windows' pairs for each particular batch of boules allows implementation of reliable comparative analysis of composition and characteristics of the grown crystals.

The windows are utilized for determination of: *i*) Ca-proportion, (1-*x*), in the final solid solutions by using of *X*-ray diffractionless analyzer BARS-3; *ii*) light transmission *t*-spectra, measured within UV-IR range by means of high-sensitive spectrophotometer, a Varian Cary type 100. Series of *t*-measurements are also obtained at discrete wavelengths (λ = 248.3 nm, 510.6 nm and 6.45 μm) by applying so called Valour Lasers Irradiation technique (VLIr) (Mouchovski et al., 2011). The corresponding light extinction losses per unit optical path are calculated from formulas grounded on Lambert-Beer law approximation where *t* is presented as a sum of absorption and scattering parts and window's plane surface reflectivity is being excluded.

The stress-induced birefringence in the windows is determined by means of polariscope/polarimeter PKS-250.

The concentration and distribution of residual impurities along the height of grown boules is determined applying Atomic Absorption Spectroscopy (AAS), Solid Sampling Electrothermal Atomic Absorption Spectrometry (SS-ETAAS) and Neutron Activation Analysis (NAA) – for the RE elements – techniques (Detcheva & Hassler, 2001; Detcheva & Havezov, 1994, 2001, 2005, as cited in Mouchovski, 2007).

For assessing the CF position, *xCF*, along the FU it is applied originally developed indirect technique, which involved determination of a Quenched Interface (QI) in multicameral crucible loaded by portions of concentrated grained fluorspar (Mouchovski et al., 1996a).

Empirically derived formulas, given elsewhere (Mouchovski, 2007a), relate *xCF* to *x1*-variable (expressing crucible movement from the starting position) and the changes in the set *T*regimes of furnace heaters. The origin of the basic axial coordinate, *z*, lies on FU crosssection separating AdZ from Z2. The height of crystallizing part of the boules is determined by the difference (*xCF* – *xcon*) where *xcon* defines the distance passed by the plane section of the conical cameras' tips. All variables are reduced appropriately to take dimensionless form. The boules' height *hboule* is expressed in *x1\** units as partition from crucible withdrawal for a given run by formulas:

$$\propto \mathbf{1}^\* = (z\_{incm} - z)\mathbf{\hat{l}}\_{crmw} \tag{15}$$

and correspondingly:

82 Supercooling

basic fluorite. The PP includes consecutive chemical treatment of grained fluorspar portions into HCl and HF acids followed by high-*T* treatment in the presence of melt scavenger (up to 2 wt.% PbF2 or/and ZnF2 additives) so that the purity of the produced polycrystalline/sintered precursors attains level of 99.6 wt.% (Mouchovski et al., 1999;

Crystal growth by BS technique, where the constant speed of crucible withdrawal is stepwise changed between 0.2 and 0.6 cm per hour, is accomplished into two types' multicameral crucibles, the specific construction of which assist for restricting considerably penetration into CZ of undesired oxygen/water vapour ionised molecule from the vacuum chamber environment. Specifically, the sizes of the openings (channels) into inner cameras' lids are adjusted of order of the mean free path of gaseous ions so that their back movement

Both types' multicameral crucibles consist of central nest surrounded by several axisymmetric peripheral cameras (Mouchovski et al., 1996). The first type crucible ("Tube support") contains 9 cameras-inserts (with diameter of 2.48 cm), placed in parallel and close to each other in peripheral tubular compartment. The free spaces around and above the inserts are, essentially, additional mass-transport resistance as regards all gaseous species penetrating from vacuum environment. Opposite to that, the 8 cameras (with diameter of 2.56 cm) of the other type crucible ("Revolver support") are surrounded my solid graphite mass, the relatively large thermal conductivity of which ensures the radial heat flux throughout the load (graphite walls surrounding crystallizing molten portions of fluorspar

The thermal field in such complex load follows the set *T*-programs for furnace power supply of tailor-made Bridgman-Stockbarger Growth System (BSGS). Its FU consists of close package graphite screens wherein upper and lower heaters differentiate hot (Z1) and cold (Z2) zones, respectively, separated by diaphragm (SD), differentiating relatively long adiabatic zone (AdZ) wherein the radial heat-exchange should expect being insignificant. Supplementary, system of molybdenum shields, part of which is related to moving crucible, has been introduced into the FU. The altering configuration of this system allows much more precise regulation of the thermal field into the FU and into the load, respectively

Products of crystal growth are batch of simultaneously grown boules wherefrom are prepared pairs of parallel optical windows, taken from one and the same sections of the cylindrical body of the boules. The windows are finished according to requirements for laser grade CaF2 (Mouchovski et al., 2011) that reduces considerably the reflectivity radiation losses from the parallel planes surfaces within and below UV region. Approximately equal distance (0.6–1 cm) between windows' pairs for each particular batch of boules allows implementation of reliable comparative analysis of composition

