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Thermal properties of polymers by non-steady-state techniques
ARTICLE IN PRESS
POLYMER TESTING Polymer Testing 26 (2007) 556–566 www.elsevier.com/locate/polytest
Test Method
Thermal properties of polymers by non-steady-state techniques Wilson Nunes dos Santos Department of Materials Engineering, Federal University of Sa˜o Carlos, Via Washington Luiz, km 235—CP 676, Sa˜o Carlos-SP, CEP 13565-905, Brazil Received 5 January 2007; accepted 10 February 2007
Abstract The nature of the molecular structure of polymers makes the properties of such materials markedly temperature dependant. Modelling heat transfer under steady-state or transient conditions is of fundamental importance in engineering design as well as for the tailoring of thermal and mechanical behaviour of materials. Thermal conductivity, thermal diffusivity and specific heat, namely the thermal properties, are the three most important physical properties of a material that are needed for heattransfer calculations. Nowadays, several different techniques for the determination of the thermal diffusivity and thermal conductivity may be found in the literature. Recently, transient techniques have become the preferable way for measuring thermal properties of materials. In this work, two absolute and non-steady-state methods are employed in the experimental determination of thermal properties of some selected polymers: the laser flash technique and the hot-wire technique. In the hotwire technique, samples were prepared from the extrusion process in the shape of a rectangular parallelepiped, with the molten mass for each sample being approximately 1500 g. Thermal conductivity, thermal diffusivity and specific heat were simultaneously determined from the same experimentally determined thermal transient. In the laser flash technique, disc shaped samples were prepared, either by hot pressing approximately 50 mg of material, or by cutting the discs from long cylindrical bars. In this technique, only the thermal diffusivity is determined from the experimentally determined thermal transient. The thermal conductivity was derived from the thermal diffusivity with the knowledge of the bulk density and the specific heat. Specific heat was experimentally determined using a modulated differential scanning calorimeter, and the density was obtained from the PVT curve at each temperature. Since the sample mass ratio between both techniques is approximately 3 104, and the speed of the cooling process may generate different morphologies, this phenomenon may be partially responsible for any discrepancy between both techniques for a specific polymer. Experimental results obtained by both techniques are checked against each other, as well as, when possible, with data found in literature. r 2007 Elsevier Ltd. All rights reserved. Keywords: Thermal properties; Non-steady-state techniques; Hot-wire technique; Laser flash technique; Melt polymers
1. Introduction The nature of the molecular structure of polymers makes the properties of such materials markedly temperature dependant. Modelling heat transfer Tel.: +55 16 3351 8529; fax: +55 16 3361 5404.
E-mail address: [email protected]. 0142-9418/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.polymertesting.2007.02.005
under steady-state or transient conditions is of fundamental importance in engineering design as well as for the tailoring of thermal and mechanical behaviour of materials. Thermal conductivity, thermal diffusivity and specific heat, namely the thermal properties, are the three most important physical properties of a material that are needed for heat-transfer calculations. These three properties
ARTICLE IN PRESS W.N. dos Santos / Polymer Testing 26 (2007) 556–566
are related by the equation: a¼
k , rcp
(1)
where a is the thermal diffusivity (m2/s), k the thermal conductivity (W/m K), r the bulk density (kg/m3) and cp the specific heat at constant pressure (J/kg K). Nowadays, several different techniques for the determination of the thermal diffusivity and thermal conductivity may be found in the literature [1–4]. Steady-state methods are those in which the desired property is measured in a steady-state heat exchange, and the calorimetric techniques employed in the determination of the thermal conductivity are examples of this category of methods. Non-steadystate methods are those in which the property is measured according to a transient regime of heat exchange. The laser flash technique is an example of a non-steady-state method. The non-steady-state methods may be further divided into two different categories: periodic heat flux and transient heat flux methods. In the periodic heat flux methods, the edge of a bar or a plate is alternately heated for a half period T/2, and cooled for the next T/2 period. The temperature measured at any point in the interior of the specimen varies periodically. When the steady state is reached, the thermal diffusivity may be evaluated from the temperature recorded at two selected fixed points. Angstrom’s method is an example of this technique for the determination of the thermal diffusivity. In the transient heat flux methods, like the laser flash technique, a heat pulse of short duration is incident on the front face of a specimen, and the thermal diffusivity is evaluated from the recorded temperature history on the rear face. The thermal conductivity may be derived from the thermal diffusivity if specific heat and bulk density are known, or alternatively, by using standard bodies. Recently, transient techniques have become the preferable way for measuring thermal properties of materials. In this work two absolute and non-steady-state methods are employed in the experimental determination of thermal properties of some selected polymers: the laser flash technique and the hot-wire technique. 2. Experimental 2.1. The laser flash technique The laser flash technique is a non-steady-state (transient heat flux), absolute and direct method for
557
thermal diffusivity measurements, which was introduced in 1961 by Parker et al. [5]. In this technique, a uniform heat pulse of short duration compared to the transit time throughout the sample is incident on the front face of a disc specimen, and the temperature rise on the rear face is recorded. If no heat losses occur, the increased normalised temperature on the rear face is given by V ¼1þ2
1 X ð1Þn expðn2 oÞ,
(2)
n¼1
where o ¼ p2at/L2; V ¼ T/Tm, is the dimensionless temperature increase of the rear face; T, the instantaneous temperature increase of the rear face of the specimen; Tm ¼ Q/rcpL, the maximum temperature increase of the rear face; Q, the input energy on the front face; r, the bulk density; cp, the specific heat at constant pressure; L, the specimen length; t, the time; and a, the thermal diffusivity. It is a common practice [5] to employ the halfmaximum temperature rise time t1/2 on the rear face of the sample, for which o is equal to 1.38, to determine the thermal diffusivity a, which is given by the following expression: a¼
1:38L2 . p2 t1=2
(3)
