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The thermal diffusivity of two replica resins
The thermal diffusivity of two replica resins
Th. A. M. Spierings 1, E Bosman 2, M. C. R. B. Peters 1, A. J. M. Plasschaert 1 1Department of Cariology and Endodontology, University of Nijmegen, 2Department of Oral Patho-physiology, University of Utrecht, The
Netherlands
Spierings T h A M , Bosman F, Peters MCRB, Plasschaert AJM. The thermal diffusivity of two replica resins. Dent Mater 1988: 4: 51-54. Abstract - Two replica resins i.e. a polyester and an epoxy resin are examined on the value of thermal diffusivity and on their suitability for pouring replicas. The first mixture of both resins was molded into cubic specimens in which center a thermocouple was embedded. The change in temperature at the surface, B(t), as well as at the center of the specimen, T(x, t), was recorded as a function of time. The logarithm of these d a t a define a straight line by which the thermal diffusivity could be calculated: First, the thermal diffusivity has been analyzed for both resins. The mean thermal diffusivity was 1.01 + 0.06 10-Tm2/s for the polyester, whereas the respective value of the epoxy resin was 1.30 + 0.02 10-Tm2/s. The polyester resin proved to be unsuitable for pouring replicas, thus, only the thermal diffusivity of the epoxy resin was analyzed in a 2nd and 3rd mixture. The method used appears to be suitable for direct analysis of the thermal diffusivity.
Theoretical analysis of transient thermal phenomena in restored teeth has shown that the temperature field depends on the cavity geometry and composition of teeth, as well as on the thermal diffusivity of the materials used (1, 2). The extent to which these and other parameters will result in different temperature patterns can be studied with physical and mathematical simulations. Because of the complex structure and the numerous variations of the natural teeth, experimental studies are preferred to be performed in replicas composed of one material, which is easy to handle. Besides, the thermal properties of this replica resin are not influenced by structures like those of the dentin and the enamel. The thermal diffusivity (a) of a material depends on 3 parameters, which can be expressed as: a = t h e r m a l conductivity~density x specific h e a t
By determining these parameters, the thermal diffusivity can be calculated. However, it is rather difficult to determine the specific heat of poor conductors like resins. Therefore, a direct method to determine the thermal diffusivity is applied in this study. In the literature, 3 direct methods are described using 2 (A, B) or 1 thermocouple (C). A ) Minesaki et al. (7) embedded 2 thermocouples within a cuboid specimen at various distances of 4*
the loading site. However, the interference of these couples with the recorded temperature can be disputed. B) Tibbetts et al. (8) and Voth et al. (9) fixed 2 thermocouples within a layered specimen at the interface of various materials. The discontinuous heat transport at the interface could result in a heat pattern in which t h e interference of the second thermocouple is negligible, Thus, this method is considered to be suitable to determine the thermal diffusivity through materials composed of different layers. C) Braden (1) and Civjan et al. (10) placed 1 thermocoupie within the specimen recording the change in temperature relative to the surroundings. The specimen was immersed in the same mercury bath in which the reference of the thermocoupie was placed. In our pilot study, a thermal fluctuation of the mercury was noticed, influencing the final results. Therefore, in the present study this phenomenon is considered in more detail. The purpose of this study is to determine the thermal diffusivity of a polyester* and an epoxy resin* since these data are not reported in literature. The resins are tested on their suitability for pouring test models after which the thermal diffusivity will be de-
Key words: thermal diffusivity, resin. Th. A. M. Spierings, Department of Cariology and Endodontolegy, University of Nijmegen, P.O. Box 9101, 6500 HB Nijrnegen, The Netherlands.
Received March 10, 1986; accepted April 23, 1987.
termined for both resins using cubic specimens. Next, the choice of resin is agreed upon the suitability for pouring replicas and their value of the thermal diffusivity. A second and third mix of this resin will be molded into samples of a cuboid shape. The given thermal diffusivity will be helpful in further thermal analysis of teeth using replicas composed of this particular resin.
