Powder Metallurgy and Metal Ceramics, Vol. 40, Nos. 1-2, 2001

THEORY AND TECHNOLOGY OF SINTERING, HEAT, AND CHEMICAL HEAT-TREATMENT PROCESSES SYNTHESIS AND SINTERING OF NANOCRYSTALLINE BARIUM TITANATE POWDER UNDER NON-ISOTHERMAL CONDITIONS. VI. STRUCTURE, GRAIN BOUNDARIES, AND DIELECTRIC PROPERTIES OF BARIUM TITANATE OBTAINED BY VARIOUS SINTERING METHODS A. V. Ragulya, V. V. Skorokhod, and A. V. Polotai UDC 621.762 The grain boundary structure of barium titanate obtained by controlled-rate sintering (CRS) and highpressure sintering (HPS), and the dependence of dielectric properties on grain size and consolidation method were studied. It was shown that sintering without the application of pressure leads to a diffusion-controlled formation of equilibrium grain boundaries with minimal impedance factor, which minimally decrease the dielectric constant of the ceramics. HPS results in the formation of non-equilibrium grain boundaries which have a large free volume, and which substantially decrease the dielectric constant. The Curie- Weiss constant was analyzed from the viewpoint of a matrix structural model, and a “brick-wall” model. Keywords: nanocrystalline barium titanate, non-isothermal sintering, rate-controlled sintering, sintering under high pressure, structure. The dielectric constant and tangent of dielectric losses of barium titanate are very sensitive to changes in particle or grain size, and also to the structure of agglomerates, domains, and grain boundaries. The dimensions of the enumerated structural elements are determined by the conditions under which the material is produced, and also by the type of specimen (monocrystal, polycrystal, film, or powder). In the case of powder, the size effect most thoroughly studied is manifested in two ways: with decreasing particle size the domain structure substantially changes, and if the size drops below a critical value the high-temperature paraelectric phase becomes stable. In large particles twinning lowers the lattice elastic energy and polarization energy; in this case a complex domain structure is formed, characterized by different types (90 and 180 °) of domain walls [1]. Domain walls, however, possess an excess energy which cannot be compensated by a decrease in crystal size. Since the surface of particles is free of external stress, excess elastic energy relaxes. In small particles the number of domains and domain walls is decreased by the effect of the highly curved surfaces, and at a certain critical size the particles become monodomain [1]. Further particle size decrease is accompanied by a phase transformation  the tetragonal lattice is replaced by a cubic one. The critical size for this transformation strongly depends on the degree of powder agglomeration and on the shape of the particles [2]; in various sources it is given as 67-300 nm [3-6]. Obviously, the degree of agglomeration and critical size depend on the method of powder synthesis. The smaller the particle size, the more difficult it is to avoid cluster and agglomerate formation. When this occurs some particles are squeezed between others, and a portion of the crystal surfaces is placed under mechanical stress by neighboring crystals. This effect has not been allowed for in all attempts to determine the critical particle size; it has been estimated as 44 nm, but the calculations did not take into account the depolarizing effect of excess surface charge [6]. The state of particles in an agglomerate is midway between that of free particles and grains in a polycrystal. The size effect in a polycrystalline ceramics differs from that in free particles. In a polycrystal the grains are subjected to mechanical stresses which originate at the grain boundaries as a result of the cubic−tetragonal transformation on cooling. These elastic stresses are relaxed by the formation of a complex domain structure [1]. As is the case for free particles, Institute for Problems of Materials Science, National Academy of Sciences of Ukraine, Kiev. Translated from Poroshkovaya Metallurgiya, Nos. 1-2(417), pp. 32-43, January-February, 2001. Original article submitted December 16, 2000. 1068-1302/01/0102-0025$25.00

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2001 Plenum Publishing Corporation

