218 7 Electrical Properties Table 7.1.Effectiveness of self-heating from a temperature of 19C Material Maximum Time to reach Power Volume temperature half of the (W) resistivity (C) maximum (2cm) temperature rise 1.Steel fiber 60 6min 5.6 0.85 (0.7 vol%)cement 2.Carbon fiber 6 4 min 1.8 100 (1.0vol%)cement 3.Graphite particle 24 4min 0.27 410 (37 vol%)cement 4.Carbon fiber 134 2 min 6.5 0.11 (uncoated)mat 5.Ni/Cu/Ni-coated 79 14s 3.0 0.07 carbon fiber mat 6.Carbon fiber epoxy- 89 16s 0.59 b matrix interlaminar interface 7.Flexible graphite 980 4s 94 7.5×10-4 Not a structural material bThe relevant quantity is the contact resistivity rather than the volume resistivity The power in Table 7.1 is the electrical power input,which is essentially equal to the heat power output after an initial period in which the material itself is being heated.The power is governed by the ability of the material to sustain current and voltage.This ability is enhanced by a decrease in resistivity. Although the resistivity is not the only criterion that governs the effectiveness of a material for self-heating,it is the dominant criterion,particularly in relation to the power and the maximum temperature.In general,the selection of a self- heating structural material depends on the requirements in terms of the maximum temperature,the power response time,and the mechanical properties.For cement- based structures,steel fiber cement(No.1 in Table 7.1)is recommended.For a continuous fiber polymer-matrix composite,carbon fiber mat(No.4in Table 7.1) is recommended for use as an interlayer.For spatially distributed heating,the carbon fiber epoxy-matrix interlaminar interface is recommended. Flexible graphite cannot be incorporated into a structural composite due to its mechanical weakness and impermeability to resin.However,it can be placed on a structural material,and its flexibility allows it to conform to the topography of the structural material. 7.5.2.5 Summary of Self-heating Structural Materials Self-heating structural materials in the form of cement-matrix and polymer- matrix composites have been engineered by using electrically conductive fibers (continuous or discontinuous)and interlayers.Both the volume of the composite and the interlaminar interface can be used as heating elements.The interlaminar
218 7 Electrical Properties Table 7.1. Effectiveness of self-heating from a temperature of 19°C Material Maximum Time to reach Power Volume temperature half of the (W) resistivity (°C) maximum (Ωcm) temperature rise 1. Steel fiber 60 6 min 5.6 0.85 (0.7vol%) cement 2. Carbon fiber 56 4 min 1.8 100 (1.0vol%) cement 3. Graphite particle 24 4 min 0.27 410 (37vol%) cement 4. Carbon fiber 134 2 min 6.5 0.11 (uncoated) mat 5. Ni/Cu/Ni-coated 79 14 s 3.0 0.07 carbon fiber mat 6. Carbon fiber epoxy- 89 16 s 0.59 –b matrix interlaminar interface 7. Flexible graphitea 980 4 s 94 7.5 × 10−4 a Not a structural material b The relevant quantity is the contact resistivity rather than the volume resistivity The power in Table 7.1 is the electrical power input, which is essentially equal to the heat power output after an initial period in which the material itself is being heated. The power is governed by the ability of the material to sustain current and voltage. This ability is enhanced by a decrease in resistivity. Although the resistivity is not the only criterion that governs the effectiveness of a material for self-heating, it is the dominant criterion, particularly in relation to the power and the maximum temperature. In general, the selection of a selfheating structural material depends on the requirements in terms of the maximum temperature, the power response time, and the mechanical properties. For cementbased structures, steel fiber cement (No. 1 in Table 7.1) is recommended. For a continuous fiber polymer–matrix composite, carbon fiber mat (No. 4 in Table 7.1) is recommended for use as an interlayer. For spatially distributed heating, the carbon fiber epoxy-matrix interlaminar interface is recommended. Flexible graphite cannot be incorporated into a structural composite due to its mechanical weakness and impermeability to resin. However, it can be placed on a structural material, and its flexibility allows it to conform to the topography of the structural material. 7.5.2.5 Summary of Self-heating Structural Materials Self-heating structural materials in the form of cement-matrix and polymer– matrix composites have been engineered by using electrically conductive fibers (continuous or discontinuous) and interlayers. Both the volume of the composite and the interlaminar interface can be used as heating elements. The interlaminar