The windows are utilized for determination of: *i*) Ca-proportion, (1-*x*), in the final solid solutions by using of *X*-ray diffractionless analyzer BARS-3; *ii*) light transmission *t*-spectra, measured within UV-IR range by means of high-sensitive spectrophotometer, a Varian Cary type 100. Series of *t*-measurements are also obtained at discrete wavelengths (λ = 248.3 nm, 510.6 nm and 6.45 μm) by applying so called Valour Lasers Irradiation technique (VLIr)

inside the cameras being under control by Knudsen diffusion (Mouchovski, 2007a).

and several connected free spaces) being correspondingly released.

(Mouchovski, 2007a). The details are discussed in sub-section 7.2.

and characteristics of the grown crystals.

Mouchovski, 2007a).

$$z = z\_{incm} - l\_{crmvv} \times \chi\_1\*\tag{16}$$

where *zincon* is the initial position of cameras' tips wherefrom the crystallization begins to propagate at the chosen starting position of the crucible into Z1 and *lcrmov* represents the distance whereto the crucible moves downward into the FU.

The real *T*-gradient along the FU is assessed approximating the calculated axial *T*-gradient obtained after differentiating the axial *T*-profile of the furnace, measured in empty crucible (Mouchovski et al., 1996).

The real CR is estimated as the set SC is being corrected accordingly the varying shift in *IFL/S* position during crucible movement.

The liquidus – solidus curves for particularly studied Ca1-xSrxF2 system (Bulgarian fluorspar as starting material) are obtained by DTA measurements accomplished using a Stenton Red Croft STA instrument in argon flow and using graphite crucibles. The measured points lie within the experimental error on curves estimated by using formula, adopting recently specified phase diagram for the system in case both end members are chemical compounds [see Fig. 3 in Mouchovski et al., 2011]:

ahead the CF should exclude mono-crystal growth to proceed. Instead, unsteady thermodynamic conditions will initiate single crystal growth from the heterogeneous region of CaF2 – CaO phase diagram. As a result, it appears a great number of fine fluorite crystals with higher temperature of crystallization than *Tcrys* at the CF, which produce an intensive light-scattering and the grown crystals turn out white-milk in colour, being fully opaque

To stabilize the growing equilibrium in accordance with criterion (18), one can lower appropriately *Vcryst*, that is, the SC. However its excessive decrease turns out unacceptable for industrial crystal production and, besides, may cause considerable evaporation and/or decomposition of the molten material with following break in stoichiometry owing to loss of F- from anionic sub-lattice. Thus for eliminating the CMSc due to the presence of CaO, mostly oxygen contaminants, involved in Ca oxidation, have to be removed before *T* to exceed 880 oC whereat temperatures the rate of chemical oxidation to increase dramatically. Alternatively, the solubility of mostly oxygen-containing contaminants in the solid phase may be risen up considerably by introducing definite amount of tri-valence RE ions into the lattice since their extra-charge leads to local compensation of the negative charge of O2– and they turn out incorporated into anion sub-lattice. Combining the two methods discussed, one may anticipate the established constitutional *T*-gradient to remain satisfactory low so that it may be efficiently compensated by ensuring opposite steep axial *T*-gradient along the

According to criterion (18), the presence of RE or any other metal impurities, the effective segregation coefficient of which remains closed to unity, should not contribute for enhancing the CMSc effect in case their concentration does not exceed several hundreds of ppm in starting fluorspar whereat they could not be treated as second solid solution component. However, in many cases when UV-grade CaF2 has been grown, the RE impurities turns out rather undesired being embedded into the lattice since they act as optical active centres of specific light absorption. At this junction, not embedment into the lattice, but efficient removal of these impurities from the CZ appears obligatory for growing

It should be taken into consideration that melt supercooling, not related to the purity in CZ, can arise being caused by appearance and constant rise up of radiation flow throughout the growing transparent crystal, surpassing increasingly the conductive flow towards the cold

Two growing runs are performed by using correspondingly the two types of multicamera crucibles placed in BSGS thermal field determined by two limiting cases as regards the mutual disposition of the movable part (batch of rings fixed on crucible stem) and fixed part (a thin liner, 15 cm long, mounted through the bore of SD, 2.47 cm thick) of the MoShS. Thus, 9 rings participate in run*1* whereas no rings are slept on the stem in run*2*. Besides the SC is chosen different for the two runs altering in step-wise manner (between 2 and 3 mm per hour – for run*1*) or being constant (equal to 6 mm per hour – for run*2*) – **Fig. 1**. Such SC regimes are thought being in conjunction with the two limiting cases for MoShS effect. These cases concern

within UV–VIS range.