Material and methods
Within an isotropic, homogeneous, rectangular, solid parallelepiped with the dimensions 2L, 2H, and 2W, with initial temperature To and constant surface temperature of Be, the temperature T(x,t) depends on time t and position x = (x,y,z) according to Carslaw (12). However, when the paraUelepiped is placed in a bath with temperature B(t), a correction factor D(x,t) is needed (see Appendix). Under the given conditions and for sufficiently large values of time, the temperature at any point within the parallelepiped can be derived from equation A.5 (see Appendix): T(x,t) - B(t)
-- A . e -k~
[1]
TO - B(t) * Type GTS + MEK Hardener (5%), Romar-Voss, Roggel, The Netherlands. * Araldite D + Hardener HY 956 (20%), Ciba-Geigy, Basel, Switzerland.
in which A = 64 a~x ~y 7;" cos y s cos
a~z cos
52
Spierings et al.
and where ko =
Ln ( T(x_ ,t) -B(t))/(To- B(t) ) 00-
4
L-~ + ~
+
[21
. . . . . . . . E p o x y Resin - -
The thermal diffusivity 'a' is expressed in m~s-~. If the temperature of the central part of the specimen and of the surface are known as well as between the initial temperatures, the logarithm of [(T(x,t) - B(t))/(To - B(t))] can be plotted as a function of time. In formula:
Polyester Resin
-0.5-
-1.0-
ln[T(x,t) - B(t))] - ln[To - B(t)l = I n A - kot [31 -1.5--
Apart from the initial stages, a straight line should be obtained. So, the slope of this straight line is alike -ko: d[ln(T(x,t)- B ( t ) ) - l n ( To - B(t))] -ko-
dt
[41
As mentioned before, the temperature of the bath was not constant. Hence, temperature B is taken as a function of time. In the Appendix, it is shown that equation [1] satisfies for sufficient large values of time. Therefore, the thermal diffusivity of the cuboid specimens will be obtained by substitution of formula [4] in equation [2]. Sample
preparation
Cuboid specimens (0.0150 + 0.0002 m) were prepared in silicone rubber molds * held in sections of copper pipe (length 0.05 m, diameter 0.03 m). The hot junction of a copper-constantan thermocouple ~ (42 ~tV/~ has been embedded in its center (Fig. 1). Both resins - the polyester and the
Th i.i]'r;,
i
g.@z.,
Fig. 1. The mold of a cuboid positive within the center the hot junction of the thermocouple.
* Wacker Silicone: RTV-M457 + Hardener T40, Wacker-chemie, GMSH-Miinchen, FRG. A1-Cu-20/A1-Cu-20, welded, Drijfhout BV, Amsterdam, The Netherlands.
..
%", Fig. 2. Representative plot of ln[(T(x,t) B(t))/(To - B(t))] for the epoxy and polyester resin versus time.
.~ "~
time (sec.)---,-
-20
'
0
I
20.0
epoxy resin - were mixed according to the manufacturers' instructions, molded into 3 samples each. To ensure complete set, the resin specimens were dismantled 60 h after initial mixing and kept at room temperature, thereafter. After finishing the first set of measurements, a second and third mix of the respective resin was molded, in order to assess the effect of possible variations in mixture on the thermal diffusivity. Experimental
'
measurements
The initial temperature of the specimen was 25.0~ When the temperature of the specimen was constant for at least 5 min, it was quickly transferred to a cold mercury bath at 0.0~ The vessel with mercury was surrounded by an insulated jacket of ice water. The specimen was kept in the bath for at least 3 min using an insulated metallic sample holder. This procedure was repeated 3 and 5 times for each sample of the first and of the 2nd and 3rd set, respectively. The thermocouple in the center of the specimen as well as a couple in the mercury bath were connected to a twopen recorder II via an ice junction. The thermocouple output (in mV) was recorded versus (vs) time. A chart speed of 500 mm/min allowed readings for every 6 s. The readings were converted to ~ resulting in the temperature values (T(x,t), B(t), and To). By means of the method of the least squares, the slope of equation [3] could be calculated. Such a procedure of data processing
I
40.0
'
I
60.0
'
I
80.0
~
"'.
'
100.0
is time consuming and does not exclude reading errors. Therefore, during the third set of measurements, the thermocouple output was amplified~ and recorded vs time using a tape recorder**. The data obtained were processed by analog to digital conversion resulting in temperatures and plots according equation [3] from which the slope ( - ko) was calculated. The experimental procedure used makes it possible to find the thermal diffusivity according to Braden (1) which is calculated for the first mixtures. Results
For both resins, a representative ln[T (x,t) - B(t))/(To - B(t)] vs time plot is shown in Fig. 2. Similar curves were obtained for each individual specimen. Apart from an initial stage of about 30 s, a straight line was obtained according to the theory. The mean diffusivity for 3 measurements of each specimen of the first set is 1.01 + 0.06 10-Tm2s-1 for the polyester and 1.30 + 0.02 10-7m2s -I for the epoxy resin (Table 1). The thermal diffusivity has been recalculated according to Braden (1). These data are given as well in Table 1. II Two Channel Recorder BD9, Kipp & Zonen, Delft, The Netherlands. Differential Amplifier 26A - 2601 R2601 Mainframe, Tektronix Inc., Beaverton, Oregon 97005, USA. **3968 Instrumentation Recorder, HewlettPackard, Palo Alto, CA 94304, USA.