The thermal conductivity may be derived from the thermal diffusivity with the help of Eq. (1). In this case, the specific heat and the sample bulk density must be previously known. There are some advantages in calculating the thermal conductivity starting from the thermal diffusivity: (1) the equation for the thermal diffusivity calculation is independent of the heat flux and the temperature gradient, (2) heat losses may be treated analytically and determined during the experiment, (3) data acquisition is very rapid, and (4) the use of small samples permit the preparation of homogeneous specimens. Details of this technique have been discussed on other occasions [6–8]. 2.2. The hot-wire technique The hot-wire technique, described by Schieirmacher in 1888 [9], is an absolute, non-steady state (transient heat flux) and direct method, and therefore it makes use of standards unnecessary. Nowadays, the hot-wire method is considered as an effective and accurate means of determining the thermal conductivity of ceramic materials, and
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recently has been used in the determination of thermal properties of polymers [3]. However, it is not possible to use this method for electrically conductive materials, unless some process of electric insulation between the hot wire and the sample is developed. It is a very useful technique, since it is possible to measure thermal conductivity at room temperature, which is impossible when using calorimetric methods. In addition, with the hot-wire technique the concept of ‘‘mean temperature’’ between hot and cold face of a sample on thermal conductivity calculations is eliminated, since the measurement is carried out at a fixed temperature. The temperature gradient across the sample is very low, and this is another virtue of this technique, since an ideal method for measuring thermal conductivity would be the one capable of measuring this property across a zero temperature gradient throughout the sample. In the mathematical formulation of the method, the hot wire is assumed to be an ideal (mass ¼ 0, and therefore heat capacity ¼ 0) infinitely thin and long heat source (diameter ¼ 0), which is in an infinite surrounding material whose thermal conductivity is to be determined [10]. Applying a constant electric current through the wire, a constant amount of heat per unit time and unit length is released by the wire and propagates throughout the material. When using the hot-wire parallel technique, one of the four possible variations of the hot-wire method, the thermal conductivity, is calculated according to the following equation [3]: q0 rcp r2 k¼ Ei , (4) 4pTðtÞ 4kt where k is the thermal conductivity of the material (W/(m K)); q0 , the linear power density (W/m); r, the material bulk density (kg/m3); cp, the specific heat of the material (J/(kg K)); r, the distance between hot wire and thermocouple (m); t, the elapsed time after beginning of heat release (s); T(t), the temperature rise registered by the thermocouple related to the initial reference temperature (K); and Ei(x) is the exponential integral function. The recommended distance r between the hot wire and the thermocouple is from 15 to 17 mm. The calculations, starting from the recorded temperature transient in the sample, are carried out by using a non-linear least-squares fitting method [11]. Both thermal conductivity and specific heat in Eq. (4) are fitted in order to obtain the best possible approx-
imation between the thermal transient experimentally determined and that predicted by the theoretical model. In this case, these two thermal properties, thermal conductivity and specific heat, are simultaneously determined from the same experimental transient. So, with the knowledge of the density, thermal diffusivity is then calculated by using Eq. (1). Details of this technique have been described on several occasions [12–14]. 3. Samples preparation 3.1. The laser flash technique Seven sets of commercial polymer samples in the shape of discs, 10 mm diameter and 0.3–1 mm thickness were prepared either by hot pressing the pellets of the solid polymer, or by cutting the discs from long cylindrical bars. A special stainless-steel mould was developed for this purpose. All samples were coated on both faces with a very thin layer of colloidal graphite, a step required in this technique, to improve their emissivity. With the purpose of eliminating the sample transparency, a thin layer of gold was sputtered on the front face of the specimen prior to the graphite coating. The mass of each sample was approximately 50 mg. The time required for the thermal equilibrium to be reached in each temperature measurement lies between 2 and 10 min. Since measurements were carried out from room temperature up to approximately 50 1C above the melting point, a special sapphire crucible was used as a sample holder in order to contain the molten polymer sample. Sapphire is a suitable material for this purpose because it is transparent to the laser beam and it does not react with the polymer. Colloidal graphite was sprayed onto the crucible in the regions shown in Fig. 1 to prevent laser light passing through the sides of the crucible and emerging from the rear to be detected by the infrared detector. 3.2. The hot-wire technique Seven sets of samples were prepared through the extrusion process starting from the powder or pellets of the solid polymer. A special mould of stainless steel with inner dimensions of (220 100 65) mm in the shape of a rectangular parallelepiped, provided with ceramic insulators between the hot wire, thermocouple and the mould
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4. Results
was employed to store the melt extruded polymer, whose thermal properties will be measured. The mass of each sample was approximately 1500 g. The time required for thermal equilibrium to be reached in each temperature measurement in this technique is approximately 8 h. Once thermal equilibrium is attained, the experimental measurement process is started, decreasing the temperature from the maximum value to room temperature within the desired temperature interval. Fig. 2 shows the stainless-steel mould used in this work. Details of sample preparation are given elsewhere [2,3,15]. Seven thermoplastic polymers of major commercial importance were selected—three amorphous thermoplastics: polystyrene (PS), high-impact polystyrene (HIPS), and polymethylmethacrylate (PMMA); three commodity semi-crystalline thermoplastics: low-density polyethylene (LDPE), highdensity polyethylene (HDPE) and polypropylene (PP); and an engineering thermoplastic: poly(hexamethylene-adipimide)-Pa-6,6.
Thermal conductivity (W/mK)
Fig. 2. Stainless-steel mould.
0.4
hot wire manufacturer literature laser flash
0.3
Tg
0.2 0.1
Solid
Viscous
0 0
50
100
150
200
250
300
Temperature (°C) Fig. 3. PS: thermal conductivity.
3000 Specific heat (J/kgK)
Fig. 1. Sapphire crucible.
Measurements were carried out from room temperature up to approximately 50 1C above the polymer melting point. In the hot wire technique, thermal conductivity and specific heat were simultaneously determined in the same experiment, and the thermal diffusivity was calculated using Eq. (1). In the laser flash technique, only the thermal diffusivity is experimentally determined in this experiment. Specific heat was determined using a modulated differential scanning calorimeter, and the polymer density at each temperature was obtained from the PVT curve for the corresponding polymer. Thermal conductivity was also derived with the aid of Eq. (1). Figs. 3–23 show the experimental results obtained. The total error in the calculations of the thermal properties using the equipments and the procedures proposed in this work is estimated, for both techniques, to lie within 75% [2,3].
hot Wire manufacturer modulated dsc
2500
Tg
2000 1500
Solid
Viscous
1000 0
50
100
150
200
Temperature (°C) Fig. 4. PS: specific heat.
