- Email: [email protected]
Method for exploring glass-forming regions in new systems
J O U R N A L OF
ELSEVIER
Journal of Non-Crystalline Solids 201 (1996) 256-261
Letter to the Editor
Method for exploring glass-forming regions in new systems Wancheng Zhou * Northwestern Polytechnical Unicersitv Department of Materials, Science and Engineering, Xian, Shaanxi 710072, People's Republic of China
Received 6 September 1995; revised 26 March 1996
Abstract A method for exploring glass-forming regions in new systems is proposed. The method is based on the fact that compositions near a good glass-forming region have better glass-forming tendencies than those further from the glass-forming region, and that the glass-forming tendency varies continuously with the composition. In this method, the experiment is designated to examine the glass-forming tendencies of compositions on straight lines. When the glass-forming tendencies of compositions on the first line are examined, the second line is set to go through the best glass-forming composition of the first line. After the compositions on the second line are examined, a third line is set to go through the best glass-forming composition on the second line. In the same way, the experiment continues until a glass-forming region is located. Because the experimental results of the previous step are used to guide the experiment in the next step, this method allows us to save experiments and find glass-forming regions.
1. Introduction In research for developing new glasses, an important issue is to find glass-forming regions in a new system. To search for glass-forming regions in a new system, a commonly used method is to do experiments on compositions all over the system with a certain increment. This is a great amount of work. In a ternary system, if the percentage of each component varies from 0 to 100% with an increment of 10%, one will need to do experiments on 66 different compositions. After some good glass-forming compositions have been found, some more experiments within smaller range and with smaller increments are
needed to define the glass-forming region more accurately. To find glass-forming regions with a smaller number of experiments, a better method is needed. Although people have used different methods to explore glass-forming regions in their research, detailed reports on the methods are rare. In this paper, we describe a method for exploring glass-forming regions. In this method, the results from the previous step are used to guide the experiment in the next step, and therefore, it allows us to find glass-forming regions with smaller number of experiments.
2. Basic idea of the method * Tel.: +86-29 849 2374; fax: +86-29 525 0199; e-mail: [email protected].
As we know, a good glass-forming region in a system is always surrounded by a less good glass-
0022-3093/96/$15.00 Copyright © 1996 Elsevier Science B.V. All rights reserved. PII S0022-3093(96)00410-3
W. Zhou / Journal of Non- Co'stalline Solids 201 (1996) 256-261
fa) Fig. 1. Geometrical relations in a set of concentric circles (A) and in ellipses with the same ratio of major axis to minor axis and with overlapping major axis and minor axis (B).
forming region. In fact, if we explore in more details, the glass-forming tendency in any system changes continuously when the composition moves away from the best glass-forming region. This character allows us to save many experiments by using the results we get from the experiment of the previous step to decide the experiment of the next step. Suppose there is a set of concentric circles and the radius difference of any two adjacent circles is small enough. When we draw a straight line anywhere in any direction in this set of concentric circles, the line will get in contact with one of the circles. If we draw a second straight line through the point of contact and perpendicular to the first one, the second line will go through the center of the concentric circles. Fig. I(A) shows some of the circles in the set and the two straight lines. Suppose a set of ellipses has the same ratio of major axis to minor axis, the major and minor axes of all the ellipses are overlapping, and the difference between the lengths of the major (or minor) axes of any two adjacent ellipses is small enough. If we draw two parallel lines anywhere in any direction in the ellipse set, each of the lines will get in contact with one of the ellipses. When we draw a third straight line through the two points of contact, the line will go through the center of the ellipses. Fig. I(B) schematically shows this geometrical relation, and the derivation of this relation is given in Appendix A. In many systems, a glass-forming region can be simplified to a circle or an ellipse that is surrounded by regions with different glass-forming abilities, forming a set of circles or ellipses. Since the glassforming tendency varies continuously, the radii of
257
the circles and the lengths of the ellipses vary continuously. In the case of true circle set or true ellipse set glass-forming regions with the same center, we can develop a method to find the best glass-forming region in the set (the center circle or ellipse) according to the above geometrical relations. Although the actual glass-forming regions in most systems are neither true circles nor true ellipses, the above geometrical relations can still help us in searching for glass-forming regions.