Thermal diffusivity of resins Table 1. The mean thermal diffusivity for polyester and epoxy resin (n recordings per sample A, B and C) and their corresponding standard deviation.
The difference between the data of the various sets of epoxy resin (Tables 1, 2) may be partly due to the inhomogeneity of the mixture, the amount of hardener added and the incorporation of macroscopically invisible air-bubbles. Statistical analysis of the data has shown that the difference within one mix and between mixes is not significant. According to the manufacturers' instructions the polyester resin should be suitable for making replicas. However, the mixture had to be put under vacuum to avoid air-bubbles in the specimens. Moreover, the surface of the polyester resin did not completely set in the presence of oxygen and a nitrogenous surrounding had to be used. The processing of the epoxy resin was found to be suitable for pouring replicas, which will be used in future studies concerning thermal analysis of teeth under in vitro and in vivo conditions. In this study, the optimized method appears to be suitable for direct analysis of the thermal diffusivity.
Thermal diffusivity a _+ sd (10 .7 m2s-1)
Sample n A B C
3 3 3
fi
Polyester resin* 1st mixture
Epoxy resin* 1st mixture
1.01_+0.03 (0.92_+0.01) 1.06_+0.07 (0.95+_0.03) 0.95_+0.04 (0.89_+0.02)
1.31+0.09 (1.21_+0.06) 1.28_+0.07(1.20_+0.06) 1.32+0.13 (1.19_+0.09)
1.01_+0.06 (0.92_+0.02)
1.30_+0.02(1.20+_0.06)
* Data between parentheses calculated according to Braden (1).
Because of the complex handling of the polyester resin, the epoxy resin was chosen to be used in further studies for the construction of replicas. The results of the second and third mixture of epoxy resin are noted in Table 2. The data sets concern the mean thermal diffusivity of 5 measurements of each specimen. Statistical analysis of the epoxy resin by means of an analysis of variance revealed that the difference within and between the mixtures is not significant for p < 0.05. Discussion The recorded thermal fluctuation in the mercury bath of 0.13 to 0.39~ optimized the results by taking this thermal fluctuation into account. If instead of ln[(T(x,t) - B(t))/(To - B(t))] the plot of ln[(T(x,t) - B(t))/(To - Bo)] was chosen, in which the denominator is constant according to Braden (1), the value of k o obtained should have been in the range of 6 - 1 0 % lower (Table 1). The approximation made by using equation (1) instead of (A.3) (see Appendix) introduced an error of maximal 2%. In view of the accuracy of the measurements and the inhomogeneities of the specimens, however, these corrections might be neglected.
For the upward pressure of mercury, the sample holder is more suitable than a ceramic rod as used by Civjan et al. (10). The sample holder used in this study causes a reduced mercury-resin interface, which is assumed to have a negligible influence. The shape of the plots in Fig. 2 are in agreement with the plots presented by Braden (1). Both plots show in the initial stages of about 30 s a curved line. According to the theory, straight lines were obtained for large values of time. Thus, the first part of the curve is excluded in determining the slope of the line.
Appendix A n isotropic, homogeneous, rectangular, solid parallelepiped specimen with dimensions 2L, 2W, and 2H, with temperature To at time t = 0, is placed in a bath with temperature B(t). Assume that the surface temperature of the specimen equals temperature B(t). The temperature at point x = (x,y,z) within the specimen is given by
'~dB(~)
T ( x , t ) - B ( o ) = (To-B(o)). q0(x,t) + ] - - ~ o
A B C
Thermal diffusivity a + sd (10 -7 m2s-1) for epoxy resin n
2nd mixture
3rd mixture
5 5 5
1.37+0.08 1.31+0.03 1.31+0.06
1.30+0.08 1.41+0.05 1.37+0.06
1.33+0.03
1.36+0.06
(A.1)
1=o m = o n=o
(21+l)~x
Sample
(1-qo(x,t-x)d~
64 ~ = ~ ( - 1 ) '+m+n in which rO(x,t) = ~ ' ~ ~ ~2 ( 2 1 + l ) ( 2 m + l ) ( Z n + l )
9C
Table 2. The mean thermal diffusivity for epoxy resin of n recordings per sample (A, B, and C) and their corresponding standard deviation.
