Transient Thermal Gradients in Barium Titanate Positive Temperature Coefficient (PTC) Thermistors David S. Smith, * Noureddine Ghayoub, Isabelle Charissou, Olivier Bellon, and Pierre Abe ´lard * Laboratoire de Mate ´riaux Ce ´ramiques et Traitements de Surface (CNRS UPRESA 6015), Ecole Nationale Supe ´rieure de Ce ´ramique Industrielle (ENSCI), 87065 Limoges, France Arthur H. Edwards Department of Electrical Engineering, University of North Carolina at Charlotte, Charlotte, North Carolina 28223 Barium titanate positive temperature coefficient (PTC) ce- ramic disk thermistors can suffer major mechanical dam- age if inhomogeneous heating occurs under voltage. The steady-state and transient temperature distributions for thin disk samples (radius of 5 mm, thickness of 2 mm) have been studied with an infrared microscope, using a spatial resolution of 35 μm. The transient temperature distribution is observed to be particularly sensitive to the electrical boundary conditions during the initial heating period after application of a voltage. Small variations in electrode sym- metry can lead to axial asymmetric thermal gradients up to 25 K/mm across the entire rim when an ac voltage of 100 V is applied. A finite-difference model in two dimensions, based on solution of the heat equation with a local- temperature-dependent Joule heating-source term, has been developed to describe the axial and radial transient temperature distributions in the cylindrical geometry. The predictions reveal current concentration at the edge of an electrode when the metal layer coverage is slightly smaller than that of the opposite face. This phenomenon results in stronger localized heating in a ring that initiates the ther- mal gradient. I. Introduction P OSITIVE temperature coefficient (PTC) thermistors are based on n-type semiconducting barium titanate (BaTiO 3 ) in a polycrystalline form. According to the model of Heywang, in the paraelectric state, potential barriers occur at the grain boundaries and yield a large electrical resistance. 1,2 On cooling below the Curie temperature (T c ) (120°C), the transition to the ferroelectric state leads to a screening of the potential barriers and the resistance decreases to several ohms. The PTC effect, which spans several orders of magnitude, is exploited for cur- rent control, for example, in overload protection. A serious problem with PTC thermistors in disk form is that mechanical damage can occur in response to strong applied voltages. Several types of failure can be distinguished, includ- ing splitting of the disk along a central plane perpendicular to the axis (delamination), irregular cracking, and chipping of ceramic surface layers around the contact wire. 3–5 When a large voltage is applied to a thermistor, the initial heating is very rapid until the current inrush is throttled by the PTC effect. Thus, damage is attributed to the formation of hot spots in the ceramic, which results in thermal shock. 6 To date, de- lamination has been studied in the greatest detail. Lubitz 4 used a computer simulation, based on the one-dimensional heat equation, to predict the appearance of a hotter center (over- heating) during the electrical impulsion for thick samples (7 mm) subject to air cooling. The approach has been extended to two dimensions and the induced thermal stresses by Dewitte and co-workers. 5,7 The nonlinear behavior of the potential bar- riers also has been considered. 3,8 In another contribution, Mader et al. 9 demonstrated that, for large-grain BaTiO 3 sub- jected to impulsions of 350 V, the heating was localized at the grain boundaries, but once the power in the electrical impulsion was attenuated, relaxed to a homogeneous temperature distri- bution across the grain in <1 ms. The present paper is concerned with the initiation and relax- ation of thermal gradients within the thermistor. Although some previous experimental studies of hot thermistors with localized buried thermocouples in the ceramic or with a ther- mograph have been conducted, 4,10 the spatial resolution was limited to 200 m. This work examines the thermal response across thin thermistors (thickness of 2 mm) at a local scale using an infrared (IR) microscope with a spatial resolution of 35 m. In particular, in a significant number of samples, we have observed displacement of overheating to one of the edge planes, which is relevant to the surface chipping mode. Other workers modeled the case where an inhomogeneous ceramic is responsible for a variation in resistivity 4,10,11 and, hence, at the origin of temperature gradients. As an alternative (but comple- mentary) approach, the present paper examines the sensitivity of the transient temperature distribution to the external thermal and electrical boundary conditions. In particular, experimental results are reported on the influence of external factors such as contact wires, convection conditions, electrode coverage, and the use (or lack thereof) of encapsulant. Finally, a two-dimen- sional model for a cylindrical geometry is developed to de- scribe the formation of thermal gradients in the disk thermistor. II. Experimental Procedure (1) Samples BaTiO 3 thermistor disk components 10 mm in diameter and 2 mm thick were supplied by LCC-Thomson (Dijon, France) for testing. The BaTiO 3 ceramic was a standard commercial formulation with a grain size in the range of 3–5 m. The room-temperature resistance (R o ) was 10 ± 2 , and a steep increase in resistance occurs at temperatures >110°C (Fig. 1). The disk faces were coated with composite sputtered electrode layers that contained nickel, chromium, and silver to a thick- ness of ∼0.3 m. A small gap of 0.1–0.3 mm to the rim around the electrode circumference was typically uncovered. Table I describes the variation of portions in the complete thermistor components that gives some modification of the external con- B. M. Kulwicki—contributing editor Manuscript No. 192933. Received June 9, 1997; approved October 21, 1997. Supported in part by the French Ministry of Research and Technology. * Member, American Ceramic Society. J. Am. Ceram. Soc., 81 [7] 1789–96 (1998) J ournal 1789