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smaller grains contain fewer domains, and there is a critical size at which the polydomain structure transforms to monodomain. However, this occurs at a substantially smaller size in grains than in particles. The critical grain size at which the ferroelectric − paraelectric transformation occurs has been determined. Not long ago nanocrystalline barium titanate with density near 98% of theoretical value and grain size of 40 nm was obtained by sintering at high pressures (up to 8 GPa). This material had ferroelectric properties (for example, polarization hysteresis, dielectric constant of approximately 1000 at room temperature, and Curie-Weiss constant C = 1.7⋅105 K [7]). Isolated barium titanate particles with a grain size of 20 nm, crystallized out of a glass matrix (the particle − glass interfaces were stressed), demonstrated a ferroelectric response [8]. Thus, the critical behavior of barium titanate (ferroelectric − paraelectric transition and change of monodomain to polydomain structure) is determined by the size of the BaTiO3 crystals and boundary conditions under which they exist, true for particles as well as polycrystals. The particle or grain size and boundary conditions (mechanical stress, depolarizing field, etc.) are determined by the method of powder or polycrystal preparation. In order to explain the size effect in particles, a geometrical model of a particle with tetragonal interior and a surface layer of cubic phase was earlier proposed [9, 10]. It was assumed that under the action of surface tension a gradient of the lattice parameter c would form near the surface of small particles, and the ratio c/a approaches unity; together with this the polarizability and dielectric constant would decrease. However, quasi-hydrostatic compression cannot lead to the formation of a gradient. Furthermore, the effect of depolarization was not taken into account. The depolarization field which exists in ferroelectric crystals is independent of their size, but depends on the boundary conditions. Thus, in free particles it is maximum because the doubly-charged layer on the surface which opposes the spontaneous polarization field is a maximum. In the boundaries of twins (Σ = 1) or special boundaries (Σ = 3) the polarization field approaches zero. High-angle boundaries, and also defective and with a high impurity concentration boundaries occupy an intermediate position. A “brick − wall” microstructural model in which the permeability of the grains and grain boundaries are assumed to differ by an order of magnitude has been proposed for polycrystals [11]. It was shown by calculations that the dielectric permeability of grains at room temperature is constant and quite high (∼4800), but the grain boundary is not a ferroelectric phase [12]. Since the specific contribution from grain boundaries increases with decreasing grain size, the total dielectric constant should decrease. This model was useful, but no physical reason for the lower dielectric constant of the grain boundaries was given. The polarization vector decreases in amplitude and passes through zero in the region of a twin or grain boundary, and this makes it possible to assert that the grain boundary material is not ferroelectric [13]. Grain boundaries are of various orientations with lattice sites more or less corresponding to those of the bulk crystal, and, consequently, have different mechanical strengths and abilities to retain excess elastic energy, or a high concentration of charged defects which cause a depolarization effect. Thus, the structure and volume of grain boundaries, and concentration and variety of impurities are the essential factors which determine the properties of a nano-ferroelectric. As descriptively stated by Waser, grain boundaries are “the heart of an electronic ceramics” [14] since, particularly in a ferroelectric ceramics, the interaction between domains and grain boundaries leads to a corresponding in dielectric permeability and its temperature dependence. The unique properties of grain boundaries are, first, that the density of electron states within them is totally different from that in the volume; second, added alloying elements are segregated in them; third, the transport of electronic and ionic charge carriers within them is anisotropic, and, finally, elastic stresses between grains strongly affect electrical properties through the piezoelectric and electrostriction constants of a material. The contribution of grain boundaries to the total dielectric properties of a nanocrystalline ferroelectric must be much greater than that in an ordinary polycrystal in view of the higher grain boundary area. How the elastic stress fields at grain boundaries vary, how high the depolarization field gradient is, how both fields affect the impurity composition of grain boundaries  all of these and other questions have been insufficiently studied. Miniaturization of multilayered condensers and electronic memory depend on these important physical phenomena and properties of boundaries. It is essential, therefore, to study the properties of grain boundaries. It is accepted that lowenergy grain boundaries, above all special, play a fundamental role in determining those properties which are controlled by grain boundary behavior. The fraction of grain boundaries which are low-energy depends on the method of specimen preparation and treatment. The statistical distribution of grain boundary orientations is sensitive to the thermal and mechanical history of a material. Therefore, it is necessary to chose an optimized method of powder consolidation in order to attain the best properties. In the present work an attempt is made to compare the grain boundary structures and dielectric properties of barium titanate ceramics obtained by sintering at a controlled shrinkage rate with those obtained by sintering under high pressure [15]. In the first case the grain boundaries are formed by diffusion-controlled processes, while under high pressure the processes occurring are primarily deformational. The grain boundary structures of the sintered nanoceramics were studied by high-resolution transmission electron microscopy. Investigation of the dielectric properties was also an objective of the present work. 26

