7 Electrical Properties This chapter covers the principles and applications of electrical behavior,empha- sizing those related to composite materials.This includes thermoelectric behavior and the effects of temperature and stress on electrical resistivity,as well as ap- plications such as thermoelectric energy generation,thermocouples,thermistors, heating (such as deicing),and electrical resistance-based sensing of strain,stress and damage.The materials used to make electrical connections are also covered. 7.1 Origin of Electrical Conduction Unless noted otherwise,an electrically conductive material utilizes electrons and/or holes as the charge carriers(mobile charged particles)for electrical con- duction.A hole is an electron-deficient site. In the presence ofan electric field(i.e.,a voltage gradient),a hole drifts toward the negative end of the voltage gradient due to electrostatic attraction,thus resulting in a current(defined as the charge per unit time)in the same direction.In the presence of an electric field,an electron drifts toward the positive end of the voltage gradient,thus resulting in a current(defined as the charge per unit time- not the magnitude of the charge per unit time)in the opposite direction.Hence,the currents resulting from electron drift and hole drift occur in the same direction, even though electrons and holes drift in opposite directions.This means that in a material with both holes and electrons,the total current is the sum of the current due to the holes and that due to the electrons.Drift is actually a term that refers to the movement of charge in response to an applied electric field. The drift velocity(v)is the velocity of the drift.It is actually the net velocity,since electrons or holes are quantum particles that obey the Heisenberg Uncertainty Principle,and so they constantly move,even in the absence of an applied electric field.The drift velocity is proportional to the applied electric field E,with the constant of proportionality being the mobility(),which describes the ease of movement of the carrier in the medium under consideration.Hence, v=uE. (7.1) Since the units of E are V/m and the units of v are m/s,the units of u are m2/(V s). For a given combination of carrier and medium,v depends on the temperature. 203
7 Electrical Properties This chapter covers the principles and applications of electrical behavior, emphasizing those related to composite materials. This includes thermoelectric behavior and the effects of temperature and stress on electrical resistivity, as well as applications such as thermoelectric energy generation, thermocouples, thermistors, heating (such as deicing), and electrical resistance-based sensing of strain, stress and damage. The materials used to make electrical connections are also covered. 7.1 Origin of Electrical Conduction Unless noted otherwise, an electrically conductive material utilizes electrons and/or holes as the charge carriers (mobile charged particles) for electrical conduction. A hole is an electron-deficient site. Inthepresenceofanelectricfield(i.e.,avoltagegradient),aholedriftstowardthe negative end of the voltage gradient due to electrostatic attraction, thus resulting in a current (defined as the charge per unit time) in the same direction. In the presence of an electric field, an electron drifts toward the positive end of the voltage gradient, thus resulting in a current (defined as the charge per unit time – not the magnitude of the charge per unit time) in the opposite direction. Hence, the currents resulting from electron drift and hole drift occur in the same direction, even though electrons and holes drift in opposite directions. This means that in a material with both holes and electrons, the total current is the sum of the current due to the holes and that due to the electrons. Drift is actually a term that refers to the movement of charge in response to an applied electric field. The drift velocity (v) is the velocity of the drift. It is actually the net velocity, since electrons or holes are quantum particles that obey the Heisenberg Uncertainty Principle, and so they constantly move, even in the absence of an applied electric field. The drift velocity is proportional to the applied electric field E, with the constant of proportionality being the mobility (μ), which describes the ease of movement of the carrier in the medium under consideration. Hence, v = μE . (7.1) Since the units of E are V/m and the units of v are m/s, the units of μ are m2/(Vs). For a given combination of carrier and medium, v depends on the temperature. 203