53
and where kt,m,. -
O
S
-
2L
(2m+l)=y COS
2W
a~ 2 {(21+ 1) 2 (2m+1) 2 4 \ L2 + - W - 2 +
(2n+l)=z C O S ' - -
2H
e -kl,m,nt
(2n+1)2~ H 2
]
Substituting B(t) = B(o) + v(t) into equation (A.1) yields trdv(t) T(x,t) = B(t) + (To-B(t)).q0(x,t) + v(t)-cp(x,t) -]--d~--~o q~(x,t-~)dx
Thus,
T(x,t) - B(t) To - B(t) - r
(A.2)
(A.3)
54
S p i e r i n g s et al.
where
References
v(t) D(_x,t) = 1 + - To - B(t)
1
1
~ dv(~)
- -- B(t) 9 qo(~_,t)Jo---~ "q)(x,t-~)dz To
The logarithm of (A.3) can be written as ln[T(x,t) - B(t)] - ln[To - B(t)] = In (p(x,t) + In D(x,t)
(A.4)
For the first 100 sec, it appeared that v(t) could be estimated by v(t) = b.t with b = 2.10 -3, whereas v(t) decreased again for t > 100 see. Substituting v(t) = b.t in D(x,t) yields b.t D(x,t) = 1 +
1
To - B ( o ) - b . t
To - B ( o ) - b . t
1
- -
(p(x,t)
b [ ~ ( o ) - ~(t)]
with ~ ( t ) as the primitive of cp(x,t). For large values of time, e.g. t > 50 sec, qp(x,t) can be approximated by the first term of q~(x,t) (see A . 2 and A . 3 ) , as neglecting the second and following terms of q~(x,t) results in a relative error of about 2% in ko. Now also D(x,t) can be estimated. It is found that Iln D(x,t)l < 0,01lln q0(x,t)l for t > 50 sec. This justifies the derivation of the thermal diffusivity 'a' from the expression for ln[(T(x,t) B(t))/(To - B(t))] w h e n the t e m p e r a t u r e of the bath is not totally constant during the experiment. Therefore, it is justified that the equation A.3 can be rewritten in: T(x,t) - B(t) (A.5)
= A . e k~
T O - B(t) 64 nx ny nz in which A = ~S" cos ~-~- cos ~ - ~ - cos 2--H
1 and where k o = ---~- 9
+ ~
+
Acknowledgements - This research was part
of the thesis of Th. A. M. Spierings in fulfillment of the requirements for Doctor of Philosophy degree. This research was part of the research program "Restorations and Restorative Materials". The authors wish to
(A.6)
thank Ir. G. Uyen for his contribution in data processing; Mr. P. van Woerden for assembling the thermocouples and Mr. S. Nottet for his suggestions in molding the models.
1. Braden M. Heat conduction in teeth and the effect of lining materials. J Dent Res 1964: 43: 315-22. 2. Spierings ThAM, De Vree JHP, Peters MCRB, Plasschaert AJM. The influence of restorative dental materials on heat transmission in human teeth. J Dent Res 1984: 63: 1096-110. 3. Domininghaus H. Entscheidingshilfen bei der Wahl von Kunststoffen. Plastverarbeiter 30 1979: 8: 428-9. 4. Ciba-Geigy. Umhiillungssysteme, Switzerland, Publ. nr. 24843/d, 1982. 3-22. 5. Craig RA, Peyton FA. Thermal conductivity of tooth structure, dental cements, and amalgam. J Dent Res 1961: 40: 411-8. 6. Brown WS, Dewey WA, Jacobs HR. Thermal properties of teeth. J Dent Res 1970: 49: 752-5. 7. Minesaki Y, Muroya M, Higashi R. A method for determining of thermal diffusivity of human teeth. Dent Mater J 1983: 2: 204-9. 8. Tibbetts VR, Schnell RJ, Swartz ML, Phillips RW. Thermal diffusion through amalgam and cement bases: comparison of in vitro and in vivo measurements. J Dent Res 1976; 55: 441-51. 9. Voth ED, Phillips RW, Swartz MJ. Thermal diffusion through amalgam and various liners. J Dent Res 1966; 45: 1184-90. 10. Civjan S, Barone JJ, Reinke PE, Selting WJ. Thermal properties of non-metallic restorative materials. J Dent Res 1972; 51: 1030-7. 11. Watts DC, Smith R. Thermal diffusivity in finite cylindrical specimens of dental cements. J Dent Res 1981: 60: 1972-6. 12. Carslaw HS, Jaeger JC. Conduction o f heat in solids. Oxford: The University Press, 1973: 184-6.
