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Influence of transparency and grinding on the thermal diffusivity of plasma sprayed ZrO2 with 8 wt.% Y2O3 coatings
Materials Science and Engineering, A151 (1992) L23-L26
L23
Letter
Influence of transparency and grinding on the thermal diffusivity of plasma sprayed Z r O 2 with 8 wt.% Y203coatings Alexandra Rudajevovfi Institute of Plasma Physics, P.O. Box 17, 182 11 Prague8 (Czechoslovakia)
(Received November 11, 1991 ; in revised form November 26, 1991)
Abstract Plasma-sprayed ZrO 2 with 8 wt.% Y203 coatings were found to be partly transparent to impinging radiation. The partial transparency affects the determination of thermal diffusivity by the flash method. The thermal diffusivity of the coatings decreases when their surfaces are ground. From the present work, it ensues that the thermal diffusivity of the coatings is affected by their surface properties.
1. Introduction Plasma-sprayed coatings of partly stabilized Z r O 2 with 8 wt.% Y203 (YSZ) are widely applied as thermal barriers in the aircraft and automobile industry. The application of these coatings as thermal barriers is based on their low thermal diffusivity. Knowing the value of the thermal diffusivity is important both for practical calculations and physical models. The thermal diffusivity of plasma sprayed coatings is always lower than that of compact materials because of their high porosity. The coatings' porosity depends on the technology of their preparation. The plasma technology also determines the coatings' phase composition. The individual phases of zirconium coatings, i.e. cubic, tetragonal and monoclinic phases, exhibit different thermal diffusivity values. On the basis of a comparison of the thermal diffusivity of YSZ coatings with the tetragonal phase dominating, and ZrO2 coatings stabilized by MgO and CaO (MSZ and CSZ), where cubic and monoclinic phases dominate [1-5], we can infer that the thermal diffusivity value of the tetragonal phase in YSZ coatings is lower than that of the mixture of cubic and monoclinic phases in MSZ and CSZ coatings.
Partly stabilized Z r O 2 coatings 0.15-0.40 mm thick are used in thermal barriers. In the application and determination of thermal diffusivity, coatings of such thicknesses cannot be considered equivalent to the bulk material of an identical composition. The thermal diffusivity value of these coatings is affected by surface p r o p e r t i e s . Z r O 2 belongs to a group of refractory materials that are partly transparent to impinging radiation. The transparency of coatings to radiation was studied by Mahlia [6], who stated that 0.14 mm thick coatings were nearly opaque to 1-3 ktm radiation. Liebert [7] found that 0.05 mm thick coatings transmitted as much as 82% radiation in the same spectrum region. The different results of the two authors are accounted for by Pawlowski et al. [4] in terms of different contents of Y203 (8% [6], 13% [7]) and differences in the coating technology. In determining the thermal diffusivity by flash method, the partial radiation transparency cannot be neglected since it would lead to erroneous results. The thinner the coatings, the greater the error. The study of the thermal diffusivity of YSZ coatings relative to the radiation coating transparency will be one of the aims of the present work. YSZ coatings are frequently used in practice and in the research of ground surfaces. Grinding the surface results in a phase transformation [8]. As mentioned above, different phases have different thermal diffusivity values. The influence of grinding upon the thermal diffusivity value will be the next topic of this work. The results will be discussed in connection with the phase transformation induced by grinding.
2. Experimental procedure The coatings were prepared by means of the Plasma-Technik PI-1 set from ZrO 2 with 8 wt.% Y203 (AMDRY 1085). The coatings, 0.2-0.8 mm thick, were sprayed on to a steel pad (Czechoslovak Standard 17 346), 16 mm in diameter. Ground samples were obtained by successively removing 0.2 mm thick layers with a diamond-charged grinding wheel. The surface of the coatings was blackened by carbon. The thermal diffusivity was determined by the flash method, processed for two layers [9]. Measurements were made with the apparatus described earlier [10]. A xenon pulse lamp with a pulse length of 1 ms was used for heating. Thermal diffusivity was measured up to Elsevier Sequoia
L24
A. Rudajevov6
/
Plasma-sprayed ZrO 2 with 8 wt. % ~ 0 3 coatings
950 °C. Phase analysis was carried out with a Siemens Diffractometer D 500 with database JCPDS [11] on samples whose thermal diffusivity was first measured at room temperature.