250
300
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5. Discussion
1.4 1.3 1.2 1.1 1 0.9 0.8 0.7 0.6 0.5
3000 Specific heat (J/kgK)
hot wire manufacturer literature laser flash
Tg Solid
Viscous
hot wire manufacturer modulated DSC
2500
Tg 2000 1500
Viscous solid
1000 0
50
100
150
200
250
0
Temperature (°C)
50
100
150
Fig. 7. HIPS: specific heat.
0.3 Solid 0.25
hot wire manufacturer laser flash
Tg
0.2
0.15 Viscous 0.1 0
20
40
200
Temperature (°C)
Fig. 5. PS: thermal diffusivity.
Thermal conductivity (W/mK)
Thermal diffusivity (x 10-7m2/s)
The effect of the sapphire crucible may be evaluated when results obtained with and without the crucible are checked against each other. For this purpose, a set of a standard samples was prepared and measurements have been made under the same conditions, observing the same time to reach thermal equilibrium before measurement at each temperature. Fig. 24 gives the results obtained and shows that the influence of the crucible is negligible, since the maximum deviation is 5.74%, which lies within the experimental error of 5%. According to Rauwendaal [16], results from different measurement techniques may not agree and considerable differences in properties may occur for a particular polymer as a result of variations in molecular weight distribution, additives, thermomechanical history, etc. Specifically, in
the case of thermal properties, data in complete disagreement are found in literature for many polymers. Since the thermal history exerts a considerable influence on the thermal properties, this may be the reason for some large discrepancies between results obtained by hot wire when checked against those obtained by the laser flash technique. The mass of the samples in the laser flash technique is around 50 mg and the duration of the measurement process in the temperature range studied is about 4 h, while in the hot wire technique the mass involved is around 1500 g, and the measurement process takes about 5 days. So, it should be very difficult, or even impossible to get measurements in both techniques with samples submitted to the same thermal treatment. Each measurement process itself (hot wire and laser flash) generates different thermal histories. Concerning a specific property, the more sensitive the polymer is to the thermal history, the
60
80
100 120 140 160 180 200 220 240 260 Temperature (°C)
Fig. 6. HIPS: thermal conductivity.
250
300
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1.3 hot wire manufacturer laser flash
Thermal diffusivity (x 10-7m2/s)
1.2 1.1 1 0.9 0.8
Tg
0.7 0.6
Viscous
Solid
0.5 0
20
40
60
80
100
120
140
160
180
200
220
240
260
Temperature (°C) Fig. 8. HIPS: thermal diffusivity.
Thermal conductivity (W/mK)
0.3 hot wire literature laser flash
0.25
Tg Viscous
0.2
0.15 Solid
0.1 0
20
40
60
80
100
120
140
160
180
200
220
240
Temperature (°C) Fig. 9. PMMA: thermal conductivity.
Specific heat (J/kgK)
3000 Tg
2500
Viscous
2000 hot wire Literature modulated DSC
1500 Solid 1000 0
50
100
150
200
Temperature (°C) Fig. 10. PMMA: specific heat.
250
300
greater may be the discrepancy between experimental results obtained by both techniques. However, considering the experimental results obtained, it is possible to assert that both techniques, hot wire and laser flash, are suitable techniques for the determination of thermal properties of both solid and molten polymers. Perhaps, one or other method is more suitable in a specific case, depending on the material to be tested. There are advantages and disadvantages when using one or other technique. What should be pointed as a disadvantage of the hot-wire technique when compared with the laser flash technique is the
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Thermal diffusivity (x10-7m2/s)
1.2 viscous 1.0
0.8 laser flash hot wire literature
0.6
Tg
solid 0.4 20
0
40
60
100
80
120
140
160
180
200
220
240
Temperature (°C) Fig. 11. PMMA: thermal diffusivity.
Thermal conductivity (W/mK)
0.6 hot wire literature laser flash
0.5 0.4 0.3 0.2 0.1
Melt
Solid 0 0
40
20
60
80
100
120
140
160
Temperature (°C) Fig. 12. LDPE: thermal conductivity.
25000 hot wire literature modulated DSC
Specific heat (J/kgK)
20000
15000
10000
Melt
Solid
5000
0 0
20
40
60
80
100
120
Temperature (°C) Fig. 13. LDPE: specific heat.
140
160
180
200
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Thermal diffusivity (x10-7m2/s)
1.8 laser flash hot wire literature
1.6 1.4 1.2 1.0
melt
0.8 0.6
solid
0.4 0
20
40
60
80
100
120
140
180
160
200
Temperature (°C) Fig. 14. LDPE: thermal diffusivity.
0.6
Thermal conductivity (W/mK)
hot wire literature
0.5
laser flash
0.4 Melt 0.3 Solid 0.2
0.1 0
20
40
60
80
100
120
140
160
180
200
220
180
200
Temperature (°C) Fig. 15. HDPE: thermal conductivity.
25000 hot wire literature
Specific heat (J/kgK)
20000
modulated DSC 15000 Melt 10000 Solid 5000
0 0
20
40
60
80
100
120
Temperature (°C) Fig. 16. HDPE: specific heat.
140
160
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Thermal diffusivity (x10-7m2/s)
4.0 laser flash hot wire
3.5 3.0 2.5 2.0
melt
1.5 solid 1.0 0.5 0
20
60
40
80
100
120
140
160
180
200
220
Temperature (°C) Fig. 17. HDPE: thermal diffusivity.
Thermal conductivity (W/mK)
0.3 hot wire literature laser flash
0.25
0.2
Melt
Tg
0.15 Solid 0.1 0
20
60
40
80
100
120
140
160
180
200
220
Temperature (°C) Fig. 18. PP: thermal conductivity.
