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PHASE TRANSITIONS OF BaTiO3 NANOPARTICLES STUDIED BY DIFFERENTIAL SCANNING CALORIMETRY

Permanent Link: http://ncf.sobek.ufl.edu/NCFE004833/00001

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Title: PHASE TRANSITIONS OF BaTiO3 NANOPARTICLES STUDIED BY DIFFERENTIAL SCANNING CALORIMETRY
Physical Description: Book
Language: English
Creator: Myers, Emily
Publisher: New College of Florida
Place of Publication: Sarasota, Fla.
Creation Date: 2013
Publication Date: 2013

Subjects

Subjects / Keywords: Differential Scanning Calorimetry
Barium Titanate
Phase Transitions
Genre: bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

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Abstract: Barium titanate (BaTiO3) has a multitude of applications in the electronics industry due to its high dielectic constant and ferroelectric properties. Use of this material in small-scale electronics is limited, however, by its loss of ferroelectricity below a certain critical particle size. Differential heat flow bands detected during thermal cycling were associated with BaTiO3 structural transitions. Specific heat capacities of 50 nm, 100 nm, 200 nm, and 300 nm BaTiO3 powders were calculated at every 0.5 K between 134 K and 451.5 K. The bands around the phase transitions were fitted with Split-Pearson VII functions to obtain peak position, full-width half-maximum (FWHM), and transition enthalpy. The 300 nm and 50 nm data were also compared with 260 cm-1 Raman mode data. It was found that the phase transitions are size-dependent.
Statement of Responsibility: by Emily Myers
Thesis: Thesis (B.A.) -- New College of Florida, 2013
Electronic Access: RESTRICTED TO NCF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE
Bibliography: Includes bibliographical references.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The New College of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Local: Faculty Sponsor: Sendova, Mariana

Record Information

Source Institution: New College of Florida
Holding Location: New College of Florida
Rights Management: Applicable rights reserved.
Classification: local - S.T. 2013 M99
System ID: NCFE004833:00001

Permanent Link: http://ncf.sobek.ufl.edu/NCFE004833/00001

Material Information

Title: PHASE TRANSITIONS OF BaTiO3 NANOPARTICLES STUDIED BY DIFFERENTIAL SCANNING CALORIMETRY
Physical Description: Book
Language: English
Creator: Myers, Emily
Publisher: New College of Florida
Place of Publication: Sarasota, Fla.
Creation Date: 2013
Publication Date: 2013

Subjects

Subjects / Keywords: Differential Scanning Calorimetry
Barium Titanate
Phase Transitions
Genre: bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Barium titanate (BaTiO3) has a multitude of applications in the electronics industry due to its high dielectic constant and ferroelectric properties. Use of this material in small-scale electronics is limited, however, by its loss of ferroelectricity below a certain critical particle size. Differential heat flow bands detected during thermal cycling were associated with BaTiO3 structural transitions. Specific heat capacities of 50 nm, 100 nm, 200 nm, and 300 nm BaTiO3 powders were calculated at every 0.5 K between 134 K and 451.5 K. The bands around the phase transitions were fitted with Split-Pearson VII functions to obtain peak position, full-width half-maximum (FWHM), and transition enthalpy. The 300 nm and 50 nm data were also compared with 260 cm-1 Raman mode data. It was found that the phase transitions are size-dependent.
Statement of Responsibility: by Emily Myers
Thesis: Thesis (B.A.) -- New College of Florida, 2013
Electronic Access: RESTRICTED TO NCF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE
Bibliography: Includes bibliographical references.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The New College of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Local: Faculty Sponsor: Sendova, Mariana

Record Information

Source Institution: New College of Florida
Holding Location: New College of Florida
Rights Management: Applicable rights reserved.
Classification: local - S.T. 2013 M99
System ID: NCFE004833:00001


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PHASE TRANSITIONS OF BaTiO 3 NANOPARTICLES STUDIED BY DIFFERENTIAL SCANNING CALORIMETRY BY EMILY MYERS A Thesis Submitted to the Division of Natural Sciences New College of Florida in partial fulfillment of the requirements for the degree B achelor of Arts Under the sponsorship of Dr. Mariana Sendova Sarasota, Florida May 2013

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ii Acknowledgements To Mariana Sendova for letting me work in her lab and for providing wonderful feedback on all of my writing; Brian Hosterman for answering all of my questions and helping me use the lab equipment; George Ruppeiner and Don Colladay for serving on my thesis committee; my family, for continued support; and all of my friends for their encouragement and companionship t hank you.