3. Results
The dependence of the thermal diffusivity upon the coating thickness for blackened and unblackened coating surfaces of ground samples is shown in Fig. 1. This dependence demonstrates the effect of the partial transparency of the coating on its thermal diffusivity, determined by the flash method. The dependence of the thermal diffusivity of ground and unground coatings with blackened surfaces on the coating thickness is illustrated in Fig. 2. Figure 3 shows the temperature dependence of the thermal diffusivity for a ground ....
0.007 ~,~ v
....
,
0.006 0.005 0.004
i
J
i
,
I
0
i
0-5
i
I [mml
Fig. 1. Thermal diffusivity, a, of ZrO2-8wt.%Y203 coatings as a function of the thickness, 1, of coatings: o, unblackened surfaces; e, blackened surfaces. 0.007
,
. . . . . . . . .
0.006
"
~
~
1
o.005
0.004
~ , , , 015 I [mml Fig. 2. Thermal diffusivity, a, of ZrO2-8wt.%Y:O3 coatings as a function of the thickness, 1, of coatings; 1, unground surfaces; 2, ground surfaces.
-'" 0.004
0.003 0.002
4. Discussion It can be presumed from the results given in Fig. 1 that the studied coatings are partly transparent to impinging radiation. Partial transparency cannot be neglected when determining thermal diffusivity by the flash method. The flash method is only applicable to materials where radiation absorption takes place only on the surface. The blackening of the coated surface prevents volume absorption of the radiation but the thermal diffusivity values obtained (true thermal diffusivity) relate solely to the phonon mechanism of heat transfer. In partly transparent coatings, heat transfer is affected through photon as well as phonon mechanisms, and therefore, the true thermal diffusivity value is not characteristic of the real heat transfer through the coating. The effective thermal diffusivity, which would be comprised of both heat transfer processes, would characterize the partly transparent coating correctly. However, this value is not a material constant for the mentioned coatings but depends on the coating thickness and the wavelength. The following discussion concerns true thermal diffusivity which is a material constant and will be further referred to as thermal diffusivity. TABLE 1. Phase composition of the coatings (conventional access to the evaluation) [11] Thickness of coating (mm)
0.005
t~
coating 0.8 mm thick. The dependence has an analogical pattern in unground coatings. After the temperature dependence of the thermal diffusivity had been measured up to 950 °C and the sample had cooled down to room temperature, the thermal diffusivity value increased by about 7%. The porosity of the coatings was about 10%. The results of a quantitative phase analysis of ground and unground coatings are presented in Table 1. All ground coatings are characterized by the phase transformation of the cubic portion into the tetragonal phase.
.
.
.
.
1
500
,
,
,
i
I
i
i
1000 T [ o c ]
Fig. 3. Thermal diffusivity, a, of ZrO2-8wt.%Y:O3 coatings as a function of temperature T: o, results from this work; e, results from ref. 4.
0.2 Unwound Ground 0.4 Unground Ground 0.6 Unground Ground
Phase composition (vol.%) Cubic
Tetragonal
Monoclinic
55 44
33 44
12 12
60 49
30 39
10 12
54 48
34 40
12 12
A. Rudajevov~
/
Plasma-sprayed ZrO 2 with 8 wt. % ~ 0 s coatings
It is evident from the results that surface grinding induced the diffusionless phase transformation of the cubic phase into a tetragonal phase in the coating. The Y redistribution does not take place in the tetragonal phase, and consequently, no transformation of the tetragonal phase into the monoclinic one is detected in the X-ray analysis [11]. From the ratio of the intensity of the (002) and (200) peaks, it is deduced that ferroelastic domain switching takes place [12]. The thermal diffusivity varies along different crystallographic axes. The problem of the ferroelastic domain switching and the thermal diffusivity variations will be studied further. On the basis of the present results it can be presumed that in the coatings studied here the cubic phase has a larger thermal diffusivity than the tetragonal phase. The phase transformation induced by grinding occurs only in the surface layer and cannot be neglected in the determination of the thermal diffusivity value of the plasma sprayed coatings which are several tenths of a millimetre thick. During vertical heat flow on the coated surface, the total thermal conductivity k r of a two-layer coating is characterized by the relation:
k,-k
kl "k~
(1)
where k is the thermal conductivity and V the relative volume. Index 1 refers to the layer with the induced phase transformation, index 2 to the remaining part of the coating. Thermal conductivity values are given as the product of thermal diffusivity, density and specific heat. Relation (1) can be applied to the problem of the coating's transparency to impinging radiation. In this case, V~ is the relative volume of the coating layer, whose radiation transparency is larger than zero. The thermal conductivity kl characterizes the heat transfer by means of photon and phonon processes. V2 is the relative volume of the layer of coating in which the energy is only transferred by conductance and the thermal conductivity k 2 refers to the phonon heat transfer process only. The thermal conductivity k t is, in all cases, larger than k 2. For partly transparent materials, Gardon [13] defined a so-called infinite thickness l=, which is the thickness where reflection and absorption take place in the course of irradiation of the sample in the surface only, i.e. V~ -*0 and k T ~ k 2 . It is the absorption coefficient which characterizes the infinite thickness and which depends on the wavelength of the impinging radiation. From the point of view of the experimental determination of the thermal diffusivity value of coatings thicker than the infinite thickness, the thermal diffusivities of blackened and unblackened surfaces are approximately identical. In Fig. 1 it is evident that the coatings under study have l~ >/0.8 ram.