3. Description of the method
3.1. Description of the procedure If there exists a glass-forming region in a ternary system and we examine the glass-forming tendency through an experiment with compositions along any straight line, we will obtain a curve of the glass-forming tendency along the line (Fig. 2(A) and (B)). The composition that has the best glass-forming tendency is the point of contact of the line with one of the circles (or ellipses or irregular circles). Then, we do the same experiment with compositions along a second line that goes through the point of contact (the best glass-forming composition on the first line) and perpendicular to the first line, and along a third line that goes through the best glass-forming composition on the second line and is perpendicular to the 8
A
C
D
E
(A)
(B)
B
B
A
C
(c)
A
C
[Ol
Fig. 2. Schematic explanation of the procedure for exploring glass-forming regions in ternary systems. (The numbers indicate the order of the lines.)
258
W. Zhou / Journal of Non-Crystalline Solids 201 (1996) 256-261
second line. If the glass-forming regions in the system are true circles, the best glass-forming composition on the third line will be the same as that on the second line and this composition is the center of a glass-forming region in the system (Fig. 2(C)). The glass-forming regions in most actual systems are not true circles and the best glass-forming compositions on the second line and on the third line may not be the same. In these cases, we need to do the same experiment with compositions on a fourth line. As the first line and the third line are parallel, the fourth line should go through the center of a glass-forming region in the system if it goes through the two points of contact on the first and the third lines and if the glass-forming regions are true ellipses (Fig. 2(D)). After the best glass-forming composition on the fourth line is found, an experiment with compositions on a fifth line is needed. The fifth line is perpendicular to the fourth line and goes through the best glass-forming composition on the fourth line. If the best glass-forming composition on the fifth line overlaps with that on the fourth line, this composition is the center of a glass-forming region in the system. If not, we need to do more experiments with compositions on a sixth line, a seventh line or even more lines, until we find the best glass-forming composition on a former line overlaps with that on a later line. The following step is to verify and define the glass-forming region by examining the glass-forming tendencies of compositions around the best glass-forming composition on the last line. If any composition is a better glass-forming one than that obtained in the previous step, we need to carry out more experiment with compositions on a line that goes through the best glass-forming composition of the last step and that of the former step. In this way, we can finally find and define a glass-forming egion in the system. Although the glass-forming regions of most systems are neither true circles nor true ellipses, we can get into a glass-forming region in a system by this method with three or four lines in most cases. Fig. 3 shows the application of the above method to a proposed ternary system. The proposed glassforming abilities in this system are shown with the irregular circles. The smallest circle indicates the best glass-forming one, and the larger ones correspond to lower glass-forming tendencies. We assume
B
H
E
A
F
D
GI
C
Fig. 3. Application of the method to exploration for glass-forming regions in a proposed ternary system.
that the glass-forming tendencies in this system are unknown before the experiment. To explore the glass-forming regions, we select BD as the first line, a line with A / C = 1. For this proposed system, experiment will reveal that the best glass-forming composition on this BD line is 'P'. Then, the second line EF is through 'P' and perpendicular to BD. We do not need to do experiments on all compositions along EF. By comparing the glass-forming tendencies of two compositions 'u' and 't' on EF that are on different sides of BD and with the same distances to the cross of BD and EF, we will find that 't' has a better glass-forming tendency than 'u'. This indicates that a glass-forming region is likely to be on the right-hand side of BD, and that examination of glass-forming tendencies with compositions on the left-hand side of BD is not necessary. When the best glass-forming composition on EF is found, we draw a third line HG through it and perpindicular to EF. In the same way, we can decide to ignore the compositions on the both sides of EF. The fourth line PK is through the best glass-forming compositions of the first and third line. We can see that the fourth line goes through the best glass-forming region, and this region will manifest itself when we draw a fifth line. After verification, a glass-forming region in the system is located. There are three alternations of the above procedure. One is to select the first line as A / C = 1, the second, fourth, and sixth lines parallel to AC, and each of the third, fifth, and seventh lines with a constant A / C ratio, the ratio of the best glass-forming composition on the former line, as shown in
W. Zhou / Journal of Non-Crystalline Solids 201 (1996) 256-261 R
^/
I\
B
Ac
^
01~
c tm
B
Fig. 4. Alternations of the procedure to explore glass-forming regions in ternary systems. (The numbers indicate the order of the lines.)