10000 hot wire
9000
literature
Specific heat (J/kgK)
8000
modulated DSC
Melt
7000 6000 Solid
Tg
5000 4000 3000 2000 1000 0
20
40
60
80
100
120
140
Temperature (°C) Fig. 19. PP: specific heat.
160
180
200
220
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1.6 Thermal diffusivity (x10-7m2/s)
laser flash 1.4
melt
hot wire literature
1.2 1.0 0.8
Tg
0.6 solid 0.4 20
0
40
60
80
100
120
140
160
180
200
Temperature (°C)
0.5 0.45 Tg
0.4 0.35
Melt
0.3 0.25 0.2
Solid
hot wire literature laser flash
0.15 0.1 0
50
100
150
200
250
300
Thermal diffusivity (x10-7m2/s)
Thermal conductivity (W/mK)
Fig. 20. PP: thermal diffusivity.
3.0 laser flash hot wire literature
2.5
melt
2.0 1.5 Tg
1.0
solid
0.5 0
50
100
150
200
250
300
Temperature (°C)
Temperature (°C)
Fig. 23. Nylon: thermal diffusivity. Fig. 21. Nylon: thermal conductivity.
Specific heat (J/kgK)
7000 Tg
6000
hot wire literature modulated DSC
Viscous
5000 4000 3000 2000
Solid
1000 0
50
100
150
200
250
300
Temperature (°C)
Fig. 22. Nylon: specific heat.
sample size. In the laser flash technique, samples are prepared in shape of discs 10 mm diameter and 0.1–1 mm thickness, permitting the preparation of very homogeneous specimens, and the total mass is around 50 mg, while in the case of the hot-wire technique samples are prepared in the shape of rectangular parallelepipeds or half cylinders and the
mass of material involved in this case is approximately 1500 g, giving a mass ratio of around 3 104. This fact clearly imposes a restriction concerning the application of this technique for some materials for which, either by technological or economic reasons, it is not possible to prepare samples with such dimensions. Another disadvantage of the hotwire technique when compared with the laser flash technique is also associated with the sample size— the time required for the sample to reach the thermal equilibrium at any measurement temperature. Depending on the thermal diffusivity of the polymer, sometimes 8–10 h are needed before any data acquisition can be made in order to assure that temperature distribution inside the sample is within the limits required by the theoretical model. The reason for this long time is the low thermal diffusivity of polymers and the big mass of material. If the size of the samples is decreased, this time interval certainly will also decrease, and as a consequence it will be possible to increase the range
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Thermal diffusivity (x10-7m2/s)
1.2 no crucible viscous
with crucible
1.0
solid 0.8 maximum deviation (%) = 5.74 0.6
0.4 0
20
40
60
80
100
120
140
160
180
200
220
Temperature (°C) Fig. 24. Standard sample: influence of the crucible.
of polymers for which this technique may be applied, and at the same time the total experimental time in any temperature interval will decrease considerably. However, the hot-wire technique has two advantages when compared with the laser flash technique. The first one is that in this technique, starting from the same experimentally determined thermal transient, the three thermal properties (thermal conductivity, thermal diffusivity and specific heat) are simultaneously determined, while in the laser flash technique only the thermal diffusivity is determined during the experiment. The second one is that the cost of the equipment for the hotwire technique is considerably lower when compared with the cost of a laser flash technique equipment. Acknowledgements The author acknowledges to CNPq (Proc. 300904/2005-3) and to FAPESP (Proc. 05/55556-7) for the financial support. References [1] W.N. dos Santos, O me´todo do fio quente: te´cnica em paralelo e te´cnica de superfı´ cie, Ceraˆmica 48 (306) (2002) 86. [2] W.N. dos Santos, P. Mummery, A. Wallwork, Thermal diffusivity of polymers by the laser flash technique, Polym. Test. 24 (5) (2005) 628.
[3] W.N. dos Santos, Thermal properties of melt polymers by the hot wire technique, Polym. Test. 24 (7) (2005) 932. [4] K.D. Maglic, A. Cezairliyan, V.E. Peletsky, Compendium of Thermophysical Property Measurement Methods, vol. 1, Kluwer Academic/Plenum Publishers, 1992. [5] W.J. Parker, R.J. Jenkins, C.P. Butter, G.L. Abbot, Flash method of determining thermal diffusivity, heat capacity, and thermal conductivity, J. Appl. Phys. (32) (1961) 1679. [6] R. Taylor, Construction of apparatus for heat pulse thermal diffusivity measurements from 300 to 3000 K, J. Phys. Sci. Instrum. 13 (1980) 1193. [7] J.A. Cape, G.W. Lehman, Temperature and finite pulse-time effects in the flash method for measuring thermal diffusivity, J. Appl. Phys. 34 (1963) 1909. [8] R.D. Cowan, Pulse method of measuring thermal diffusivity at high temperatures, J. Appl. Phys. 34 (1963) 1679. [9] A.L. Schieirmacher, Wiedemann Ann. Phys. 34 (1888) 38. [10] J. Boer, J. Butter, B. Grosskopf, P. Jeschke, Hot wire technique for determining high thermal conductivities, Refract. J. 55 (1980) 22. [11] W.N. dos Santos, J.S. Cintra Filho, O me´todo de fio quente, Ceraˆmica 37 (252) (1991) 101. [12] H.S. Carslaw, J.C. Jaeger, Conduction of Heat in Solids, Oxford University Press, Oxford, 1959. [13] W.E. Haupin, Hot wire method for rapid determination of thermal conductivity, Am. Ceram. Soc. Bull. 39 (3) (1960) 139. [14] W.N. dos Santos, R. Grego´rio Filho, Me´todo de fio quente na determinac- a˜o das propriedades te´rmicas de polı´ meros, Polı´ meros: Cieˆncia e Tecnologia 14 (5) (2004) 354. [15] W.N. dos Santos, J.A.M. Agnelli, P. Mummery, A. Wallwork, Effect of recycling on the thermal properties of polymers, Polym. Test. 26 (2007) 216. [16] Rauwendaal, Polymer Extrusion, fourth ed., Hanser Carl Hanser Verlag, 2001.