Experimental Methods The ceramic specimens were obtained by rate controlled sintering (RCS) and high-pressure sintering (HPS) [15] of nanocrystalline barium titanate powder synthesized by the decomposition of barium-titanyl oxalate at a controlled rate, as described in [16-20]. The structure of the material was studied with the aid of a transmission electron microscope (FE HRTEM Hitachi HF2000) having a resolution capability of 0.18 nm and a ultimate magnification of 1.5 M. The specimens were prepared by the standard methods accepted for ceramics. The initial specimens 3 mm in diameter and 200 µm thick were ground using a rotating disc with diamond paste to a central thickness of 20 µm. Then the specimen thickness was decreased to 8-10 µm by gentle polishing. Finally, the central region thickness was reduced up to perforation by ion etching. The etching procedure was as follows: a flux of argon ions Ar+ was directed at the specimen from two sides with an imparted energy of 4 keV. The ion current was 5 mA, and the angle of attack varied from 12 to 20 °. The grain-boundary geometry was revealed in the transmission electron microscope by the method described in [21]. The misorientation of adjacent grains was determined from local diffraction patterns. The dielectric properties of the specimens were determined by a bridge method. Gold was deposited on the lateral surfaces of the specimens in order to form electrodes. Specimen capacities were measured at frequencies of 1, 10, and 100 kHz with an error of no more than 1%. The dielectric permeability was calculated from the equation ε = (11.3 Cd)/S, where C is condenser capacity, πΦ; d is distance between electrodes, cm; S is electrode surface area, cm2. Experimental Results X-ray Diffraction Analysis. Before measuring the dielectric properties of the nanostructural barium titanate ceramics obtained by the different consolidation methods [15], an x-ray diffraction analysis of the specimens was carried out in a “Philips Xpert” precision diffractometer at room temperature in order to detect the cubic phase, and determine the c/a ratio of the lattice. The initial powder, obtained by synthesis at a controlled transformation rate [16-20], was nanocrystalline and consisted of cubic phase with an average particle size 26 nm. The barium titanate prepared by sintering in a RCS regime (final temperature ∼1280°C) had a final grain size 150 nm, and at room temperature consisted of pure tetragonal phase with c/a = 1.007. No traces of cubic phase were detected in this material. In the specimens sintered under high pressure the simultaneous presence of two phases was confirmed by x-ray diffraction analysis and electron microscopy. As the temperature of sintering under pressure increased (from 400 to 1000°C) the initially cubic particles consolidated, and upon cooling to room temperature transformed into grains of tetragonal phase. This is related to the increase in grain size which accompanied shrinkage of the specimen [15]. Broadening and partial splitting of the (200) peaks appeared in the angular range 2θ = 44-47°. These peaks were approximated by a sum of Gauss curves, by means of which the simultaneous presence of cubic and tetragonal phases was confirmed (Fig. 1). The cubic phase is stable at high pressures, and also becomes stable in small particles which are compressed by surface tension forces. In the given case the cubic-tetragonal transformation occurred in the largest grains upon cooling, while small grains remained cubic. Since the pressure was removed and the x-ray diffraction measurements carried out below the Curie point (125°C) the cubic phase was retained because of the size factor and conservation of metastable cubic state formed under high pressure. The presence of two phases in the system leads to lattice mismatch between contiguous grains and the appearance of elastic stresses along grain boundaries. A model of twophase particles or grains has been discussed in the literature for the past ten years. However this visualizes a particle (grain) in which the interior (core) is a tetragonal phase with the properties of a ferroelectric, and the surface (shell) a cubic paraelectric phase [9]. The spontaneous polarization vector actually decreases abruptly in the vicinity of a particle surface or grain boundary since it changes direction from grain to grain, and because of the action of the depolarization field. It can be assumed that the c and a parameters of the titanate lattice approach each other and dipoles vanish. Furthermore, change of lattice parameters and the appearance of an interphase boundary has never been observed within a single particle or grain. It appears that x-ray diffraction analysis indicates the simultaneous presence of two phases because particles of two phases coexist, rather than two-phase particles. Since the electron beam changes the structure of the specimen, direct observation of two-phase nanoparticles or grains is not possible in a transmission electron microscope. High-Resolution Electron Microscopy. The grain structure of sintered ceramics is described in [15]. Grain boundary structures were studied by high-resolution transmission electron microscopy. The objective was to determine how the grain boundary structure depends on the method of consolidating the polycrystalline ceramics. By “grain structure” is understood the orientation of boundary planes, orientation of grains forming the boundaries, and presence or absence of second-phase (amorphous) layers along the boundaries [21, 22]. It is practically impossible to exactly determine the orientation of grain boundary planes from local diffraction patterns in nanocrystalline materials with grain size below 100 nm due to the absence 27

Fig. 1. Change of the C(200) diffraction reflection with increase in the temperature of sintering under high pressure.

a

b

c

d

Fig. 2. Grain boundaries in nanocrystalline barium titanate sintered by RCS (a, b) and HPS (c, d).