Fig. 4(A). Another alternation is to select the first line as A / C = 1 and each of the following lines being perpendicular to its former line and through the best glass-forming composition on its former line, as shown in Fig. 4(B). The third alternation is to examine the glass-forming tendencies of seven compositions in the center of the system in the first step as shown in Fig. 4(C), and draw the first line through the center one and the best glass-forming one of the seven. The following lines can be decided through any of the above procedures. For a system containing four components A - B C-D, we can simplify it to a quasi-ternary system A - B - C by keeping the amount D constant and using the above procedure. When the best glass-forming region of the quasi-ternary system has been found, the ratio of A:B:C can be kept constant to study compositions with different amounts of D. With the best glass-forming amount of D found in the previous step, another quasi-ternary system A - B - C is studied. This procedure is continued until a glass-forming region in the four-component system is located. This procedure can be extended to systems containing more components.
259
(1) Starting from the second line, each line is divided in two parts by the former line and the best glass-forming composition is on one part. By comparing the glass-forming tendencies of two compositions on the line which are on different sides of the former line and with the same distances to the cross of the two lines, we will know which part of the line contains the best glass-forming composition, and we can ignore the other part of the line, as mentioned in the fourth paragraph of Section 3. (2) The selection of the first line is important. For an entirely unexplored system, the first line can be selected to go through one of the components and keep equal amounts of the other two as this line contains the most information about the system. If we know that one of the components is a glass former, this component is the one that the first line have to go through. If we know that one of the binary systems has a glass-forming region, the first line can go through the center of the binary glassforming region and the opposite component.
3.3. Advantages and disadvantages of the method From the above procedure, we can see that a better glass-forming composition will be found on every additional line. Therefore, if we do enough experiments, we will finally find the best glass-forming region. For most systems, this method allows us to locate glass-forming regions with less experiments and thus save materials and time. The disadvantage of this method is that some glass-forming region(s) may be missing if a system has more than one glass-forming region. One way to prevent this is to examine the glass-forming tendencies of some selected compositions in the unexplored area. In Fig. 3, for example, we can draw a line with A / C = 3, a line that goes through B and the middle of AD, and examine the glass-forming tendencies of compositions on it. By comparing the glass-forming tendencies with compositions on BD, we will know whether there are more glass-forming regions in the ABD area or not.
3.2. Waysfor saving experiments
4. Example of applications
In this method, there are several ways to save experiments and finding glass-forming regions faster.