204 7 Electrical Properties 7.2 Volume Electrical Resistivity The electric current(or simply "current,"I,with units of ampere,or A)is defined as the charge (in coulomb,or C)flowing through the cross-section per unit time. It is the charge,not the magnitude of charge.Thus,the direction of the current is the same as the direction of positive charge flow and is opposite to the direction of negative charge flow. The current density is the charge flow per unit cross-sectional area per unit time.The units of current density are A/m2. The presence of a current in a material requires the presence of charges that can move in response to the applied voltage.These mobile charges are called carriers (i.e.,carriers ofelectricity).The carrier concentration(with units ofm3)is defined as the number of carriers per unit volume. An electrical conductor may be an electronic conductor(electrons are the main carriers),an ionic conductor (ions are the main carriers),or a mixed conductor (both ions and electrons are the main carriers).In this context,electrons constitute a class ofcarriers that include both electrons and holes.Holes are electron-deficient sites that are present in semiconductors and semimetals,but not in conventional metals.Holes are positively charged,since the removal of a(negatively charged) electron leaves something that is positively charged. The current due to mobile charges with a charge g per carrier is given by I=nqvA, (7.2) where n is the carrier concentration.When the carriers are electrons,q=-1.6 x 10-1C.Equation 7.2 derives from the simple argument that each electron drifts by a distance v in one second,and that nvA electrons (the number of electrons in a volume of vA)move through a particular cross-section in one second,as illustrated in Fig.7.1.Dividing Eq.7.2 by A gives I/A nqv. (7.3) Area A Current Figure 7.1.An electric current in a wire of cross-sectional area A.The current results from the drift of a type of charge carrier with a charge g per carrier at a drift velocity of v
204 7 Electrical Properties 7.2 Volume Electrical Resistivity The electric current (or simply “current,” I, with units of ampere, or A) is defined as the charge (in coulomb, or C) flowing through the cross-section per unit time. It is the charge, not the magnitude of charge. Thus, the direction of the current is the same as the direction of positive charge flow and is opposite to the direction of negative charge flow. The current density is the charge flow per unit cross-sectional area per unit time. The units of current density are A/m2. The presence of a current in a material requires the presence of charges that can move in response to the applied voltage. These mobile charges are called carriers (i.e., carriers of electricity). The carrier concentration (with units of m−3) is defined as the number of carriers per unit volume. An electrical conductor may be an electronic conductor (electrons are the main carriers), an ionic conductor (ions are the main carriers), or a mixed conductor (both ions and electrons are the main carriers). In this context, electrons constitute a class of carriers that include both electrons and holes. Holes are electron-deficient sites that are present in semiconductors and semimetals, but not in conventional metals. Holes are positively charged, since the removal of a (negatively charged) electron leaves something that is positively charged. The current due to mobile charges with a charge q per carrier is given by I = nqvA , (7.2) where n is the carrier concentration. When the carriers are electrons, q = −1.6 × 10−19 C. Equation 7.2 derives from the simple argument that each electron drifts by a distance v in one second, and that nvA electrons (the number of electrons in a volume of vA) move through a particular cross-section in one second, as illustrated in Fig. 7.1. Dividing Eq. 7.2 by A gives I/A = nqv . (7.3) v Current Area A Figure 7.1. An electric current in a wire of cross-sectional area A. The current results from the drift of a type of charge carrier with a charge q per carrier at a drift velocity of v