7.6 Effect of Temperature on the Electrical Resistivity 219 interface between continuous carbon fiber laminae is attractive since it is amenable to providing a two-dimensional array of heating elements.A cement-matrix com- posite containing 0.7 vol%steel fibers(8 um diameter)and a mat of discontinuous uncoated carbon fibers for use as an interlayer are effective for self-heating.How- ever,the effectiveness of such a system is low compared to flexible graphite,which is not a structural material. 7.6 Effect of Temperature on the Electrical Resistivity 7.6.1 Scientific Basis For a given material,the volume electrical resistivity depends on the temperature. This dependence means that the resistivity can be employed as an indicator of the temperature;in other words,by measuring the resistance,one can obtain the temperature.This type of temperature sensor is known as a thermistor. The dependence of the volume electrical resistivity on temperature is expressed by the temperature coefficient of electrical resistivity(a),which is defined as (△g)/eo=c△T, (7.49) where po is the resistivity at 20C and Ap is the change in resistivity relative to o when the temperature is increased or decreased from 20C by AT.The units of a areC-1.Since △p=P-Po, (7.50) Equation 7.49 can be rewritten as P=Po(1+cx△T). (7.51) Thus,the plot of o vs.AT gives a line with a slope of oa.This line tends to be straight only for metals.However,temperature sensing based on this phenomenon does not require the curve to be a straight line.The greater the magnitude of a, the more sensitive the thermistor. For metals,a is positive;in other words,the resistivity increases with increasing temperature.This is because the amplitudes of the thermal vibrations of the atoms in the metal increase with increasing temperature,thereby decreasing the mobility u(as defined in Eq.7.1).A decrease in mobility in turn results in a decrease in the conductivity(Eq.7.7);in other words,an increase in the resistivity.The carrier concentration n of a metal does not change with temperature,due to the availability of the valence electrons as mobile charges without any need to excite them.When the temperature is 0K,there is no thermal energy to provide thermal vibrations, so the resistivity at this temperature is not due to thermal vibrations but rather the scattering of the electrons(carriers)at any defects present in the material.The resistivity at 0K is known as the residual resistivity(Fig.7.8).The greater the defect concentration,the higher the residual resistivity
7.6 Effect of Temperature on the Electrical Resistivity 219 interface between continuous carbon fiber laminae is attractive since it is amenable to providing a two-dimensional array of heating elements. A cement-matrix composite containing 0.7vol% steel fibers (8μm diameter) and a mat of discontinuous uncoated carbon fibers for use as an interlayer are effective for self-heating. However, the effectiveness of such a system is low compared to flexible graphite, which is not a structural material. 7.6 Effect of Temperature on the Electrical Resistivity 7.6.1 Scientific Basis For a given material, the volume electrical resistivity depends on the temperature. This dependence means that the resistivity can be employed as an indicator of the temperature; in other words, by measuring the resistance, one can obtain the temperature. This type of temperature sensor is known as a thermistor. The dependence of the volume electrical resistivity on temperature is expressed by the temperature coefficient of electrical resistivity (α), which is defined as (Δρ)/ρo = αΔT , (7.49) where ρo is the resistivity at 20°C and Δρ is the change in resistivity relative to ρo when the temperature is increased or decreased from 20°C by ΔT. The units of α are °C−1. Since Δρ = ρ − ρo , (7.50) Equation 7.49 can be rewritten as ρ = ρo(1 + αΔT) . (7.51) Thus, the plot of ρ vs. ΔT gives a line with a slope of ρoα. This line tends to be straight only for metals. However, temperature sensing based on this phenomenon does not require the curve to be a straight line. The greater the magnitude of α, the more sensitive the thermistor. For metals, α is positive; in other words, the resistivity increases with increasing temperature. This is because the amplitudes of the thermal vibrations of the atoms in the metal increase with increasing temperature, thereby decreasing the mobility μ (as defined in Eq. 7.1). A decrease in mobility in turn results in a decrease in the conductivity (Eq. 7.7); in other words, an increase in the resistivity. The carrier concentration n of a metal does not change with temperature, due to the availability of the valence electrons as mobile charges without any need to excite them. When the temperature is 0K, there is no thermal energy to provide thermal vibrations, so the resistivity at this temperature is not due to thermal vibrations but rather the scattering of the electrons (carriers) at any defects present in the material. The resistivity at 0K is known as the residual resistivity (Fig. 7.8). The greater the defect concentration, the higher the residual resistivity