The above method has been applied to the exploration for high-density fluoride glasses in the HfF4-
LETTER T O THE EDITOR
260
w. Zhou / Journal of Non-Crystalline Solids 201 (1996) 256-261
YbF3-ZnF2-BaF 2 system, and we have found a good glass-forming region with experiments on only 15 compositions. We know a good fluoride glass composition of 28ThFa-28YbF3-28ZnF2-16BaF 2 [1], and the difference of the two systems is that we use HfF4 instead of ThF4. Thus, we set the content of BaF 2 at 16 mol%, the same percentage as in Ref. [1], and study the quasi-ternary system of HfF4-YbF3-ZnF 2. The first line is kept at a ratio of Y b / Z n = 1. With experiment on seven compositions on the first line, we find the best glass-forming composition on the line is at HfF4 = 20 mol%. The second line is set at HfF4 = 20 mol%, a line that is perpendicular to the first line. On this second line, the first two compositions studied are Y b / Z n = 4 0 / 2 4 and 24/40, two points with the same distance to the cross of the first and second line. When we find that Y b / Z n = 2 4 / 4 0 is better glass-forming than the other composition, the next two compositions to be studied are Y b / Z n = 2 0 / 4 4 and 28/36. The best Y b / Z n ratio on the line is proved to be 28/36. The next step is to study the amount of BaF 2 keeping the ratio of H f / Y b / Z n as constant at the best value found. The first two compositions on this line to be studied are 12 and 20 mol% BaF 2. As the glass-forming tendency of the composition of 12 mol% BaF 2 is lower than that of the 16 mol% and that of the 20 mol% is higher than the two, more experiment is needed only for compositions with higher BaF 2 content. The next two compositions to be studied are 24 and 28 mol% BaF 2, and we find 24 mol% BaF 2 is the best glass-forming one of the compositions studied. Thus we find a good glass-forming composition of 18HfF425YbFa-33ZnF2-24BaF 2 with experiment on only 15 compositions. More experiment proves that this composition is in a good glass-forming region. Property measuements show that the glasses in this region have high densities and very good optical transparencies on the untraviolet range [2].
results from the previous step can be used to guide the experiments of the following steps. The method can be applied to ternary systems and can be extended to systems containing four or more components.
Acknowledgements The author thanks Dr Steve W. Martin for his help in this study.
Appendix A As in Fig. 5(A), the equation of an individual ellipse is X2
y2
a--T + -'b-T= 1.
If we use c to represent the ratio of the major axis to the minor axis, i.e., c = a/b, Eq. (1) will be rewritten as X2
y2
(cb)-----g + -~- = 1.
(2)
The slope of the tangent line of the ellipse at a point (X, Y) is dY
X
d-X-
c2y "
(3)
If we draw a straight line OA through the center of the ellipse in any direction, the equation of the line will be Y= kX and thus the straight line and the ellipse crosses at (X1, Y1). From Eq. (2) and the relation of Y = kX, X 1 and Y 1 can be expressed as X l = b c ( k 2 c 2 + 1) - ' / 2 ,
(4)
Y I = b c k ( k 2 c 2 + 1) - . / 2
(5)
Y
5. Summary The proposed method allows us to locate glassforming regions in a new system with a small number of experiments and thus to save materials and time. It is based on the idea that the experimental
(1)
fa) Fig. 5. Schematicexplanation of relationsin ellipses.
W. Zhou / Journal of Non-Crystalline Solids 201 (1996) 256-261
From Eqs. (3)-(5) the slope of the tangent line of the ellipse at (X1, Y1) can be expressed as 1 (dY/dX)x=x,.r=r,
-
c2 k •
(6)
Eq. (6) shows that the slope of the tangent line of an ellipse at the point that the ellipse crosses with a OA line which goes through the center of the ellipse, is dependent only on the slope of the OA line and the ratio of the major axis to the minor axis, independent of the lengths of the axes. Therefore, for a set of ellipses that have the same ratio of major axis to minor axis and have overlapping major and minor axes, the tangent lines of all the ellipses at the points where the ellipses cross with the same straight line
261
which goes through the center of the ellipses, are parallel. In other words, for a set of ellipses that have the same ratio of major axis to minor axis and have overlapping major and minor axes, when each of a set of parallel lines contacts with one of the ellipses, a line that links all the contacting points will be a straight line and its extension line will go through the center of the ellipses Fig. 5(B).
References [1] G. Fonteneau, H. Slim and J. Lucas, J. Non-Cryst. Solids 50 (1982) 61. [2] W. Zhou, S. W. Martin, J. Hauptman and D. Schwellenbach, J. Non-Cryst. Solids 184 (1995) 84.