L25
The upper curve of Fig. 3 represents the temperature dependence of the thermal diffusivity of the coatings studied here, the lower one represents the dependence given in ref. 4. In the cited work, measurements of the thermal diffusivity were made by the flash method in ground samples 1 mm thick with 10% porosity and 9 5% tetragonal phase content. It is evident from the results and the discussion of the present work that the determination of the thermal diffusivity of partly transparent coatings is not simple and that it is necessary to analyse the conditions under which the given values are determined before applying them in practical calculations and physical models. The correct value, including both photon and phonon processes of heat transfer, cannot be determined by the existing experimental methods.
5. Conclusion
It is shown that the YSZ plasma sprayed coatings are partly transparent to impinging radiation. This partial transparency cannot be neglected in coatings that are several tenths of a millimetre thick when determining the thermal diffusivity by the flash method. Grinding the coatings causes a decrease in the thermal diffusivity value. The decrease is larger when the coating thickness is smaller. During grinding, the cubic portion transforms to the tetragonal phase. The cubic phase of the YSZ coatings has a larger thermal diffusivity value than the tetragonal phase. In addition, the thermal diffusivity changes induced by grinding are connected with the ferroelastic domain switching. It ensues from the work that when determining the thermal diffusivity of coatings several tenths of a millimetre thick, we have to consider the influence of the surface properties upon the total thermal diffusivity value. When comparing and applying the experimentally established thermal diffusivity values, it is important to consider not only the method of measuring but also the method of preparing the samples.
Acknowledgment
The author wishes to thank Dr. Zeman and Dr. Cepera for the X-ray measurements.
References 1 W. J. Buykx and M. V. Swain, Proc. 2nd Int. Conf. on the Science Technology of Zirconia, Stuttgart, 1983, pp. 518-527. 2 D. Bernard, M. Vardele and P. Fauchais, Colloque de Physique, 51 (1990) C5-331-341.
L26
A. Rudajevov6
/
Plasma-sprayed ZrO 2 with 8 wt. % Y,. OF coatings
3 J. R. Brandon and R. Taylor, Surf. Coat. Technol., 39/40 (1989) 143-151. 4 L. Pawlowski, D. Lombard, A. Mahlia, Ch. Martin and P. Fauchais, High Temp.-High Press., 16 (1984) 347-359. 5 R. Brandt, High Temp.-High Press., 13 (1981 ) 79-88. 6 A. Mahlia, Thesis oflll Cycle, University of Limoges, 1983. 7 C.H. Liebert, Thin Solid Films, 53 (1978) 235-240. 8 R. Stevens, Zirconia and Zirconia Ceramics, Magnesium Elektron Ltd., 1986.
9 K. B. Larson and K. Koyama, J. Appl. Phys., 39 (9) (1968) 4408-4416. 10 S. Rudajevov~i, Silik6ty, 32 (1989) 343-350. 11 J. Zeman and M. (~epera, Report vO 070 Brno, 001-2151, 1991. 12 K. Mehta, J. F. Jue and A. V. Vikar, J. Am. Ceram. Soc., 73 (6)(1990) 1777-1779. 13 R. Gardon, J. Am. Ceram. Soc., 39 (8)(1956) 278-287.