220 7 Electrical Properties Resistivity p P (O) Residual resistivity at T=OK due to defect scattering Temperature(K)→ Figure7.8.Dependence of the volume electrical resistivity on the temperature near OK,showing the residual resistivity atOK Semiconductors (e.g.,silicon and germanium)and semimetals(e.g.,graphite) together constitute a class of materials known as metalloids (or semimetals). For metalloids,a is negative.This is because,for these materials,n increases significantly with increasing temperature.This trend in n is due to the thermal excitation of electrons to form mobile carriers.For a semiconductor,the valence electrons need to be excited across the energy band gap in order for them to become mobile.The larger the energy band gap,the greater the effect of temperature on the resistivity.For a semimetal there is no energy band gap,only a small band overlap, thus resulting in a small effect of temperature on the resistivity.The increase in n with increasing temperature predominates over the decrease in mobility u with increasing temperature(Eq.7.7),thus resulting in an overall effect in which the conductivity increases with increasing temperature (i.e.,the resistivity decreases with increasing temperature). For metals,the value of a is typically around +0.004/C.For semiconductors and semimetals,a is negative,but its magnitude is higher for semiconductors than for semimetals due to the energy band gap present in a semiconductor. Thus,the magnitude of a is smaller for carbon (a semimetal)than for silicon or germanium (semiconductors).The magnitude of a is larger for silicon than germanium due to the larger energy band gap for silicon(1.12eV,compared to 0.68eV for germanium).The energy band gap(also known as the energy gap)is the energy between the top of the valance band and the bottom of the conduction band above the valence band,and describes the energy needed to excite an electron from the valence band to the conduction band so that the electron becomes free to respond to an applied electric field. For a composite with an electrically nonconductive matrix and a conductive discontinuous filler(particles or fibers)where the CTE of the matrix is higher than that of the filler,increasing the temperature causes more thermal expansion of the matrix than the filler.As a consequence,the degree of contact between adjacent filler units(i.e.,adjacent particles or adjacent fibers)is reduced,causing the volume resistivity of the composite to increase.The resistivity typically increases abruptly
220 7 Electrical Properties Figure 7.8. Dependence of the volume electrical resistivity on the temperature near 0K, showing the residual resistivity at 0K Semiconductors (e.g., silicon and germanium) and semimetals (e.g., graphite) together constitute a class of materials known as metalloids (or semimetals). For metalloids, α is negative. This is because, for these materials, n increases significantly with increasing temperature. This trend in n is due to the thermal excitation of electrons to form mobile carriers. For a semiconductor, the valence electrons need to be excited across the energy band gap in order for them to become mobile. The larger the energy band gap, the greater the effect of temperature on the resistivity. For a semimetal there is no energy band gap, only a small band overlap, thus resulting in a small effect of temperature on the resistivity. The increase in n with increasing temperature predominates over the decrease in mobility μ with increasing temperature (Eq. 7.7), thus resulting in an overall effect in which the conductivity increases with increasing temperature (i.e., the resistivity decreases with increasing temperature). For metals, the value of α is typically around +0.004/°C. For semiconductors and semimetals, α is negative, but its magnitude is higher for semiconductors than for semimetals due to the energy band gap present in a semiconductor. Thus, the magnitude of α is smaller for carbon (a semimetal) than for silicon or germanium (semiconductors). The magnitude of α is larger for silicon than germanium due to the larger energy band gap for silicon (1.12eV, compared to 0.68eV for germanium). The energy band gap (also known as the energy gap) is