7.2 Volume Electrical Resistivity 205 The current density (J,with units of A/m2)is defined as the current(I)per unit cross-sectional area (A): J=I/A. (7.4) Hence,Eq.7.3 can be written as J=nqv. (7.5) Dividing Eq.7.5 by the electric field E and using Eq.7.1,we get J/E=nqp· (7.6) The electrical conductivity (o)is defined as J/E.In other words,it is defined as the current density per unit electric field.Hence,Eq.7.6 becomes o=nqμ. (7.7) From Eq.7.7,the units of o are 1/(m),i.e.,m-,since the units of n arem, the units of g are C(coulomb),and the units of u are m2/Vs).Alternate units for o are Sm-,where S(short for siemens)equals-1.Note that,by definition, ampere coulomb/second (7.8) and,from Ohm's law, volt/ampere. (7.9) The electrical resistivity (o)is defined as 1/o.Hence,the units of e are m.The electrical resistivity is also known as the volume electrical resistivity in order to emphasize that it relates to the property of a volume of the material.It is also known as the specific resistance. The electrical resistivity ois related to the electricalresistance(R)by theequation R=ol/A, (7.10) where I is the length of the material in the direction of the current.This length is perpendicular to the cross-sectional area A.Equation 7.10 means that the re- sistance R depends on the geometry such that it is proportional to l and inversely proportional to A.On the other hand,the resistivity o is independent of the geome- try and is thus a material property.Similarly,o is a material property.Equation 7.7 allows the calculation of o from n,q and u. The rearrangement of Eq.7.10 gives o=RA/I. (7.11)
7.2 Volume Electrical Resistivity 205 The current density (J, with units of A/m2) is defined as the current (I) per unit cross-sectional area (A): J = I/A . (7.4) Hence, Eq. 7.3 can be written as J = nqv . (7.5) Dividing Eq. 7.5 by the electric field E and using Eq. 7.1, we get J/E = nqμ . (7.6) The electrical conductivity (σ) is defined as J/E. In other words, it is defined as the current density per unit electric field. Hence, Eq. 7.6 becomes σ = nqμ . (7.7) From Eq. 7.7, the units of σ are 1/(Ωm), i.e., Ω−1 m−1, since the units of n are m−3, the units of q are C (coulomb), and the units of μ are m2/(Vs). Alternate units for σ are Sm−1, where S (short for siemens) equals Ω−1. Note that, by definition, ampere = coulomb/second (7.8) and, from Ohm’s law, Ω = volt/ampere . (7.9) The electrical resistivity (ρ) is defined as 1/σ. Hence, the units of ρ are Ωm. The electrical resistivity is also known as the volume electrical resistivity in order to emphasize that it relates to the property of a volume of the material. It is also known as the specific resistance. Theelectricalresistivity ρ isrelatedtotheelectricalresistance(R)bytheequation R = ρl/A , (7.10) where l is the length of the material in the direction of the current. This length is perpendicular to the cross-sectional area A. Equation 7.10 means that the resistance R depends on the geometry such that it is proportional to l and inversely proportional toA. On the other hand, the resistivity ρ is independent of the geometry and is thus a material property. Similarly, σ is a material property. Equation 7.7 allows the calculation of σ from n, q and μ. The rearrangement of Eq. 7.10 gives ρ = RA/l . (7.11)
206 7 Electrical Properties Hence, O=V/RA. (7.12) Since,by definition,o =J/E, J/E V/RA. (7.13) Based on Eq.7.4,Eq.7.13 becomes I/E=VR. (7.14) Since,by definition,E V/I,Eq.7.14 becomes V=IR, (7.15) which is known as Ohm's law.Thus,the equation o =J/E is the same as Ohm's law. 7.3 Calculating the Volume Electrical Resistivity of a Composite Material The volume electrical resistivity of a composite material can be calculated from the resistivities and volume fractions of all of the components.Various mathematical models are available for this calculation.The simplest model is known as the rule of mixtures(ROM),as described below for two configurations:the parallel configuration(each component is continuous and oriented in the direction of the current)and the series configuration(each component is continuous and oriented in the direction perpendicular to the current). 7.3.1 Parallel Configuration Consider an electric current I flowing in a composite material under a voltage difference of V over a distance of l,such that the composite material consists of two components,labeled 1 and 2,as illustrated in Fig.5.16a.The cross-sectional areas are A and Az for Component 1 (all the strips of Component 1 together)and Component 2(all the strips of Component 2 together),respectively.The electrical resistivities are and for Components 1 and 2,respectively.The current(ie., the charge per unit time)I through the composite is given by I=1+12, (7.16) where li is the current in Component 1(all the strips of Component 1 together) and I2 is the current in Component 2(all the strips of Component 2 together). Using Ohm's law, I1=V/R (7.17)