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iii Table of Contents Acknowledgements ii Abstract iv Introduction 1 Chapter 1: Physical Properties and Applications of Barium Titanate (BaTiO 3 ) 3 1.1 Dielectrics and Ferroelectrics 3 1.2 Types of Phase Transitions 7 1.3 BaTiO 3 Structure and Pha se Transitions 9 1.4 Synthesis of BaTiO 3 11 1.5 Size Effects 12 1.6 Heat Capacity 14 1.7 Experimental Methods for Studying BaTiO 3 Phase Transitions 1 7 Chapter 2: Experiment 25 2.1 Materials and Apparatus 2 5 2.2 Experimental P rocedure 26 2.3 Specific Heat Measurement 30 2.4 Phase Transition Band Analysis 35 Chapter 3: Results and Discussion 40 3.1 Raw Heat Flow Data for BaTiO 3 Particles 40 3.2 Heat Capacity of BaTiO 3 and Indium 42 3.3 DSC Raman Compari son 48 3.4 Size Dependent Parameter Comparison 50 C hapter 4: C onclusions and Future Experiments 53 4.1 Conclusions 53 4.2 Future Experiments 54 References 57

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iv PHASE TRANSITIONS OF BaTiO 3 NANOPARTICLES STUDIED BY DIFFERENTIAL S CANNING CALORIMETRY Emily Myers New College of Florida, 2013 ABSTRACT B arium titanate (BaTiO 3 ) ha s a multitude of applications in the electronics industry due to its high dielectic constant and ferroelectric properties. Use of this material in small s cale electronics is limited, however, by its loss of ferroelectricity below a certain critical particle size. Differential heat flow bands detected during thermal cycling were associated with BaTiO 3 structural transitions. Specific h eat capacities of 50 n m 100 nm 200 nm and 300 nm BaTiO 3 powders were calculated at every 0.5 K between 134 K and 451.5 K. The bands around the phase transitions were fitted with Split Pearson VII functions to obtain peak position full width half maximum (FWHM) and transition enthalpy The 300 nm and 50 nm data were also compared with 260 cm 1 Raman mode data. It was found that the phase transitions are size dependent. Dr. Mariana Sendova Division of Natural Sciences

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1 Introduction Ferroelectric ceramics like barium titanate (BaTiO 3 ) have gained importance in the electronics industry due to their applications as capacitors, thermistors, and chemical sensors. 1,2 Their high dielectric constant and ferroelectric propertie s prove useful in the manufacture of such devices because they are alterable by applied heat or electric field; capacitors enclosing ferroelectric materials can exploit this tunability of ferroelectrics to alter capacitance. Barium titanate capacitors hav e been studied for utilization in dynamic random access memory (DRAM) as well. 3 For the use of BaTiO 3 in memory storage, the value of a ferroelectric capacitor is stored in its polarization and read with the application of an electric field that reverses its state. All of these devices must be small in order to keep up with the current trends towards miniaturization of electronics. Ceramic capacitors in particular contain layers of ferroelectric materials less than 1 m in thickness, and can be expected to be made even thinner in coming years. 4 Unfortunately, the dielectric constant of polycrystalline barium titanate decreases for grain sizes below 1000 nm. Below 100 nm, BaTiO 3 is cubic at room temperature, meaning t hat it no longer exhibits spontaneous polarization. Both of these factors severely limit its usefulness in small scale ferroelectric applications. 3 Attempts have been made to produce BaTiO 3 thin films with minimal defects in order to maintain the materia determine how useful materials like barium titanate can be in microelectronics. This study attempts to compare the phase transition behaviors of BaTiO 3 powders ranging from 50 nm to 300 nm in size. Data taken with a differential scanning calorimeter was used to calculate the heat capacities of these particles over a large

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2 temperature range. The heat capacit y as a function of temperature w as utilized to locate and analyze the powder Factors taken into consideration included transition temperature, transition enthalpy, band width, and band intensity Differences observed in the heat capacity signals between the larger and smaller particles studied helped to pinpoint the effects of particle size on the ferroelectric to paraelectric transition temperature. Knowledge of the size dependency of capacitors and other electronic devices.

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3 Chapter 1: Physical Properties and Applications of Barium Titanate ( BaTiO 3 ) 1.1 Dielectrics and Ferroelectrics A dielectric material is an electrical insulator a material that does not conduct electric cur rent Although any insulator is technically a dielectric, is used to describe insulators that polarize in applied electric fields. This polarization allows materials with a high dielectric constant to impede travelling electro magne tic waves In order to explain polarization, it seems useful to consider electric dipoles: Figure 1. 1 A physical dipole A physical dipole i s formed by two particles of equal and opposite charge q separated by distance d The dipole mo ment p of these particles is a vector from the negative towards the positive charge : (1) Similarly, if the centers of the net positive and net negative charge within a given region are displaced the dipole moment of the charges in that r egion is given by Eq. 1 with d equal to the distance between the positive and negative charge centers.