POLYMER TESTING Polymer Testing 26 (2007) 556–566 www.elsevier.com/locate/polytest
Test Method
Thermal properties of polymers by non-steady-state techniques Wilson Nunes dos Santos Department of Materials Engineering, Federal University of Sa˜o Carlos, Via Washington Luiz, km 235—CP 676, Sa˜o Carlos-SP, CEP 13565-905, Brazil Received 5 January 2007; accepted 10 February 2007
Abstract The nature of the molecular structure of polymers makes the properties of such materials markedly temperature dependant. Modelling heat transfer under steady-state or transient conditions is of fundamental importance in engineering design as well as for the tailoring of thermal and mechanical behaviour of materials. Thermal conductivity, thermal diffusivity and specific heat, namely the thermal properties, are the three most important physical properties of a material that are needed for heattransfer calculations. Nowadays, several different techniques for the determination of the thermal diffusivity and thermal conductivity may be found in the literature. Recently, transient techniques have become the preferable way for measuring thermal properties of materials. In this work, two absolute and non-steady-state methods are employed in the experimental determination of thermal properties of some selected polymers: the laser flash technique and the hot-wire technique. In the hotwire technique, samples were prepared from the extrusion process in the shape of a rectangular parallelepiped, with the molten mass for each sample being approximately 1500 g. Thermal conductivity, thermal diffusivity and specific heat were simultaneously determined from the same experimentally determined thermal transient. In the laser flash technique, disc shaped samples were prepared, either by hot pressing approximately 50 mg of material, or by cutting the discs from long cylindrical bars. In this technique, only the thermal diffusivity is determined from the experimentally determined thermal transient. The thermal conductivity was derived from the thermal diffusivity with the knowledge of the bulk density and the specific heat. Specific heat was experimentally determined using a modulated differential scanning calorimeter, and the density was obtained from the PVT curve at each temperature. Since the sample mass ratio between both techniques is approximately 3 104, and the speed of the cooling process may generate different morphologies, this phenomenon may be partially responsible for any discrepancy between both techniques for a specific polymer. Experimental results obtained by both techniques are checked against each other, as well as, when possible, with data found in literature. r 2007 Elsevier Ltd. All rights reserved. Keywords: Thermal properties; Non-steady-state techniques; Hot-wire technique; Laser flash technique; Melt polymers
1. Introduction The nature of the molecular structure of polymers makes the properties of such materials markedly temperature dependant. Modelling heat transfer Tel.: +55 16 3351 8529; fax: +55 16 3361 5404.
E-mail address: [email protected]. 0142-9418/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.polymertesting.2007.02.005
under steady-state or transient conditions is of fundamental importance in engineering design as well as for the tailoring of thermal and mechanical behaviour of materials. Thermal conductivity, thermal diffusivity and specific heat, namely the thermal properties, are the three most important physical properties of a material that are needed for heat-transfer calculations. These three properties
ARTICLE IN PRESS W.N. dos Santos / Polymer Testing 26 (2007) 556–566
are related by the equation: a¼
k , rcp
(1)
where a is the thermal diffusivity (m2/s), k the thermal conductivity (W/m K), r the bulk density (kg/m3) and cp the specific heat at constant pressure (J/kg K). Nowadays, several different techniques for the determination of the thermal diffusivity and thermal conductivity may be found in the literature [1–4]. Steady-state methods are those in which the desired property is measured in a steady-state heat exchange, and the calorimetric techniques employed in the determination of the thermal conductivity are examples of this category of methods. Non-steadystate methods are those in which the property is measured according to a transient regime of heat exchange. The laser flash technique is an example of a non-steady-state method. The non-steady-state methods may be further divided into two different categories: periodic heat flux and transient heat flux methods. In the periodic heat flux methods, the edge of a bar or a plate is alternately heated for a half period T/2, and cooled for the next T/2 period. The temperature measured at any point in the interior of the specimen varies periodically. When the steady state is reached, the thermal diffusivity may be evaluated from the temperature recorded at two selected fixed points. Angstrom’s method is an example of this technique for the determination of the thermal diffusivity. In the transient heat flux methods, like the laser flash technique, a heat pulse of short duration is incident on the front face of a specimen, and the thermal diffusivity is evaluated from the recorded temperature history on the rear face. The thermal conductivity may be derived from the thermal diffusivity if specific heat and bulk density are known, or alternatively, by using standard bodies. Recently, transient techniques have become the preferable way for measuring thermal properties of materials. In this work two absolute and non-steady-state methods are employed in the experimental determination of thermal properties of some selected polymers: the laser flash technique and the hot-wire technique. 2. Experimental 2.1. The laser flash technique The laser flash technique is a non-steady-state (transient heat flux), absolute and direct method for
557
thermal diffusivity measurements, which was introduced in 1961 by Parker et al. [5]. In this technique, a uniform heat pulse of short duration compared to the transit time throughout the sample is incident on the front face of a disc specimen, and the temperature rise on the rear face is recorded. If no heat losses occur, the increased normalised temperature on the rear face is given by V ¼1þ2
1 X ð1Þn expðn2 oÞ,
(2)
n¼1
where o ¼ p2at/L2; V ¼ T/Tm, is the dimensionless temperature increase of the rear face; T, the instantaneous temperature increase of the rear face of the specimen; Tm ¼ Q/rcpL, the maximum temperature increase of the rear face; Q, the input energy on the front face; r, the bulk density; cp, the specific heat at constant pressure; L, the specimen length; t, the time; and a, the thermal diffusivity. It is a common practice [5] to employ the halfmaximum temperature rise time t1/2 on the rear face of the sample, for which o is equal to 1.38, to determine the thermal diffusivity a, which is given by the following expression: a¼
1:38L2 . p2 t1=2
(3)