Th. A. M. Spierings 1, E Bosman 2, M. C. R. B. Peters 1, A. J. M. Plasschaert 1 1Department of Cariology and Endodontology, University of Nijmegen, 2Department of Oral Patho-physiology, University of Utrecht, The
Netherlands
Spierings T h A M , Bosman F, Peters MCRB, Plasschaert AJM. The thermal diffusivity of two replica resins. Dent Mater 1988: 4: 51-54. Abstract - Two replica resins i.e. a polyester and an epoxy resin are examined on the value of thermal diffusivity and on their suitability for pouring replicas. The first mixture of both resins was molded into cubic specimens in which center a thermocouple was embedded. The change in temperature at the surface, B(t), as well as at the center of the specimen, T(x, t), was recorded as a function of time. The logarithm of these d a t a define a straight line by which the thermal diffusivity could be calculated: First, the thermal diffusivity has been analyzed for both resins. The mean thermal diffusivity was 1.01 + 0.06 10-Tm2/s for the polyester, whereas the respective value of the epoxy resin was 1.30 + 0.02 10-Tm2/s. The polyester resin proved to be unsuitable for pouring replicas, thus, only the thermal diffusivity of the epoxy resin was analyzed in a 2nd and 3rd mixture. The method used appears to be suitable for direct analysis of the thermal diffusivity.
Theoretical analysis of transient thermal phenomena in restored teeth has shown that the temperature field depends on the cavity geometry and composition of teeth, as well as on the thermal diffusivity of the materials used (1, 2). The extent to which these and other parameters will result in different temperature patterns can be studied with physical and mathematical simulations. Because of the complex structure and the numerous variations of the natural teeth, experimental studies are preferred to be performed in replicas composed of one material, which is easy to handle. Besides, the thermal properties of this replica resin are not influenced by structures like those of the dentin and the enamel. The thermal diffusivity (a) of a material depends on 3 parameters, which can be expressed as: a = t h e r m a l conductivity~density x specific h e a t
By determining these parameters, the thermal diffusivity can be calculated. However, it is rather difficult to determine the specific heat of poor conductors like resins. Therefore, a direct method to determine the thermal diffusivity is applied in this study. In the literature, 3 direct methods are described using 2 (A, B) or 1 thermocouple (C). A ) Minesaki et al. (7) embedded 2 thermocouples within a cuboid specimen at various distances of 4*
the loading site. However, the interference of these couples with the recorded temperature can be disputed. B) Tibbetts et al. (8) and Voth et al. (9) fixed 2 thermocouples within a layered specimen at the interface of various materials. The discontinuous heat transport at the interface could result in a heat pattern in which t h e interference of the second thermocouple is negligible, Thus, this method is considered to be suitable to determine the thermal diffusivity through materials composed of different layers. C) Braden (1) and Civjan et al. (10) placed 1 thermocoupie within the specimen recording the change in temperature relative to the surroundings. The specimen was immersed in the same mercury bath in which the reference of the thermocoupie was placed. In our pilot study, a thermal fluctuation of the mercury was noticed, influencing the final results. Therefore, in the present study this phenomenon is considered in more detail. The purpose of this study is to determine the thermal diffusivity of a polyester* and an epoxy resin* since these data are not reported in literature. The resins are tested on their suitability for pouring test models after which the thermal diffusivity will be de-
Key words: thermal diffusivity, resin. Th. A. M. Spierings, Department of Cariology and Endodontolegy, University of Nijmegen, P.O. Box 9101, 6500 HB Nijrnegen, The Netherlands.
Received March 10, 1986; accepted April 23, 1987.
termined for both resins using cubic specimens. Next, the choice of resin is agreed upon the suitability for pouring replicas and their value of the thermal diffusivity. A second and third mix of this resin will be molded into samples of a cuboid shape. The given thermal diffusivity will be helpful in further thermal analysis of teeth using replicas composed of this particular resin.