the energy between the top of the valance band and the bottom of the conduction band above the valence band, and describes the energy needed to excite an electron from the valence band to the conduction band so that the electron becomes free to respond to an applied electric field. For a composite with an electrically nonconductive matrix and a conductive discontinuous filler (particles or fibers) where the CTE of the matrix is higher than that of the filler, increasing the temperature causes more thermal expansion of the matrix than the filler. As a consequence, the degree of contact between adjacent filler units (i.e., adjacent particles or adjacent fibers) is reduced, causing the volume resistivity of the composite to increase. The resistivity typically increases abruptly
7.6 Effect of Temperature on the Electrical Resistivity 221 when the increase in temperature has reached a sufficiently large value.This means that the curve of resistivity vs.temperature is not linear.When the filler volume fraction is around the percolation threshold,the increase in resistivity is particularly large.This is because the resistivity is particularly sensitive to the degree of contact between adjacent filler units when the filler volume fraction is around the percolation threshold.This phenomenon is useful for temperature- activated switching-the deactivation of a circuit(due to the high resistivity of the composite used in series in the circuit)when the temperature is higher than a critical value.The composite therefore serves as a fuse that protects the electronics from excessive temperatures. 7.6.2 Structural Materials Used as Thermistors 7.6.2.1 Polymer-Matrix Structural Composites Used as Thermistors and Thermal Damage Sensors Continuous fiber polymer-matrix composites are structural composites,in con- trast to the relative weaknesses of short-fiber polymer-matrix composites.The interlaminar interface of a continuous carbon fiber polymer-matrix composite is associated with a contact (two-dimensional)resistivity that decreases quite re- versibly upon heating,as shown in Fig.7.9,thereby allowing it to be a thermistor. The contact resistivity of the interface decreases with increasing temperature be- cause of the energy needed for electrons to jump from one lamina to the next through the interlaminar interface.The higher the temperature,the greater the thermal energy available,and the larger the proportion of electrons that manage to jump across the interface.By using the configuration shown in Fig.7.6,one can 0006 Heating Cooling 0.003+ 0.001 50 70 90 110 130 150 Temperature (C) Figure 7.9.Variation in the contact electrical resistivity with temperature during heating and cooling at 0.15C/min of the crossply interlaminar interface of a continuous carbon fiber epoxy-matrix composite made with a curing pressure of 0.33 MPa(from[3])
7.6 Effect of Temperature on the Electrical Resistivity 221 when the increase in temperature has reached a sufficiently large value. This means that the curve of resistivity vs. temperature is not linear. When the filler volume fraction is around the percolation threshold, the increase in resistivity is particularly large. This is because the resistivity is particularly sensitive to the degree of contact between adjacent filler units when the filler volume fraction is around the percolation threshold. This phenomenon is useful for temperatureactivated switching – the deactivation of a circuit (due to the high resistivity of the composite used in series in the circuit) when the temperature is higher than acriticalvalue.Thecompositethereforeservesasafusethatprotectstheelectronics from excessive temperatures. 7.6.2 Structural Materials Used as Thermistors 7.6.2.1 Polymer-Matrix Structural Composites Used as Thermistors and Thermal Damage Sensors Continuous fiber polymer-matrix composites are structural composites, in contrast to the relative weaknesses of short-fiber polymer-matrix composites. The interlaminar interface of a continuous carbon fiber polymer-matrix composite is associated with a contact (two-dimensional) resistivity that decreases quite reversibly upon heating, as shown in Fig. 7.9, thereby allowing it to be a thermistor. The contact resistivity of the interface decreases with increasing temperature because of the energy needed for electrons to jump from one lamina to the next through the interlaminar interface. The higher the temperature, the greater the thermal energy available, and the larger the proportion of electrons that manage to jump across the interface. By using the configuration shown in Fig. 7.6, one can Figure 7.9. Variation in the contact electrical resistivity with temperature during heating and cooling at 0.15°C/min of the crossply interlaminar interface of a continuous carbon fiber epoxy-matrix composite made with a curing pressure of 0.33 MPa (from [3])