The thermal conductivity may be derived from the thermal diffusivity with the help of Eq. (1). In this case, the specific heat and the sample bulk density must be previously known. There are some advantages in calculating the thermal conductivity starting from the thermal diffusivity: (1) the equation for the thermal diffusivity calculation is independent of the heat flux and the temperature gradient, (2) heat losses may be treated analytically and determined during the experiment, (3) data acquisition is very rapid, and (4) the use of small samples permit the preparation of homogeneous specimens. Details of this technique have been discussed on other occasions [6–8]. 2.2. The hot-wire technique The hot-wire technique, described by Schieirmacher in 1888 [9], is an absolute, non-steady state (transient heat flux) and direct method, and therefore it makes use of standards unnecessary. Nowadays, the hot-wire method is considered as an effective and accurate means of determining the thermal conductivity of ceramic materials, and
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recently has been used in the determination of thermal properties of polymers [3]. However, it is not possible to use this method for electrically conductive materials, unless some process of electric insulation between the hot wire and the sample is developed. It is a very useful technique, since it is possible to measure thermal conductivity at room temperature, which is impossible when using calorimetric methods. In addition, with the hot-wire technique the concept of ‘‘mean temperature’’ between hot and cold face of a sample on thermal conductivity calculations is eliminated, since the measurement is carried out at a fixed temperature. The temperature gradient across the sample is very low, and this is another virtue of this technique, since an ideal method for measuring thermal conductivity would be the one capable of measuring this property across a zero temperature gradient throughout the sample. In the mathematical formulation of the method, the hot wire is assumed to be an ideal (mass ¼ 0, and therefore heat capacity ¼ 0) infinitely thin and long heat source (diameter ¼ 0), which is in an infinite surrounding material whose thermal conductivity is to be determined [10]. Applying a constant electric current through the wire, a constant amount of heat per unit time and unit length is released by the wire and propagates throughout the material. When using the hot-wire parallel technique, one of the four possible variations of the hot-wire method, the thermal conductivity, is calculated according to the following equation [3]: q0 rcp r2 k¼ Ei , (4) 4pTðtÞ 4kt where k is the thermal conductivity of the material (W/(m K)); q0 , the linear power density (W/m); r, the material bulk density (kg/m3); cp, the specific heat of the material (J/(kg K)); r, the distance between hot wire and thermocouple (m); t, the elapsed time after beginning of heat release (s); T(t), the temperature rise registered by the thermocouple related to the initial reference temperature (K); and Ei(x) is the exponential integral function. The recommended distance r between the hot wire and the thermocouple is from 15 to 17 mm. The calculations, starting from the recorded temperature transient in the sample, are carried out by using a non-linear least-squares fitting method [11]. Both thermal conductivity and specific heat in Eq. (4) are fitted in order to obtain the best possible approx-
imation between the thermal transient experimentally determined and that predicted by the theoretical model. In this case, these two thermal properties, thermal conductivity and specific heat, are simultaneously determined from the same experimental transient. So, with the knowledge of the density, thermal diffusivity is then calculated by using Eq. (1). Details of this technique have been described on several occasions [12–14]. 3. Samples preparation 3.1. The laser flash technique Seven sets of commercial polymer samples in the shape of discs, 10 mm diameter and 0.3–1 mm thickness were prepared either by hot pressing the pellets of the solid polymer, or by cutting the discs from long cylindrical bars. A special stainless-steel mould was developed for this purpose. All samples were coated on both faces with a very thin layer of colloidal graphite, a step required in this technique, to improve their emissivity. With the purpose of eliminating the sample transparency, a thin layer of gold was sputtered on the front face of the specimen prior to the graphite coating. The mass of each sample was approximately 50 mg. The time required for the thermal equilibrium to be reached in each temperature measurement lies between 2 and 10 min. Since measurements were carried out from room temperature up to approximately 50 1C above the melting point, a special sapphire crucible was used as a sample holder in order to contain the molten polymer sample. Sapphire is a suitable material for this purpose because it is transparent to the laser beam and it does not react with the polymer. Colloidal graphite was sprayed onto the crucible in the regions shown in Fig. 1 to prevent laser light passing through the sides of the crucible and emerging from the rear to be detected by the infrared detector. 3.2. The hot-wire technique Seven sets of samples were prepared through the extrusion process starting from the powder or pellets of the solid polymer. A special mould of stainless steel with inner dimensions of (220 100 65) mm in the shape of a rectangular parallelepiped, provided with ceramic insulators between the hot wire, thermocouple and the mould
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4. Results
was employed to store the melt extruded polymer, whose thermal properties will be measured. The mass of each sample was approximately 1500 g. The time required for thermal equilibrium to be reached in each temperature measurement in this technique is approximately 8 h. Once thermal equilibrium is attained, the experimental measurement process is started, decreasing the temperature from the maximum value to room temperature within the desired temperature interval. Fig. 2 shows the stainless-steel mould used in this work. Details of sample preparation are given elsewhere [2,3,15]. Seven thermoplastic polymers of major commercial importance were selected—three amorphous thermoplastics: polystyrene (PS), high-impact polystyrene (HIPS), and polymethylmethacrylate (PMMA); three commodity semi-crystalline thermoplastics: low-density polyethylene (LDPE), highdensity polyethylene (HDPE) and polypropylene (PP); and an engineering thermoplastic: poly(hexamethylene-adipimide)-Pa-6,6.
Thermal conductivity (W/mK)
Fig. 2. Stainless-steel mould.
0.4
hot wire manufacturer literature laser flash
0.3
Tg
0.2 0.1
Solid
Viscous
0 0
50
100
150
200
250
300
Temperature (°C) Fig. 3. PS: thermal conductivity.
3000 Specific heat (J/kgK)
Fig. 1. Sapphire crucible.
Measurements were carried out from room temperature up to approximately 50 1C above the polymer melting point. In the hot wire technique, thermal conductivity and specific heat were simultaneously determined in the same experiment, and the thermal diffusivity was calculated using Eq. (1). In the laser flash technique, only the thermal diffusivity is experimentally determined in this experiment. Specific heat was determined using a modulated differential scanning calorimeter, and the polymer density at each temperature was obtained from the PVT curve for the corresponding polymer. Thermal conductivity was also derived with the aid of Eq. (1). Figs. 3–23 show the experimental results obtained. The total error in the calculations of the thermal properties using the equipments and the procedures proposed in this work is estimated, for both techniques, to lie within 75% [2,3].
hot Wire manufacturer modulated dsc
2500
Tg
2000 1500
Solid
Viscous
1000 0
50
100
150
200
Temperature (°C) Fig. 4. PS: specific heat.