FU and by setting moderately low SC.

crystals with needed optical characteristics.

**7.2 Interface stability during the growth of Ca1-xSrxF2 crystals**

furnace zone.

**7.2.1 Growing runs**

$$T\_{\text{liq}\circ\text{sol}}(\text{Mod}\,\text{l}) = T\_{\text{liq}\circ\text{sol}}(\text{Klim}\,\text{m}) - (\text{l} - \text{x})\,\Delta T\_{x=0} \tag{17}$$

Here a linear decrease is being approximated for calcium proportion (1–*x*) with the initial CaF2 m.p-difference, *ΔTx=0*, equal to 1691 – 1648 = 43 K for the used types of fluorspar.

Thus obtained datasets are fitted by 8-order polynomial with correlation coefficient above 0.99.

The critical values for stability function *F(x)crit* are calculated from the phase diagram where both dependences, *m(xo)* and *Δxo = xS – sL* are indirect functions of *T*.

The compositional dependences of segregation coefficient and interdiffusion coefficient are calculated from (8) and (13) being in conjunction with found alterations of the real *T*gradient and real CR during the growth of each particular boule.

#### **7. Experimental**

#### **7.1 Interface stability during the growth of CaF2 crystals**

#### *Effect of oxygen-containing contaminants on interface stability*

In case of CaF2 crystal growth by using fluorspar, the melt of which is contaminated by oxygen-containing molecules/ions, a sequence of physical processes and chemical reactions leads to eutectics formation since the produced final compound – CaO – is not isomorphic with CaF2 and cannot dissolve into the lattice to form a solid crystal solution.

The eutectics formation may consider on the grounds on supercooling effect arisen. For the purpose the right-hand side of criterion (10) is presented by the crucial *T*-gradient along the layer ahead the CF with thickness *δi*, whereat the equilibrium between the average interdiffusion coefficient for particular impurity "*i*', *Di*, and the maximum linear CR, *Vcryst*, starts to disrupt:

$$
\Delta T\_{cry} \delta \mathbf{j} = (\mathbf{R} \, T\_{cr} \, ^2 \!/ \mathbf{L}\_{milt}) \{ \mathbf{x}\_{o(b)} (\mathbf{k}\_{cf}{}^{-1} - 1) \} (\mathbf{V}\_{crys} \, \!/ \mathbf{D}\_i) \tag{18}
$$

Substituting the proportion between the average diffusion coefficient, *Di*, and *Vcryst* for *δi* in (18), the decrease in crystallization temperature *Tcrys*, due to enrichment of the layer ahead the CF by particular impurity, is expressed by *Тcrys*-quantity according to formula:

$$
\Delta T\_{\rm cru} = (\mathbf{R} T\_{\rm cryst} \mathbf{\hat{\tau}} \mathbf{\hat{\ell}} L\_{\rm mult}) \{ \mathbf{x}\_{\rm o(b)} (\mathbf{k}\_{\rm cyl} \mathbf{\hat{\ell}}^{-1} - 1) \}\tag{19}
$$

In case CaF2 crystal growth proceeds where the used fluorspar possesses m.p. (1648±5) K and *Lmolt* = 7100 cal/mole, then (*RТcryst2/Lmolt*) = (760 ± 2) K. Thus, when the principal contaminant is insoluble CaO with *keff* 1, it is fulfilled (*keff–1 – 1*) 1. At this junction even at relatively small CaO concentration, the product [*xo(b)*(*keff–1 – 1*)] may attain significant value, exceeding of two order of magnitude *xo(b)*. Using arbitrary: *keff* 0.01 and *xo(b)* = 510*–*4 mole parts, the calculated *T*-fall is significant: *Тcrys* 39 K.

On the other hand, *Di* for О*2-* dissolved in CaF2 melt is too low ( 10*–11* m*2*/s) so that these anions are immobile in practical compared to F*-* anions, which, possessing nearly the same effective ionic radius, bear twice less negative electrical charge. At this junction, within the usually attained interval for *Vcryst* that follows approximately the SC (2–10 mm/h), *δi* stays of order of 10*–*3 cm (10 m) while the crucial *T*-gradient, *Тcru/im*, is calculated equal to 3.9104 K/cm (39 K/m). Such extremely steep positive *T*-gradient along the enriched to CaO layer ahead the CF should exclude mono-crystal growth to proceed. Instead, unsteady thermodynamic conditions will initiate single crystal growth from the heterogeneous region of CaF2 – CaO phase diagram. As a result, it appears a great number of fine fluorite crystals with higher temperature of crystallization than *Tcrys* at the CF, which produce an intensive light-scattering and the grown crystals turn out white-milk in colour, being fully opaque within UV–VIS range.