Material and methods
Within an isotropic, homogeneous, rectangular, solid parallelepiped with the dimensions 2L, 2H, and 2W, with initial temperature To and constant surface temperature of Be, the temperature T(x,t) depends on time t and position x = (x,y,z) according to Carslaw (12). However, when the paraUelepiped is placed in a bath with temperature B(t), a correction factor D(x,t) is needed (see Appendix). Under the given conditions and for sufficiently large values of time, the temperature at any point within the parallelepiped can be derived from equation A.5 (see Appendix): T(x,t) - B(t)
-- A . e -k~
[1]
TO - B(t) * Type GTS + MEK Hardener (5%), Romar-Voss, Roggel, The Netherlands. * Araldite D + Hardener HY 956 (20%), Ciba-Geigy, Basel, Switzerland.
in which A = 64 a~x ~y 7;" cos y s cos
a~z cos
52
Spierings et al.
and where ko =
Ln ( T(x_ ,t) -B(t))/(To- B(t) ) 00-
4
L-~ + ~
+
[21
. . . . . . . . E p o x y Resin - -
The thermal diffusivity 'a' is expressed in m~s-~. If the temperature of the central part of the specimen and of the surface are known as well as between the initial temperatures, the logarithm of [(T(x,t) - B(t))/(To - B(t))] can be plotted as a function of time. In formula:
Polyester Resin
-0.5-
-1.0-
ln[T(x,t) - B(t))] - ln[To - B(t)l = I n A - kot [31 -1.5--
Apart from the initial stages, a straight line should be obtained. So, the slope of this straight line is alike -ko: d[ln(T(x,t)- B ( t ) ) - l n ( To - B(t))] -ko-
dt
[41
As mentioned before, the temperature of the bath was not constant. Hence, temperature B is taken as a function of time. In the Appendix, it is shown that equation [1] satisfies for sufficient large values of time. Therefore, the thermal diffusivity of the cuboid specimens will be obtained by substitution of formula [4] in equation [2]. Sample
preparation
Cuboid specimens (0.0150 + 0.0002 m) were prepared in silicone rubber molds * held in sections of copper pipe (length 0.05 m, diameter 0.03 m). The hot junction of a copper-constantan thermocouple ~ (42 ~tV/~ has been embedded in its center (Fig. 1). Both resins - the polyester and the
Th i.i]'r;,
i
g.@z.,
Fig. 1. The mold of a cuboid positive within the center the hot junction of the thermocouple.
* Wacker Silicone: RTV-M457 + Hardener T40, Wacker-chemie, GMSH-Miinchen, FRG. A1-Cu-20/A1-Cu-20, welded, Drijfhout BV, Amsterdam, The Netherlands.
..
%", Fig. 2. Representative plot of ln[(T(x,t) B(t))/(To - B(t))] for the epoxy and polyester resin versus time.
.~ "~
time (sec.)---,-
-20
'
0
I
20.0
epoxy resin - were mixed according to the manufacturers' instructions, molded into 3 samples each. To ensure complete set, the resin specimens were dismantled 60 h after initial mixing and kept at room temperature, thereafter. After finishing the first set of measurements, a second and third mix of the respective resin was molded, in order to assess the effect of possible variations in mixture on the thermal diffusivity. Experimental
'
measurements
The initial temperature of the specimen was 25.0~ When the temperature of the specimen was constant for at least 5 min, it was quickly transferred to a cold mercury bath at 0.0~ The vessel with mercury was surrounded by an insulated jacket of ice water. The specimen was kept in the bath for at least 3 min using an insulated metallic sample holder. This procedure was repeated 3 and 5 times for each sample of the first and of the 2nd and 3rd set, respectively. The thermocouple in the center of the specimen as well as a couple in the mercury bath were connected to a twopen recorder II via an ice junction. The thermocouple output (in mV) was recorded versus (vs) time. A chart speed of 500 mm/min allowed readings for every 6 s. The readings were converted to ~ resulting in the temperature values (T(x,t), B(t), and To). By means of the method of the least squares, the slope of equation [3] could be calculated. Such a procedure of data processing
I
40.0
'
I
60.0
'
I
80.0
~
"'.
'
100.0
is time consuming and does not exclude reading errors. Therefore, during the third set of measurements, the thermocouple output was amplified~ and recorded vs time using a tape recorder**. The data obtained were processed by analog to digital conversion resulting in temperatures and plots according equation [3] from which the slope ( - ko) was calculated. The experimental procedure used makes it possible to find the thermal diffusivity according to Braden (1) which is calculated for the first mixtures. Results
For both resins, a representative ln[T (x,t) - B(t))/(To - B(t)] vs time plot is shown in Fig. 2. Similar curves were obtained for each individual specimen. Apart from an initial stage of about 30 s, a straight line was obtained according to the theory. The mean diffusivity for 3 measurements of each specimen of the first set is 1.01 + 0.06 10-Tm2s-1 for the polyester and 1.30 + 0.02 10-7m2s -I for the epoxy resin (Table 1). The thermal diffusivity has been recalculated according to Braden (1). These data are given as well in Table 1. II Two Channel Recorder BD9, Kipp & Zonen, Delft, The Netherlands. Differential Amplifier 26A - 2601 R2601 Mainframe, Tektronix Inc., Beaverton, Oregon 97005, USA. **3968 Instrumentation Recorder, HewlettPackard, Palo Alto, CA 94304, USA.