L23
Letter
Influence of transparency and grinding on the thermal diffusivity of plasma sprayed Z r O 2 with 8 wt.% Y203coatings Alexandra Rudajevovfi Institute of Plasma Physics, P.O. Box 17, 182 11 Prague8 (Czechoslovakia)
(Received November 11, 1991 ; in revised form November 26, 1991)
Abstract Plasma-sprayed ZrO 2 with 8 wt.% Y203 coatings were found to be partly transparent to impinging radiation. The partial transparency affects the determination of thermal diffusivity by the flash method. The thermal diffusivity of the coatings decreases when their surfaces are ground. From the present work, it ensues that the thermal diffusivity of the coatings is affected by their surface properties.
1. Introduction Plasma-sprayed coatings of partly stabilized Z r O 2 with 8 wt.% Y203 (YSZ) are widely applied as thermal barriers in the aircraft and automobile industry. The application of these coatings as thermal barriers is based on their low thermal diffusivity. Knowing the value of the thermal diffusivity is important both for practical calculations and physical models. The thermal diffusivity of plasma sprayed coatings is always lower than that of compact materials because of their high porosity. The coatings' porosity depends on the technology of their preparation. The plasma technology also determines the coatings' phase composition. The individual phases of zirconium coatings, i.e. cubic, tetragonal and monoclinic phases, exhibit different thermal diffusivity values. On the basis of a comparison of the thermal diffusivity of YSZ coatings with the tetragonal phase dominating, and ZrO2 coatings stabilized by MgO and CaO (MSZ and CSZ), where cubic and monoclinic phases dominate [1-5], we can infer that the thermal diffusivity value of the tetragonal phase in YSZ coatings is lower than that of the mixture of cubic and monoclinic phases in MSZ and CSZ coatings.
Partly stabilized Z r O 2 coatings 0.15-0.40 mm thick are used in thermal barriers. In the application and determination of thermal diffusivity, coatings of such thicknesses cannot be considered equivalent to the bulk material of an identical composition. The thermal diffusivity value of these coatings is affected by surface p r o p e r t i e s . Z r O 2 belongs to a group of refractory materials that are partly transparent to impinging radiation. The transparency of coatings to radiation was studied by Mahlia [6], who stated that 0.14 mm thick coatings were nearly opaque to 1-3 ktm radiation. Liebert [7] found that 0.05 mm thick coatings transmitted as much as 82% radiation in the same spectrum region. The different results of the two authors are accounted for by Pawlowski et al. [4] in terms of different contents of Y203 (8% [6], 13% [7]) and differences in the coating technology. In determining the thermal diffusivity by flash method, the partial radiation transparency cannot be neglected since it would lead to erroneous results. The thinner the coatings, the greater the error. The study of the thermal diffusivity of YSZ coatings relative to the radiation coating transparency will be one of the aims of the present work. YSZ coatings are frequently used in practice and in the research of ground surfaces. Grinding the surface results in a phase transformation [8]. As mentioned above, different phases have different thermal diffusivity values. The influence of grinding upon the thermal diffusivity value will be the next topic of this work. The results will be discussed in connection with the phase transformation induced by grinding.
2. Experimental procedure The coatings were prepared by means of the Plasma-Technik PI-1 set from ZrO 2 with 8 wt.% Y203 (AMDRY 1085). The coatings, 0.2-0.8 mm thick, were sprayed on to a steel pad (Czechoslovak Standard 17 346), 16 mm in diameter. Ground samples were obtained by successively removing 0.2 mm thick layers with a diamond-charged grinding wheel. The surface of the coatings was blackened by carbon. The thermal diffusivity was determined by the flash method, processed for two layers [9]. Measurements were made with the apparatus described earlier [10]. A xenon pulse lamp with a pulse length of 1 ms was used for heating. Thermal diffusivity was measured up to Elsevier Sequoia
L24
A. Rudajevov6
/
Plasma-sprayed ZrO 2 with 8 wt. % ~ 0 3 coatings
950 °C. Phase analysis was carried out with a Siemens Diffractometer D 500 with database JCPDS [11] on samples whose thermal diffusivity was first measured at room temperature.