222 7 Electrical Properties 6.6 6.5 6.3+ 0.00250.00250.0026000270.00270.00280.00280.002900029 1T(11K) Figure 7.10.Arrhenius plot of log contact conductivity versus inverse absolute temperature during heating at 0.15C/min for the crossply interlaminar interface of a continuous carbon fiber epoxy-matrix composite made with a curing pressure of 0.33MPa(om[3]) obtain a two-dimensional array of thermistors,thereby enabling spatially resolved temperature sensing. Corresponding Arrhenius plots of log contact conductivity (inverse of con- tact resistivity)versus inverse absolute temperature during heating are shown in Fig.7.10.From the slope(negative)of the Arrhenius plot,which is quite linear,the activation energy can be calculated by using the equation Slope =-E/(2.3kg), (7.52) where kg is the Boltzmann constant,T is the absolute temperature(in K)and E is the activation energy,which is thus found to be 0.118eV.This activation energy is the energy needed for an electron to jump from one lamina to the next.Exciting electrons to this energy enables conduction in the through-thickness direction. Figure 7.11 shows the sensing of both temperature and thermal damage through the measurement of the contact electrical resistivity (the geometry-independent quantity equal to the product of the contact resistance and the contact area)of the interlaminar interface.The resistivity decreases upon heating during each heating cycle due to the activation energy associated with the movement ofelectrons across the interface.This is the thermistor effect,which allows temperature sensing.Minor thermal damage at the maximum temperature of the hottest cycle causes a spike in the resistivity,allowing damage sensing. Polymer-matrix composites with conductive short fibers or particles can also function as thermistors,due to the increase in spacing between adjacent filler units as the polymer matrix(which has a high CTE compared to the filler)expands upon heating.The increase in spacing means that the chance of filler units touching one another to form a continuous conductive path is decreased,thereby causing the resistivity of the composite to increase.Instead of increasing smoothly with increasing temperature,the resistivity tends to increase abruptly as the temperature
222 7 Electrical Properties Figure 7.10. Arrhenius plot of log contact conductivity versus inverse absolute temperature during heating at 0.15°C/min for the crossply interlaminar interface of a continuous carbon fiber epoxy-matrix composite made with a curing pressure of 0.33MPa (from [3]) obtain a two-dimensional array of thermistors, thereby enabling spatially resolved temperature sensing. Corresponding Arrhenius plots of log contact conductivity (inverse of contact resistivity) versus inverse absolute temperature during heating are shown in Fig. 7.10. From the slope (negative) of the Arrhenius plot, which is quite linear, the activation energy can be calculated by using the equation Slope = −E/(2.3 kB) , (7.52) where kB is the Boltzmann constant, T is the absolute temperature (in K) and E is the activation energy, which is thus found to be 0.118eV. This activation energy is the energy needed for an electron to jump from one lamina to the next. Exciting electrons to this energy enables conduction in the through-thickness direction. Figure 7.11 shows the sensing of both temperature and thermal damage through the measurement of the contact electrical resistivity (the geometry-independent quantity equal to the product of the contact resistance and the contact area) of the interlaminar interface. The resistivity decreases upon heating during each heating cycle due to the activation energy associated with the movement of electrons across theinterface.Thisisthethermistoreffect,whichallowstemperaturesensing.Minor thermal damage at the maximum temperature of the hottest cycle causes a spike in the resistivity, allowing damage sensing. Polymer-matrix composites with conductive short fibers or particles can also function as thermistors, due to the increase in spacing between adjacent filler units as the polymer matrix (which has a high CTE compared to the filler) expands upon heating. The increase in spacing means that the chance of filler units touching one another to form a continuous conductive path is decreased, thereby causing the resistivity of the composite to increase. Instead of increasing smoothly with increasingtemperature,theresistivitytendstoincreaseabruptlyasthetemperature