250
300
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5. Discussion
1.4 1.3 1.2 1.1 1 0.9 0.8 0.7 0.6 0.5
3000 Specific heat (J/kgK)
hot wire manufacturer literature laser flash
Tg Solid
Viscous
hot wire manufacturer modulated DSC
2500
Tg 2000 1500
Viscous solid
1000 0
50
100
150
200
250
0
Temperature (°C)
50
100
150
Fig. 7. HIPS: specific heat.
0.3 Solid 0.25
hot wire manufacturer laser flash
Tg
0.2
0.15 Viscous 0.1 0
20
40
200
Temperature (°C)
Fig. 5. PS: thermal diffusivity.
Thermal conductivity (W/mK)
Thermal diffusivity (x 10-7m2/s)
The effect of the sapphire crucible may be evaluated when results obtained with and without the crucible are checked against each other. For this purpose, a set of a standard samples was prepared and measurements have been made under the same conditions, observing the same time to reach thermal equilibrium before measurement at each temperature. Fig. 24 gives the results obtained and shows that the influence of the crucible is negligible, since the maximum deviation is 5.74%, which lies within the experimental error of 5%. According to Rauwendaal [16], results from different measurement techniques may not agree and considerable differences in properties may occur for a particular polymer as a result of variations in molecular weight distribution, additives, thermomechanical history, etc. Specifically, in
the case of thermal properties, data in complete disagreement are found in literature for many polymers. Since the thermal history exerts a considerable influence on the thermal properties, this may be the reason for some large discrepancies between results obtained by hot wire when checked against those obtained by the laser flash technique. The mass of the samples in the laser flash technique is around 50 mg and the duration of the measurement process in the temperature range studied is about 4 h, while in the hot wire technique the mass involved is around 1500 g, and the measurement process takes about 5 days. So, it should be very difficult, or even impossible to get measurements in both techniques with samples submitted to the same thermal treatment. Each measurement process itself (hot wire and laser flash) generates different thermal histories. Concerning a specific property, the more sensitive the polymer is to the thermal history, the
60
80
100 120 140 160 180 200 220 240 260 Temperature (°C)
Fig. 6. HIPS: thermal conductivity.
250
300
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1.3 hot wire manufacturer laser flash
Thermal diffusivity (x 10-7m2/s)
1.2 1.1 1 0.9 0.8
Tg
0.7 0.6
Viscous
Solid
0.5 0
20
40
60
80
100
120
140
160
180
200
220
240
260
Temperature (°C) Fig. 8. HIPS: thermal diffusivity.
Thermal conductivity (W/mK)
0.3 hot wire literature laser flash
0.25
Tg Viscous
0.2
0.15 Solid
0.1 0
20
40
60
80
100
120
140
160
180
200
220
240
Temperature (°C) Fig. 9. PMMA: thermal conductivity.
Specific heat (J/kgK)
3000 Tg
2500
Viscous
2000 hot wire Literature modulated DSC
1500 Solid 1000 0
50
100
150
200
Temperature (°C) Fig. 10. PMMA: specific heat.
250
300
greater may be the discrepancy between experimental results obtained by both techniques. However, considering the experimental results obtained, it is possible to assert that both techniques, hot wire and laser flash, are suitable techniques for the determination of thermal properties of both solid and molten polymers. Perhaps, one or other method is more suitable in a specific case, depending on the material to be tested. There are advantages and disadvantages when using one or other technique. What should be pointed as a disadvantage of the hot-wire technique when compared with the laser flash technique is the
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Thermal diffusivity (x10-7m2/s)
1.2 viscous 1.0
0.8 laser flash hot wire literature
0.6
Tg
solid 0.4 20
0
40
60
100
80
120
140
160
180
200
220
240
Temperature (°C) Fig. 11. PMMA: thermal diffusivity.
Thermal conductivity (W/mK)
0.6 hot wire literature laser flash
0.5 0.4 0.3 0.2 0.1
Melt
Solid 0 0
40
20
60
80
100
120
140
160
Temperature (°C) Fig. 12. LDPE: thermal conductivity.
25000 hot wire literature modulated DSC
Specific heat (J/kgK)
20000
15000
10000
Melt
Solid
5000
0 0
20
40
60
80
100
120
Temperature (°C) Fig. 13. LDPE: specific heat.
140
160
180
200
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Thermal diffusivity (x10-7m2/s)
1.8 laser flash hot wire literature
1.6 1.4 1.2 1.0
melt
0.8 0.6
solid
0.4 0
20
40
60
80
100
120
140
180
160
200
Temperature (°C) Fig. 14. LDPE: thermal diffusivity.
0.6
Thermal conductivity (W/mK)
hot wire literature
0.5
laser flash
0.4 Melt 0.3 Solid 0.2
0.1 0
20
40
60
80
100
120
140
160
180
200
220
180
200
Temperature (°C) Fig. 15. HDPE: thermal conductivity.
25000 hot wire literature
Specific heat (J/kgK)
20000
modulated DSC 15000 Melt 10000 Solid 5000
0 0
20
40
60
80
100
120
Temperature (°C) Fig. 16. HDPE: specific heat.
140
160
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Thermal diffusivity (x10-7m2/s)
4.0 laser flash hot wire
3.5 3.0 2.5 2.0
melt
1.5 solid 1.0 0.5 0
20
60
40
80
100
120
140
160
180
200
220
Temperature (°C) Fig. 17. HDPE: thermal diffusivity.
Thermal conductivity (W/mK)
0.3 hot wire literature laser flash
0.25
0.2
Melt
Tg
0.15 Solid 0.1 0
20
60
40
80
100
120
140
160
180
200
220
Temperature (°C) Fig. 18. PP: thermal conductivity.