3. Results
The dependence of the thermal diffusivity upon the coating thickness for blackened and unblackened coating surfaces of ground samples is shown in Fig. 1. This dependence demonstrates the effect of the partial transparency of the coating on its thermal diffusivity, determined by the flash method. The dependence of the thermal diffusivity of ground and unground coatings with blackened surfaces on the coating thickness is illustrated in Fig. 2. Figure 3 shows the temperature dependence of the thermal diffusivity for a ground ....
0.007 ~,~ v
....
,
0.006 0.005 0.004
i
J
i
,
I
0
i
0-5
i
I [mml
Fig. 1. Thermal diffusivity, a, of ZrO2-8wt.%Y203 coatings as a function of the thickness, 1, of coatings: o, unblackened surfaces; e, blackened surfaces. 0.007
,
. . . . . . . . .
0.006
"
~
~
1
o.005
0.004
~ , , , 015 I [mml Fig. 2. Thermal diffusivity, a, of ZrO2-8wt.%Y:O3 coatings as a function of the thickness, 1, of coatings; 1, unground surfaces; 2, ground surfaces.
-'" 0.004
0.003 0.002
4. Discussion It can be presumed from the results given in Fig. 1 that the studied coatings are partly transparent to impinging radiation. Partial transparency cannot be neglected when determining thermal diffusivity by the flash method. The flash method is only applicable to materials where radiation absorption takes place only on the surface. The blackening of the coated surface prevents volume absorption of the radiation but the thermal diffusivity values obtained (true thermal diffusivity) relate solely to the phonon mechanism of heat transfer. In partly transparent coatings, heat transfer is affected through photon as well as phonon mechanisms, and therefore, the true thermal diffusivity value is not characteristic of the real heat transfer through the coating. The effective thermal diffusivity, which would be comprised of both heat transfer processes, would characterize the partly transparent coating correctly. However, this value is not a material constant for the mentioned coatings but depends on the coating thickness and the wavelength. The following discussion concerns true thermal diffusivity which is a material constant and will be further referred to as thermal diffusivity. TABLE 1. Phase composition of the coatings (conventional access to the evaluation) [11] Thickness of coating (mm)
0.005
t~
coating 0.8 mm thick. The dependence has an analogical pattern in unground coatings. After the temperature dependence of the thermal diffusivity had been measured up to 950 °C and the sample had cooled down to room temperature, the thermal diffusivity value increased by about 7%. The porosity of the coatings was about 10%. The results of a quantitative phase analysis of ground and unground coatings are presented in Table 1. All ground coatings are characterized by the phase transformation of the cubic portion into the tetragonal phase.
.
.
.
.
1
500
,
,
,
i
I
i
i
1000 T [ o c ]
Fig. 3. Thermal diffusivity, a, of ZrO2-8wt.%Y:O3 coatings as a function of temperature T: o, results from this work; e, results from ref. 4.
0.2 Unwound Ground 0.4 Unground Ground 0.6 Unground Ground
Phase composition (vol.%) Cubic
Tetragonal
Monoclinic
55 44
33 44
12 12
60 49
30 39
10 12
54 48
34 40
12 12
A. Rudajevov~
/
Plasma-sprayed ZrO 2 with 8 wt. % ~ 0 s coatings
It is evident from the results that surface grinding induced the diffusionless phase transformation of the cubic phase into a tetragonal phase in the coating. The Y redistribution does not take place in the tetragonal phase, and consequently, no transformation of the tetragonal phase into the monoclinic one is detected in the X-ray analysis [11]. From the ratio of the intensity of the (002) and (200) peaks, it is deduced that ferroelastic domain switching takes place [12]. The thermal diffusivity varies along different crystallographic axes. The problem of the ferroelastic domain switching and the thermal diffusivity variations will be studied further. On the basis of the present results it can be presumed that in the coatings studied here the cubic phase has a larger thermal diffusivity than the tetragonal phase. The phase transformation induced by grinding occurs only in the surface layer and cannot be neglected in the determination of the thermal diffusivity value of the plasma sprayed coatings which are several tenths of a millimetre thick. During vertical heat flow on the coated surface, the total thermal conductivity k r of a two-layer coating is characterized by the relation:
k,-k
kl "k~
(1)
where k is the thermal conductivity and V the relative volume. Index 1 refers to the layer with the induced phase transformation, index 2 to the remaining part of the coating. Thermal conductivity values are given as the product of thermal diffusivity, density and specific heat. Relation (1) can be applied to the problem of the coating's transparency to impinging radiation. In this case, V~ is the relative volume of the coating layer, whose radiation transparency is larger than zero. The thermal conductivity kl characterizes the heat transfer by means of photon and phonon processes. V2 is the relative volume of the layer of coating in which the energy is only transferred by conductance and the thermal conductivity k 2 refers to the phonon heat transfer process only. The thermal conductivity k t is, in all cases, larger than k 2. For partly transparent materials, Gardon [13] defined a so-called infinite thickness l=, which is the thickness where reflection and absorption take place in the course of irradiation of the sample in the surface only, i.e. V~ -*0 and k T ~ k 2 . It is the absorption coefficient which characterizes the infinite thickness and which depends on the wavelength of the impinging radiation. From the point of view of the experimental determination of the thermal diffusivity value of coatings thicker than the infinite thickness, the thermal diffusivities of blackened and unblackened surfaces are approximately identical. In Fig. 1 it is evident that the coatings under study have l~ >/0.8 ram.