206 7 Electrical Properties Hence, σ = l/RA . (7.12) Since, by definition, σ = J/E, J/E = l/RA . (7.13) Based on Eq. 7.4, Eq. 7.13 becomes I/E = l/R . (7.14) Since, by definition, E = V/l, Eq. 7.14 becomes V = IR , (7.15) which is known as Ohm’s law. Thus, the equation σ = J/E is the same as Ohm’s law. 7.3 Calculating the Volume Electrical Resistivity of a Composite Material The volume electrical resistivity of a composite material can be calculated from the resistivities and volume fractions of all of the components. Various mathematical models are available for this calculation. The simplest model is known as the rule of mixtures (ROM), as described below for two configurations: the parallel configuration (each component is continuous and oriented in the direction of the current) and the series configuration (each component is continuous and oriented in the direction perpendicular to the current). 7.3.1 Parallel Configuration Consider an electric current I flowing in a composite material under a voltage difference of V over a distance of l, such that the composite material consists of two components, labeled 1 and 2, as illustrated in Fig. 5.16a. The cross-sectional areas are A1 and A2 for Component 1 (all the strips of Component 1 together) and Component 2 (all the strips of Component 2 together), respectively. The electrical resistivities are ρ1 and ρ2 for Components 1 and 2, respectively. The current (i.e., the charge per unit time) I through the composite is given by I = I1 + I2 , (7.16) where I1 is the current in Component 1 (all the strips of Component 1 together) and I2 is the current in Component 2 (all the strips of Component 2 together). Using Ohm’s law, I1 = V/R1 (7.17)
7.3 Calculating the Volume Electrical Resistivity of a Composite Material 207 and 2=V/R2, (7.18) where R is the resistance of Component 1(all the strips of Component I together) and R2 is the resistance of Component 2(all the strips of Component 2 together). Hence,Eq.7.16 becomes I=V[(1/R)+(1/R2)月. (7.19) Based on Eq.7.10, R1=PlA1, (7.20) and R2=el/A2. (7.21) Based on Eqs.7.20 and 7.21,Eq.7.19 becomes I=(V/0[(A1/e)+(A2/e2)】. (7.22) On the other hand,from Ohm's law, I=V/R, (7.23) where R is the resistance of the composite.Based on Eq.7.10,R is given by R=ol/A, (7.24) where p is the resistivity of the composite and A is the total cross-sectional area of the composite.Combining Eqs.7.23 and 7.24 gives I=VA/el. (7.25) Combining Eqs.7.22 and 7.25 gives A/el=(V/0[(A/e)+(A2/e2】, (7.26) or A/e=[(A1/e1)+(A2/e2】. (7.27) Dividing by A gives 1/e=(A/A)1/e)+(A2/A)(1/e2)=v/e1+v2/P2, (7.28) where v and v2 are the volume fractions of Components 1 and 2,respectively. Equation 7.28 implies that,for the parallel configuration,the reciprocal of the resistivity of the composite is the weighted average of the reciprocal of the resistiv- ities of the two components,where the weighting factors are the volume fractions of the two components.In other words, 0=y101+V202, (7.29) where o is the electrical conductivity of the composite material,and o and o2 are the conductivities of Components 1 and 2,respectively.Equation 7.29 is a mani- festation of the rule of mixtures
7.3 Calculating the Volume Electrical Resistivity of a Composite Material 207 and I2 = V/R2 , (7.18) where R1 is the resistance of Component 1 (all the strips of Component 1 together) and R2 is the resistance of Component 2 (all the strips of Component 2 together). Hence, Eq. 7.16 becomes I = V [(1/R1) + (1/R2)] . (7.19) Based on Eq. 7.10, R1 = ρ1l/A1 , (7.20) and R2 = ρ2l/A2 . (7.21) Based on Eqs. 7.20 and 7.21, Eq. 7.19 becomes I = (V/l)[(A1/ρ1)+(A2/ρ2)] . (7.22) On the other hand, from Ohm’s law, I = V/R , (7.23) where R is the resistance of the composite. Based on Eq. 7.10, R is given by R = ρl/A , (7.24) where ρ is the resistivity of the composite and A is the total cross-sectional area of the composite. Combining Eqs. 7.23 and 7.24 gives I = VA/ρl . (7.25) Combining Eqs. 7.22 and 7.25 gives VA/ρl = (V/l)[(A1/ρ1)+(A2/ρ2)] , (7.26) or A/ρ = [(A1/ρ1)+(A2/ρ2)] . (7.27) Dividing by A gives 1/ρ = (A1/A)(1/ρ1)+(A2/A)(1/ρ2) = v1/ρ1 + v2/ρ2 , (7.28) where v1 and v2 are the volume fractions of Components 1 and 2, respectively. Equation 7.28 implies that, for the parallel configuration, the reciprocal of the resistivity of the composite is the weighted average of the reciprocal of the resistivities of the two components, where the weighting factors are the volume fractions of the two components. In other words, σ = v1σ1 + v2σ2 , (7.29) where σ is the electrical conductivity of the composite material, and σ1 and σ2 are the conductivities of Components 1 and 2, respectively. Equation 7.29 is a manifestation of the rule of mixtures