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of Kikuchi lines, for whose reflection the transmitting crystal must be “sufficiently thick”. Three pairs of Kikuchi lines are necessary to correctly orient a crystal in space relative to the impinging beam. Furthermore, portions of the transmitted edge in which grains are not overlapped and Moire patterns are absent are encountered less frequently, the smaller the grain. Therefore, the angular misorientation of grains was determined from point diffractograms assuming that the electron beam was directed approximately parallel to the (001) axis. The orientation of contiguous grains was determined from bright-field reflections under the condition that diffraction contrast was present in both grains, for which the technique of specimen inclination and rotation under the microscope beam was used. Foreign grain boundary phases were detected by electron-loss spectroscopy, and also by the analysis of bright- and dark-field reflections. Impurity phases along grain boundaries were not detected by any of the methods. This is a consequence of the high purity of the initial barium titanate powder and sintered ceramics, and also the sintering conditions. The grain boundaries of ceramics sintered in a RCS regime are pure, perfect, and low-angle. Typical boundaries are shown in Fig. 2a, b. The given boundary is a plane formed throughout its entire extent by one and the same crystallographic planes of the adjoining crystals. It has a high density of coinciding sites (Σ = 17) and a constant angle of misorientation (26.5°). Boundaries of this type are equilibrium boundaries [23]. The grain region attributable to the boundary is narrow, has a well distinguished periodic structure, and does not contain amorphous phase. Such a boundary is not a Scottky barrier, and does not create any significant depolarization field in the ferroelectric. Steps are visible in Fig. 2b. There is no indication of the presence of low-density grain boundaries. Two types of grain boundaries were found in the specimens sintered under high pressure: coherent twins, and high-angle boundaries which are usually non-equilibrium and incoherent because they had atomic structures with low atom densities, varying misorientation angles at a constant angle of misorientation between the grains, and, consequently, exhibit multiple fragmentation into facets with more or less perfect sections of lattice coincidence. Similar observations were made in [24] on specimens of other oxides (MgO, ZnO, etc.). A large fraction of boundaries had the periodic atomic structure of a lattice of coinciding sites with Σ = 3 and 11. The small radius of curvature of nanograins is the reason for frequent variation of the crystallographic orientation of the boundary between one pair of grains, and the absence of extended grain boundaries. Barium titanate is characterized by the low stress required to displace and regroup twins  0.01 MN/m2 [25]. Therefore, the appearance of twin boundaries in barium titanate is the probable, but not the only reason that the boundaries are periodic and not disordered. A possible reason for the appearance of twin boundaries is the alignment of initial nuclei in contact along planes of identical crystallographic orientation to form agglomerates during synthesis. However, the statistics of grain misorientation in a nanostructure is determined not only by twins relocating to a surface or boundary during grain growth, but also by grain reconstruction via shear or rotation. In spite of the high specimen density (95-98% of theoretical) the grain boundaries were imperfect, and may be better viewed as mechanical contacts rather than real crystal boundaries (Fig. 2c, d). An indirect proof of the imperfection of such grain boundaries is their low strength; specimens easily fractured under the electron beam, causing substantial experimental difficulty. Small pores (less than 2 nm) were seen at ternary junctions between grains (Fig. 2d), indicating insufficient plastic deformation at regions of contact under high pressure. The above characteristics are typical of non-equilibrium grain boundaries [22, 23] which exhibit variable atom coordination and, consequently, are highly defective and have a high free volume. Such a polycrystalline structure in a ferroelectric leads to stronger depolarization fields, lower dielectric permeability, and increased dielectric losses. Non-equilibrium grain boundary configurations are the likely cause of the low mechanical strength of the specimens. Closed pores and non-equilibrium grain boundaries become sites for relaxation of the excess elastic energy which accumulates during the cubic-tetragonal transformation. It is reasonable to assume that in this case the stress state of grains and the level of internal stresses differ from those in a sintered ceramics with an equilibrium pore-free structure. Since internal stresses are partially relaxed by grain boundary cracking, the dielectric constant is lower in specimens sintered under pressure than in those sintered without pressure. Dielectric Properties. Figure 3 shows the variation of dielectric constant with temperature and grain size. It is obvious that nanocrystalline ceramics prepared by RCS and under high pressure exhibits normal ferroelectric behavior; all barium titanate phase transformation peaks are present. The dielectric constant at room temperature and in the Curie point decreases with grain size, remaining above 2000 in both RCS specimens and those sintered under high pressure. These data agree with those obtained in [7], and contradict results in [26, 27]. The dielectric constant of barium titanate sintered under high pressure is lower than that of BaTiO3 with the same average grain size (130-150 nm) and similar residual porosity (14%) sintered without applied pressure in a RCS regime. The difference can be explained by the formation of more perfect boundaries in the conventional sintering process. Another important result is constancy of the Curie temperature (Tc). This

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Fig. 3

Fig. 4

Fig. 3. Temperature dependence of the dielectric constant of barium titanate obtained by sintering at a controlled rate (1, 2) and under a pressure of 5 GPa (3, 4). Grain size: (1) 450; (2, 3) 130; (4) 100 nm. Fig. 4. Change in direction of the spontaneous polarization vector on passing from one single-domain grain to another.