Thermal diffusivity of resins Table 1. The mean thermal diffusivity for polyester and epoxy resin (n recordings per sample A, B and C) and their corresponding standard deviation.
The difference between the data of the various sets of epoxy resin (Tables 1, 2) may be partly due to the inhomogeneity of the mixture, the amount of hardener added and the incorporation of macroscopically invisible air-bubbles. Statistical analysis of the data has shown that the difference within one mix and between mixes is not significant. According to the manufacturers' instructions the polyester resin should be suitable for making replicas. However, the mixture had to be put under vacuum to avoid air-bubbles in the specimens. Moreover, the surface of the polyester resin did not completely set in the presence of oxygen and a nitrogenous surrounding had to be used. The processing of the epoxy resin was found to be suitable for pouring replicas, which will be used in future studies concerning thermal analysis of teeth under in vitro and in vivo conditions. In this study, the optimized method appears to be suitable for direct analysis of the thermal diffusivity.
Thermal diffusivity a _+ sd (10 .7 m2s-1)
Sample n A B C
3 3 3
fi
Polyester resin* 1st mixture
Epoxy resin* 1st mixture
1.01_+0.03 (0.92_+0.01) 1.06_+0.07 (0.95+_0.03) 0.95_+0.04 (0.89_+0.02)
1.31+0.09 (1.21_+0.06) 1.28_+0.07(1.20_+0.06) 1.32+0.13 (1.19_+0.09)
1.01_+0.06 (0.92_+0.02)
1.30_+0.02(1.20+_0.06)
* Data between parentheses calculated according to Braden (1).
Because of the complex handling of the polyester resin, the epoxy resin was chosen to be used in further studies for the construction of replicas. The results of the second and third mixture of epoxy resin are noted in Table 2. The data sets concern the mean thermal diffusivity of 5 measurements of each specimen. Statistical analysis of the epoxy resin by means of an analysis of variance revealed that the difference within and between the mixtures is not significant for p < 0.05. Discussion The recorded thermal fluctuation in the mercury bath of 0.13 to 0.39~ optimized the results by taking this thermal fluctuation into account. If instead of ln[(T(x,t) - B(t))/(To - B(t))] the plot of ln[(T(x,t) - B(t))/(To - Bo)] was chosen, in which the denominator is constant according to Braden (1), the value of k o obtained should have been in the range of 6 - 1 0 % lower (Table 1). The approximation made by using equation (1) instead of (A.3) (see Appendix) introduced an error of maximal 2%. In view of the accuracy of the measurements and the inhomogeneities of the specimens, however, these corrections might be neglected.
For the upward pressure of mercury, the sample holder is more suitable than a ceramic rod as used by Civjan et al. (10). The sample holder used in this study causes a reduced mercury-resin interface, which is assumed to have a negligible influence. The shape of the plots in Fig. 2 are in agreement with the plots presented by Braden (1). Both plots show in the initial stages of about 30 s a curved line. According to the theory, straight lines were obtained for large values of time. Thus, the first part of the curve is excluded in determining the slope of the line.
Appendix A n isotropic, homogeneous, rectangular, solid parallelepiped specimen with dimensions 2L, 2W, and 2H, with temperature To at time t = 0, is placed in a bath with temperature B(t). Assume that the surface temperature of the specimen equals temperature B(t). The temperature at point x = (x,y,z) within the specimen is given by
'~dB(~)
T ( x , t ) - B ( o ) = (To-B(o)). q0(x,t) + ] - - ~ o
A B C
Thermal diffusivity a + sd (10 -7 m2s-1) for epoxy resin n
2nd mixture
3rd mixture
5 5 5
1.37+0.08 1.31+0.03 1.31+0.06
1.30+0.08 1.41+0.05 1.37+0.06
1.33+0.03
1.36+0.06
(A.1)
1=o m = o n=o
(21+l)~x
Sample
(1-qo(x,t-x)d~
64 ~ = ~ ( - 1 ) '+m+n in which rO(x,t) = ~ ' ~ ~ ~2 ( 2 1 + l ) ( 2 m + l ) ( Z n + l )
9C
Table 2. The mean thermal diffusivity for epoxy resin of n recordings per sample (A, B, and C) and their corresponding standard deviation.