ELSEVIER
Journal of Non-Crystalline Solids 201 (1996) 256-261
Letter to the Editor
Method for exploring glass-forming regions in new systems Wancheng Zhou * Northwestern Polytechnical Unicersitv Department of Materials, Science and Engineering, Xian, Shaanxi 710072, People's Republic of China
Received 6 September 1995; revised 26 March 1996
Abstract A method for exploring glass-forming regions in new systems is proposed. The method is based on the fact that compositions near a good glass-forming region have better glass-forming tendencies than those further from the glass-forming region, and that the glass-forming tendency varies continuously with the composition. In this method, the experiment is designated to examine the glass-forming tendencies of compositions on straight lines. When the glass-forming tendencies of compositions on the first line are examined, the second line is set to go through the best glass-forming composition of the first line. After the compositions on the second line are examined, a third line is set to go through the best glass-forming composition on the second line. In the same way, the experiment continues until a glass-forming region is located. Because the experimental results of the previous step are used to guide the experiment in the next step, this method allows us to save experiments and find glass-forming regions.
1. Introduction In research for developing new glasses, an important issue is to find glass-forming regions in a new system. To search for glass-forming regions in a new system, a commonly used method is to do experiments on compositions all over the system with a certain increment. This is a great amount of work. In a ternary system, if the percentage of each component varies from 0 to 100% with an increment of 10%, one will need to do experiments on 66 different compositions. After some good glass-forming compositions have been found, some more experiments within smaller range and with smaller increments are
needed to define the glass-forming region more accurately. To find glass-forming regions with a smaller number of experiments, a better method is needed. Although people have used different methods to explore glass-forming regions in their research, detailed reports on the methods are rare. In this paper, we describe a method for exploring glass-forming regions. In this method, the results from the previous step are used to guide the experiment in the next step, and therefore, it allows us to find glass-forming regions with smaller number of experiments.
2. Basic idea of the method * Tel.: +86-29 849 2374; fax: +86-29 525 0199; e-mail: [email protected].
As we know, a good glass-forming region in a system is always surrounded by a less good glass-
0022-3093/96/$15.00 Copyright © 1996 Elsevier Science B.V. All rights reserved. PII S0022-3093(96)00410-3
W. Zhou / Journal of Non- Co'stalline Solids 201 (1996) 256-261
fa) Fig. 1. Geometrical relations in a set of concentric circles (A) and in ellipses with the same ratio of major axis to minor axis and with overlapping major axis and minor axis (B).
forming region. In fact, if we explore in more details, the glass-forming tendency in any system changes continuously when the composition moves away from the best glass-forming region. This character allows us to save many experiments by using the results we get from the experiment of the previous step to decide the experiment of the next step. Suppose there is a set of concentric circles and the radius difference of any two adjacent circles is small enough. When we draw a straight line anywhere in any direction in this set of concentric circles, the line will get in contact with one of the circles. If we draw a second straight line through the point of contact and perpendicular to the first one, the second line will go through the center of the concentric circles. Fig. I(A) shows some of the circles in the set and the two straight lines. Suppose a set of ellipses has the same ratio of major axis to minor axis, the major and minor axes of all the ellipses are overlapping, and the difference between the lengths of the major (or minor) axes of any two adjacent ellipses is small enough. If we draw two parallel lines anywhere in any direction in the ellipse set, each of the lines will get in contact with one of the ellipses. When we draw a third straight line through the two points of contact, the line will go through the center of the ellipses. Fig. I(B) schematically shows this geometrical relation, and the derivation of this relation is given in Appendix A. In many systems, a glass-forming region can be simplified to a circle or an ellipse that is surrounded by regions with different glass-forming abilities, forming a set of circles or ellipses. Since the glassforming tendency varies continuously, the radii of
257
the circles and the lengths of the ellipses vary continuously. In the case of true circle set or true ellipse set glass-forming regions with the same center, we can develop a method to find the best glass-forming region in the set (the center circle or ellipse) according to the above geometrical relations. Although the actual glass-forming regions in most systems are neither true circles nor true ellipses, the above geometrical relations can still help us in searching for glass-forming regions.