Fig. 5

Fig. 6

Fig. 5. Temperature dependence of the impedance factor and its determination for grain boundaries of barium titanate ceramics obtained by HPS (1, 2) and RCS (3, 4). Grain size: (1) 100; (2, 3) 130; (4) 750 nm. Fig. 6. Dependence of the grain-boundary impedance factor on grain size and sintering method. (1) CRS; (2, 3) HPS (curve 2 is based on data from [7], and curve 3 from the present investigation). The solid curves were calculated using equation (1) and the dashed using equation (2). 30

Fig. 7. Dependence of the dielectric permeability of nanocrystalline barium titanate on grain size and method of production. Solid curve is based on data taken from [1] and the bar graph on data from the present investigation.

conclusion agrees with the data in [7], and contradicts known work [27] in which a decrease in the paraelectric − ferroelectric transformation temperature is reported. The above proposed “quasi-two-phase” grain structure enabled the authors of [11] to formulate a “brick wall” microstructural model in which the permeability of grains and boundaries differ by an order of magnitude, and by the use of which the Curie-Weiss behavior could be satisfactorily described. According to the model, the grain consists of a ferroelectric core with ε1 = 4800, and a surrounding thin envelope with ε2 = 100. Neither of these values was determined experimentally; they were taken as the limits for coarsely dispersed ceramics and thin films, respectively. Such a structure is possible if upon passing through the grain boundary the polarization vector changes direction, and, therefore, its decrease in the grain boundary region can be described by a gradient (Fig. 4). The model permitted determination of the effective dielectric constant using the equation for a two-phase ferroelectric [11, 28]: gv 1 v1 gv 2 T − Θ = + = v1 = 2 , ε ε1 ε 2 ε2 C

(1)

where ε1 = C/(T − Θ) above the Curie point; C is Curie constant; Θ is extrapolated Curie-Weiss temperature; vi is volume fraction of phase (v1 is volume of ferroelectric, v2 is volume of envelope); g is geometry factor. The behavior of a two-phase composite with a matrix structure can be also described using the self-consistent field method [29]:     ε 2 (1 + 2v2 ) + 2ε 0 (1 − v2 ) v 2 . ε = ε1 , or ε = ε1 1 + (2) ε1 v2   ε 2 (1 − v2 ) + ε1( 2 + v2 ) +   ε 2 − ε1 3   In equations (1) and (2) the volume fractions of the phases are determined as v1 = (1 + d2/d1)−3,

(3)

where d2 is thickness of the layer in which the polarization gradient is observed. Calculated impedance factors ψ = 1/ε are given in Fig. 5. The dielectric constants of inclusion and matrix are taken as ε1 = 4800 and ε2 = 100, respectively [7], and thickness variation of the near-boundary volume (of the matrix) d2 = 0.4-

4 nm. Here barium titanate with a grain size of 0.75 µm served as a standard (the effect of grain boundary impedance in this is minimal) for calculating the effective Curie-Weiss temperature Θ = 110°C at which ψ = 0. The impedance factor of the 31

nanocrystalline ceramics was determined at 110°C by extrapolating the experimental data. Both theoretical approaches satisfactorily approximate the experimental results, both those obtained earlier [7] on specimens consolidated under high pressure, and those of the present work. The higher values of ψ correspond to ceramics sintered at high pressure, which was characterized by defective grain boundaries with a large near-boundary volume (Fig. 6). The impedance factor is minimal in ceramics sintered in a RCS regime, which reflects the high perfection of grain boundaries formed by a diffusion mechanism. Comparing the results obtained with the data of Arlt and Hennings [1] shows (Fig. 7) that the dielectric permeability at room temperature of RCS specimens with a grain size of 0.8 µm practically coincides with that of specimens prepared by traditional sintering methods. At the same time, the value of ε for RCS specimens with average grain sizes of 0.15 and 0.6 µm was 1000-1500 units higher than that of specimens prepared by hot pressing or sintering under high pressure [1, 28]. Thus, the dielectric properties are different in spite of identical specimen densities and grain sizes. The specimens differ only in the grain boundary structures produced by the method of synthesis. Therefore, it is possible to obtain high dielectric properties in nanocrystalline barium titanate by the formation of perfect grain boundaries, and to achieve this objective sintering technology is preferred over sintering at high pressure.

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