L25
The upper curve of Fig. 3 represents the temperature dependence of the thermal diffusivity of the coatings studied here, the lower one represents the dependence given in ref. 4. In the cited work, measurements of the thermal diffusivity were made by the flash method in ground samples 1 mm thick with 10% porosity and 9 5% tetragonal phase content. It is evident from the results and the discussion of the present work that the determination of the thermal diffusivity of partly transparent coatings is not simple and that it is necessary to analyse the conditions under which the given values are determined before applying them in practical calculations and physical models. The correct value, including both photon and phonon processes of heat transfer, cannot be determined by the existing experimental methods.
5. Conclusion
It is shown that the YSZ plasma sprayed coatings are partly transparent to impinging radiation. This partial transparency cannot be neglected in coatings that are several tenths of a millimetre thick when determining the thermal diffusivity by the flash method. Grinding the coatings causes a decrease in the thermal diffusivity value. The decrease is larger when the coating thickness is smaller. During grinding, the cubic portion transforms to the tetragonal phase. The cubic phase of the YSZ coatings has a larger thermal diffusivity value than the tetragonal phase. In addition, the thermal diffusivity changes induced by grinding are connected with the ferroelastic domain switching. It ensues from the work that when determining the thermal diffusivity of coatings several tenths of a millimetre thick, we have to consider the influence of the surface properties upon the total thermal diffusivity value. When comparing and applying the experimentally established thermal diffusivity values, it is important to consider not only the method of measuring but also the method of preparing the samples.
Acknowledgment
The author wishes to thank Dr. Zeman and Dr. Cepera for the X-ray measurements.
References 1 W. J. Buykx and M. V. Swain, Proc. 2nd Int. Conf. on the Science Technology of Zirconia, Stuttgart, 1983, pp. 518-527. 2 D. Bernard, M. Vardele and P. Fauchais, Colloque de Physique, 51 (1990) C5-331-341.
L26
A. Rudajevov6
/
Plasma-sprayed ZrO 2 with 8 wt. % Y,. OF coatings
3 J. R. Brandon and R. Taylor, Surf. Coat. Technol., 39/40 (1989) 143-151. 4 L. Pawlowski, D. Lombard, A. Mahlia, Ch. Martin and P. Fauchais, High Temp.-High Press., 16 (1984) 347-359. 5 R. Brandt, High Temp.-High Press., 13 (1981 ) 79-88. 6 A. Mahlia, Thesis oflll Cycle, University of Limoges, 1983. 7 C.H. Liebert, Thin Solid Films, 53 (1978) 235-240. 8 R. Stevens, Zirconia and Zirconia Ceramics, Magnesium Elektron Ltd., 1986.
9 K. B. Larson and K. Koyama, J. Appl. Phys., 39 (9) (1968) 4408-4416. 10 S. Rudajevov~i, Silik6ty, 32 (1989) 343-350. 11 J. Zeman and M. (~epera, Report vO 070 Brno, 001-2151, 1991. 12 K. Mehta, J. F. Jue and A. V. Vikar, J. Am. Ceram. Soc., 73 (6)(1990) 1777-1779. 13 R. Gardon, J. Am. Ceram. Soc., 39 (8)(1956) 278-287.