3. Description of the method
3.1. Description of the procedure If there exists a glass-forming region in a ternary system and we examine the glass-forming tendency through an experiment with compositions along any straight line, we will obtain a curve of the glass-forming tendency along the line (Fig. 2(A) and (B)). The composition that has the best glass-forming tendency is the point of contact of the line with one of the circles (or ellipses or irregular circles). Then, we do the same experiment with compositions along a second line that goes through the point of contact (the best glass-forming composition on the first line) and perpendicular to the first line, and along a third line that goes through the best glass-forming composition on the second line and is perpendicular to the 8
A
C
D
E
(A)
(B)
B
B
A
C
(c)
A
C
[Ol
Fig. 2. Schematic explanation of the procedure for exploring glass-forming regions in ternary systems. (The numbers indicate the order of the lines.)
258
W. Zhou / Journal of Non-Crystalline Solids 201 (1996) 256-261
second line. If the glass-forming regions in the system are true circles, the best glass-forming composition on the third line will be the same as that on the second line and this composition is the center of a glass-forming region in the system (Fig. 2(C)). The glass-forming regions in most actual systems are not true circles and the best glass-forming compositions on the second line and on the third line may not be the same. In these cases, we need to do the same experiment with compositions on a fourth line. As the first line and the third line are parallel, the fourth line should go through the center of a glass-forming region in the system if it goes through the two points of contact on the first and the third lines and if the glass-forming regions are true ellipses (Fig. 2(D)). After the best glass-forming composition on the fourth line is found, an experiment with compositions on a fifth line is needed. The fifth line is perpendicular to the fourth line and goes through the best glass-forming composition on the fourth line. If the best glass-forming composition on the fifth line overlaps with that on the fourth line, this composition is the center of a glass-forming region in the system. If not, we need to do more experiments with compositions on a sixth line, a seventh line or even more lines, until we find the best glass-forming composition on a former line overlaps with that on a later line. The following step is to verify and define the glass-forming region by examining the glass-forming tendencies of compositions around the best glass-forming composition on the last line. If any composition is a better glass-forming one than that obtained in the previous step, we need to carry out more experiment with compositions on a line that goes through the best glass-forming composition of the last step and that of the former step. In this way, we can finally find and define a glass-forming egion in the system. Although the glass-forming regions of most systems are neither true circles nor true ellipses, we can get into a glass-forming region in a system by this method with three or four lines in most cases. Fig. 3 shows the application of the above method to a proposed ternary system. The proposed glassforming abilities in this system are shown with the irregular circles. The smallest circle indicates the best glass-forming one, and the larger ones correspond to lower glass-forming tendencies. We assume
B
H
E
A
F
D
GI
C
Fig. 3. Application of the method to exploration for glass-forming regions in a proposed ternary system.
that the glass-forming tendencies in this system are unknown before the experiment. To explore the glass-forming regions, we select BD as the first line, a line with A / C = 1. For this proposed system, experiment will reveal that the best glass-forming composition on this BD line is 'P'. Then, the second line EF is through 'P' and perpendicular to BD. We do not need to do experiments on all compositions along EF. By comparing the glass-forming tendencies of two compositions 'u' and 't' on EF that are on different sides of BD and with the same distances to the cross of BD and EF, we will find that 't' has a better glass-forming tendency than 'u'. This indicates that a glass-forming region is likely to be on the right-hand side of BD, and that examination of glass-forming tendencies with compositions on the left-hand side of BD is not necessary. When the best glass-forming composition on EF is found, we draw a third line HG through it and perpindicular to EF. In the same way, we can decide to ignore the compositions on the both sides of EF. The fourth line PK is through the best glass-forming compositions of the first and third line. We can see that the fourth line goes through the best glass-forming region, and this region will manifest itself when we draw a fifth line. After verification, a glass-forming region in the system is located. There are three alternations of the above procedure. One is to select the first line as A / C = 1, the second, fourth, and sixth lines parallel to AC, and each of the third, fifth, and seventh lines with a constant A / C ratio, the ratio of the best glass-forming composition on the former line, as shown in
W. Zhou / Journal of Non-Crystalline Solids 201 (1996) 256-261 R
^/
I\
B
Ac
^
01~
c tm
B
Fig. 4. Alternations of the procedure to explore glass-forming regions in ternary systems. (The numbers indicate the order of the lines.)