53
and where kt,m,. -
O
S
-
2L
(2m+l)=y COS
2W
a~ 2 {(21+ 1) 2 (2m+1) 2 4 \ L2 + - W - 2 +
(2n+l)=z C O S ' - -
2H
e -kl,m,nt
(2n+1)2~ H 2
]
Substituting B(t) = B(o) + v(t) into equation (A.1) yields trdv(t) T(x,t) = B(t) + (To-B(t)).q0(x,t) + v(t)-cp(x,t) -]--d~--~o q~(x,t-~)dx
Thus,
T(x,t) - B(t) To - B(t) - r
(A.2)
(A.3)
54
S p i e r i n g s et al.
where
References
v(t) D(_x,t) = 1 + - To - B(t)
1
1
~ dv(~)
- -- B(t) 9 qo(~_,t)Jo---~ "q)(x,t-~)dz To
The logarithm of (A.3) can be written as ln[T(x,t) - B(t)] - ln[To - B(t)] = In (p(x,t) + In D(x,t)
(A.4)
For the first 100 sec, it appeared that v(t) could be estimated by v(t) = b.t with b = 2.10 -3, whereas v(t) decreased again for t > 100 see. Substituting v(t) = b.t in D(x,t) yields b.t D(x,t) = 1 +
1
To - B ( o ) - b . t
To - B ( o ) - b . t
1
- -
(p(x,t)
b [ ~ ( o ) - ~(t)]
with ~ ( t ) as the primitive of cp(x,t). For large values of time, e.g. t > 50 sec, qp(x,t) can be approximated by the first term of q~(x,t) (see A . 2 and A . 3 ) , as neglecting the second and following terms of q~(x,t) results in a relative error of about 2% in ko. Now also D(x,t) can be estimated. It is found that Iln D(x,t)l < 0,01lln q0(x,t)l for t > 50 sec. This justifies the derivation of the thermal diffusivity 'a' from the expression for ln[(T(x,t) B(t))/(To - B(t))] w h e n the t e m p e r a t u r e of the bath is not totally constant during the experiment. Therefore, it is justified that the equation A.3 can be rewritten in: T(x,t) - B(t) (A.5)
= A . e k~
T O - B(t) 64 nx ny nz in which A = ~S" cos ~-~- cos ~ - ~ - cos 2--H
1 and where k o = ---~- 9
+ ~
+
Acknowledgements - This research was part
of the thesis of Th. A. M. Spierings in fulfillment of the requirements for Doctor of Philosophy degree. This research was part of the research program "Restorations and Restorative Materials". The authors wish to
(A.6)
thank Ir. G. Uyen for his contribution in data processing; Mr. P. van Woerden for assembling the thermocouples and Mr. S. Nottet for his suggestions in molding the models.
1. Braden M. Heat conduction in teeth and the effect of lining materials. J Dent Res 1964: 43: 315-22. 2. Spierings ThAM, De Vree JHP, Peters MCRB, Plasschaert AJM. The influence of restorative dental materials on heat transmission in human teeth. J Dent Res 1984: 63: 1096-110. 3. Domininghaus H. Entscheidingshilfen bei der Wahl von Kunststoffen. Plastverarbeiter 30 1979: 8: 428-9. 4. Ciba-Geigy. Umhiillungssysteme, Switzerland, Publ. nr. 24843/d, 1982. 3-22. 5. Craig RA, Peyton FA. Thermal conductivity of tooth structure, dental cements, and amalgam. J Dent Res 1961: 40: 411-8. 6. Brown WS, Dewey WA, Jacobs HR. Thermal properties of teeth. J Dent Res 1970: 49: 752-5. 7. Minesaki Y, Muroya M, Higashi R. A method for determining of thermal diffusivity of human teeth. Dent Mater J 1983: 2: 204-9. 8. Tibbetts VR, Schnell RJ, Swartz ML, Phillips RW. Thermal diffusion through amalgam and cement bases: comparison of in vitro and in vivo measurements. J Dent Res 1976; 55: 441-51. 9. Voth ED, Phillips RW, Swartz MJ. Thermal diffusion through amalgam and various liners. J Dent Res 1966; 45: 1184-90. 10. Civjan S, Barone JJ, Reinke PE, Selting WJ. Thermal properties of non-metallic restorative materials. J Dent Res 1972; 51: 1030-7. 11. Watts DC, Smith R. Thermal diffusivity in finite cylindrical specimens of dental cements. J Dent Res 1981: 60: 1972-6. 12. Carslaw HS, Jaeger JC. Conduction o f heat in solids. Oxford: The University Press, 1973: 184-6.