10000 hot wire
9000
literature
Specific heat (J/kgK)
8000
modulated DSC
Melt
7000 6000 Solid
Tg
5000 4000 3000 2000 1000 0
20
40
60
80
100
120
140
Temperature (°C) Fig. 19. PP: specific heat.
160
180
200
220
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1.6 Thermal diffusivity (x10-7m2/s)
laser flash 1.4
melt
hot wire literature
1.2 1.0 0.8
Tg
0.6 solid 0.4 20
0
40
60
80
100
120
140
160
180
200
Temperature (°C)
0.5 0.45 Tg
0.4 0.35
Melt
0.3 0.25 0.2
Solid
hot wire literature laser flash
0.15 0.1 0
50
100
150
200
250
300
Thermal diffusivity (x10-7m2/s)
Thermal conductivity (W/mK)
Fig. 20. PP: thermal diffusivity.
3.0 laser flash hot wire literature
2.5
melt
2.0 1.5 Tg
1.0
solid
0.5 0
50
100
150
200
250
300
Temperature (°C)
Temperature (°C)
Fig. 23. Nylon: thermal diffusivity. Fig. 21. Nylon: thermal conductivity.
Specific heat (J/kgK)
7000 Tg
6000
hot wire literature modulated DSC
Viscous
5000 4000 3000 2000
Solid
1000 0
50
100
150
200
250
300
Temperature (°C)
Fig. 22. Nylon: specific heat.
sample size. In the laser flash technique, samples are prepared in shape of discs 10 mm diameter and 0.1–1 mm thickness, permitting the preparation of very homogeneous specimens, and the total mass is around 50 mg, while in the case of the hot-wire technique samples are prepared in the shape of rectangular parallelepipeds or half cylinders and the
mass of material involved in this case is approximately 1500 g, giving a mass ratio of around 3 104. This fact clearly imposes a restriction concerning the application of this technique for some materials for which, either by technological or economic reasons, it is not possible to prepare samples with such dimensions. Another disadvantage of the hotwire technique when compared with the laser flash technique is also associated with the sample size— the time required for the sample to reach the thermal equilibrium at any measurement temperature. Depending on the thermal diffusivity of the polymer, sometimes 8–10 h are needed before any data acquisition can be made in order to assure that temperature distribution inside the sample is within the limits required by the theoretical model. The reason for this long time is the low thermal diffusivity of polymers and the big mass of material. If the size of the samples is decreased, this time interval certainly will also decrease, and as a consequence it will be possible to increase the range
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Thermal diffusivity (x10-7m2/s)
1.2 no crucible viscous
with crucible
1.0
solid 0.8 maximum deviation (%) = 5.74 0.6
0.4 0
20
40
60
80
100
120
140
160
180
200
220
Temperature (°C) Fig. 24. Standard sample: influence of the crucible.
of polymers for which this technique may be applied, and at the same time the total experimental time in any temperature interval will decrease considerably. However, the hot-wire technique has two advantages when compared with the laser flash technique. The first one is that in this technique, starting from the same experimentally determined thermal transient, the three thermal properties (thermal conductivity, thermal diffusivity and specific heat) are simultaneously determined, while in the laser flash technique only the thermal diffusivity is determined during the experiment. The second one is that the cost of the equipment for the hotwire technique is considerably lower when compared with the cost of a laser flash technique equipment. Acknowledgements The author acknowledges to CNPq (Proc. 300904/2005-3) and to FAPESP (Proc. 05/55556-7) for the financial support. References [1] W.N. dos Santos, O me´todo do fio quente: te´cnica em paralelo e te´cnica de superfı´ cie, Ceraˆmica 48 (306) (2002) 86. [2] W.N. dos Santos, P. Mummery, A. Wallwork, Thermal diffusivity of polymers by the laser flash technique, Polym. Test. 24 (5) (2005) 628.
[3] W.N. dos Santos, Thermal properties of melt polymers by the hot wire technique, Polym. Test. 24 (7) (2005) 932. [4] K.D. Maglic, A. Cezairliyan, V.E. Peletsky, Compendium of Thermophysical Property Measurement Methods, vol. 1, Kluwer Academic/Plenum Publishers, 1992. [5] W.J. Parker, R.J. Jenkins, C.P. Butter, G.L. Abbot, Flash method of determining thermal diffusivity, heat capacity, and thermal conductivity, J. Appl. Phys. (32) (1961) 1679. [6] R. Taylor, Construction of apparatus for heat pulse thermal diffusivity measurements from 300 to 3000 K, J. Phys. Sci. Instrum. 13 (1980) 1193. [7] J.A. Cape, G.W. Lehman, Temperature and finite pulse-time effects in the flash method for measuring thermal diffusivity, J. Appl. Phys. 34 (1963) 1909. [8] R.D. Cowan, Pulse method of measuring thermal diffusivity at high temperatures, J. Appl. Phys. 34 (1963) 1679. [9] A.L. Schieirmacher, Wiedemann Ann. Phys. 34 (1888) 38. [10] J. Boer, J. Butter, B. Grosskopf, P. Jeschke, Hot wire technique for determining high thermal conductivities, Refract. J. 55 (1980) 22. [11] W.N. dos Santos, J.S. Cintra Filho, O me´todo de fio quente, Ceraˆmica 37 (252) (1991) 101. [12] H.S. Carslaw, J.C. Jaeger, Conduction of Heat in Solids, Oxford University Press, Oxford, 1959. [13] W.E. Haupin, Hot wire method for rapid determination of thermal conductivity, Am. Ceram. Soc. Bull. 39 (3) (1960) 139. [14] W.N. dos Santos, R. Grego´rio Filho, Me´todo de fio quente na determinac- a˜o das propriedades te´rmicas de polı´ meros, Polı´ meros: Cieˆncia e Tecnologia 14 (5) (2004) 354. [15] W.N. dos Santos, J.A.M. Agnelli, P. Mummery, A. Wallwork, Effect of recycling on the thermal properties of polymers, Polym. Test. 26 (2007) 216. [16] Rauwendaal, Polymer Extrusion, fourth ed., Hanser Carl Hanser Verlag, 2001.