Fig. 4(A). Another alternation is to select the first line as A / C = 1 and each of the following lines being perpendicular to its former line and through the best glass-forming composition on its former line, as shown in Fig. 4(B). The third alternation is to examine the glass-forming tendencies of seven compositions in the center of the system in the first step as shown in Fig. 4(C), and draw the first line through the center one and the best glass-forming one of the seven. The following lines can be decided through any of the above procedures. For a system containing four components A - B C-D, we can simplify it to a quasi-ternary system A - B - C by keeping the amount D constant and using the above procedure. When the best glass-forming region of the quasi-ternary system has been found, the ratio of A:B:C can be kept constant to study compositions with different amounts of D. With the best glass-forming amount of D found in the previous step, another quasi-ternary system A - B - C is studied. This procedure is continued until a glass-forming region in the four-component system is located. This procedure can be extended to systems containing more components.
259
(1) Starting from the second line, each line is divided in two parts by the former line and the best glass-forming composition is on one part. By comparing the glass-forming tendencies of two compositions on the line which are on different sides of the former line and with the same distances to the cross of the two lines, we will know which part of the line contains the best glass-forming composition, and we can ignore the other part of the line, as mentioned in the fourth paragraph of Section 3. (2) The selection of the first line is important. For an entirely unexplored system, the first line can be selected to go through one of the components and keep equal amounts of the other two as this line contains the most information about the system. If we know that one of the components is a glass former, this component is the one that the first line have to go through. If we know that one of the binary systems has a glass-forming region, the first line can go through the center of the binary glassforming region and the opposite component.
3.3. Advantages and disadvantages of the method From the above procedure, we can see that a better glass-forming composition will be found on every additional line. Therefore, if we do enough experiments, we will finally find the best glass-forming region. For most systems, this method allows us to locate glass-forming regions with less experiments and thus save materials and time. The disadvantage of this method is that some glass-forming region(s) may be missing if a system has more than one glass-forming region. One way to prevent this is to examine the glass-forming tendencies of some selected compositions in the unexplored area. In Fig. 3, for example, we can draw a line with A / C = 3, a line that goes through B and the middle of AD, and examine the glass-forming tendencies of compositions on it. By comparing the glass-forming tendencies with compositions on BD, we will know whether there are more glass-forming regions in the ABD area or not.
3.2. Waysfor saving experiments
4. Example of applications
In this method, there are several ways to save experiments and finding glass-forming regions faster.
The above method has been applied to the exploration for high-density fluoride glasses in the HfF4-
LETTER T O THE EDITOR
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w. Zhou / Journal of Non-Crystalline Solids 201 (1996) 256-261
YbF3-ZnF2-BaF 2 system, and we have found a good glass-forming region with experiments on only 15 compositions. We know a good fluoride glass composition of 28ThFa-28YbF3-28ZnF2-16BaF 2 [1], and the difference of the two systems is that we use HfF4 instead of ThF4. Thus, we set the content of BaF 2 at 16 mol%, the same percentage as in Ref. [1], and study the quasi-ternary system of HfF4-YbF3-ZnF 2. The first line is kept at a ratio of Y b / Z n = 1. With experiment on seven compositions on the first line, we find the best glass-forming composition on the line is at HfF4 = 20 mol%. The second line is set at HfF4 = 20 mol%, a line that is perpendicular to the first line. On this second line, the first two compositions studied are Y b / Z n = 4 0 / 2 4 and 24/40, two points with the same distance to the cross of the first and second line. When we find that Y b / Z n = 2 4 / 4 0 is better glass-forming than the other composition, the next two compositions to be studied are Y b / Z n = 2 0 / 4 4 and 28/36. The best Y b / Z n ratio on the line is proved to be 28/36. The next step is to study the amount of BaF 2 keeping the ratio of H f / Y b / Z n as constant at the best value found. The first two compositions on this line to be studied are 12 and 20 mol% BaF 2. As the glass-forming tendency of the composition of 12 mol% BaF 2 is lower than that of the 16 mol% and that of the 20 mol% is higher than the two, more experiment is needed only for compositions with higher BaF 2 content. The next two compositions to be studied are 24 and 28 mol% BaF 2, and we find 24 mol% BaF 2 is the best glass-forming one of the compositions studied. Thus we find a good glass-forming composition of 18HfF425YbFa-33ZnF2-24BaF 2 with experiment on only 15 compositions. More experiment proves that this composition is in a good glass-forming region. Property measuements show that the glasses in this region have high densities and very good optical transparencies on the untraviolet range [2].
results from the previous step can be used to guide the experiments of the following steps. The method can be applied to ternary systems and can be extended to systems containing four or more components.
Acknowledgements The author thanks Dr Steve W. Martin for his help in this study.
Appendix A As in Fig. 5(A), the equation of an individual ellipse is X2
y2
a--T + -'b-T= 1.
If we use c to represent the ratio of the major axis to the minor axis, i.e., c = a/b, Eq. (1) will be rewritten as X2
y2
(cb)-----g + -~- = 1.
(2)
The slope of the tangent line of the ellipse at a point (X, Y) is dY
X
d-X-
c2y "
(3)
If we draw a straight line OA through the center of the ellipse in any direction, the equation of the line will be Y= kX and thus the straight line and the ellipse crosses at (X1, Y1). From Eq. (2) and the relation of Y = kX, X 1 and Y 1 can be expressed as X l = b c ( k 2 c 2 + 1) - ' / 2 ,
(4)
Y I = b c k ( k 2 c 2 + 1) - . / 2
(5)
Y
5. Summary The proposed method allows us to locate glassforming regions in a new system with a small number of experiments and thus to save materials and time. It is based on the idea that the experimental
(1)
fa) Fig. 5. Schematicexplanation of relationsin ellipses.
W. Zhou / Journal of Non-Crystalline Solids 201 (1996) 256-261
From Eqs. (3)-(5) the slope of the tangent line of the ellipse at (X1, Y1) can be expressed as 1 (dY/dX)x=x,.r=r,
-
c2 k •
(6)
Eq. (6) shows that the slope of the tangent line of an ellipse at the point that the ellipse crosses with a OA line which goes through the center of the ellipse, is dependent only on the slope of the OA line and the ratio of the major axis to the minor axis, independent of the lengths of the axes. Therefore, for a set of ellipses that have the same ratio of major axis to minor axis and have overlapping major and minor axes, the tangent lines of all the ellipses at the points where the ellipses cross with the same straight line
261
which goes through the center of the ellipses, are parallel. In other words, for a set of ellipses that have the same ratio of major axis to minor axis and have overlapping major and minor axes, when each of a set of parallel lines contacts with one of the ellipses, a line that links all the contacting points will be a straight line and its extension line will go through the center of the ellipses Fig. 5(B).
References [1] G. Fonteneau, H. Slim and J. Lucas, J. Non-Cryst. Solids 50 (1982) 61. [2] W. Zhou, S. W. Martin, J. Hauptman and D. Schwellenbach, J. Non-Cryst. Solids 184 (1995) 84.