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4 Figure 1. 2 A dipole due to displacement of the anion in a lattice The polarization P of a material is defined as the to ta l dipole moment per unit volume V (2) There are three mechanisms for material polarization: induced, ionic, and orientational. Induced (e lectronic ) polarization works on the atomic or molecular level ; a n applied electric field distorts the negative electron cloud This shift creates a dipole moment between the positive nucleus and the center of the electron cloud Electronic polarization can be induced in all dielectrics Ionic polarization occurs only in materials with ionic structure, meaning that they are compo sed of positive and negative ions arranged in a lattice. An applied electric field again causes the positive and negative ions to shift in opposite directions This shift lead s to an off centering of the cations with respect to the anions and vice versa Net polarization in the direction of the electric field develops in response to the induced dipole moment. Orientational polarization occurs in molecules that have a built in permanent dipole moment, such as water ( H 2 O ) Ordinarily, t hese molecules move around randomly with dipoles facing in all directions.

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5 In the presence of an electric field, however, the dipoles rotate to align with the field and with each other, resulting in a net polarization. 5 Dielectric strength ( the polarizability of a mate rial ) dielectric constant This constant, also known as relative permittivity, is larger for dielectrics that demonstrate stronger polarization in the presence of applied fields Relative permittivity is calculated u sing capacitor s A parallel plate capacitor consists of two conducting plates separated by vacuum or dielectric material and held at different electric potentials If the potential difference between the plates is given by V and the charges on the plates are given by +Q and Q the capacitance C (in units of Farads or Coulombs/Volts) is : ( 3 ) Capacitance is also determined in terms of the capacitor dimensions and d ielectric properties of the material between the conduc ive plates For plane parallel capacitors, the dependence is given by ( 4 ) wher e A is the area of overlap of the two plates, d is the distance between the plates is the relative permittivity of the insulating material between the plates, and the constant is the permittivity of free spa ce For reference, the relative permittivity is equal to one in a vacuum. Div iding Eq. 4 for a capacitor filled with dielectric material of some ( ) by that of the same capacitor with no material between its plates ( ) gives: ( 5 )

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6 Thus, C increases by a factor of when a dielectric of that permittivity is inserted between its plates. Applying the definition of C in Eq. 3 to the result of Eq. 5 gives ( 6 a) O nly the voltage between plates is altered by insertion of a diele ctric. Thus, and Eq. 6 a simplifies to ( 6 b) If a voltmeter is connected between the plates, the dielectric constant of a material is measured by a comparison of th e voltage (or capacitance) before and after the insertion of that material between the plates. Most dielectrics have an induced polarization proportional to the strength of the external electric field E Thus, the polarization P is a linear function of E and is zero in the absence of an external electric field This linear polarization is called dielectric or paraelectric polarization. F erro electric materials, on the other hand, have a permanent nonzero polarization that is reversible with the applicatio n of a strong electric field. A ferroelectric material exhibiting a positive polarization can be switched to a negative polarization given sufficient negative field, and vice versa. T he reversible polarization depends somewhat on the tric history. This history dependence of ferroelectrics is the property of hysteresis. Below are two graphs of polarization versus applied electric field : one for a typical paraelectric and one for a ferroelectric The ferroelectric material displays a hysteresis loop (Fig. 1.3).

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7 Figure 1. 3 Paraelectric (left) and ferroelectric (right) polarization I nternal dipoles that switch direction with the application of a strong enough electric field explain ferroelectricity This s witch usually disappears above a transition temperature called the Curie temperature A bove t his temperature, the material exists in a paraelectric state a state without built in polarization 6 1.2 Types of Phase Transitions Any material can exist in vario us states, called phases solid, liquid, gas, depending on the temperature and pressure. On a microscopic level, each phase is characterized by its own equilibrium atomic or molecular arrangement. Within the solid phase, various stable atomic and ionic a rrangements can exist, depending on outside temperature and pressure. A given atomic or ionic ferroelectric arrangement is characterized by its symmetry and resulting polarization. Usually, solid phases with high symmetry (cubic) are not ferroelectric. Pha se transitions within the solid phase in ferroelectrics can be classified by the behavior of the electric dipoles in the material. In an order disorder transition, reversible or rotatable dipoles in crystals are oriented parallel to one another below th e Curie point.

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8 This parallel orientation gives rise to spontaneous polarization. Above the Curie point, however, the ordering (and therefore the spontaneous polarization) is lost. 7 P hase transitions are also assigned an order: first or second In a firs t order suffer a discontinuity at the transition temperature In second order transitions, entropy, volume, and polarization are continuous functions of temperature but the ir tempera ture derivatives show discontinuities at the transition temperature These two types of transitions have different effects on specific heat capacity as a function of temperature. Specific heat capacity is explained in detail in Section 1.6. For a crysta l undergoing a first order transition by heating, heat absorption occurs and a latent heat is observed essentially, the specific heat capacity jumps up or down discontinuously Order disorder phase transitions such as from solid to liquid phase, or liq uid to gas phase, are first order transitions For a second order transition, there is no latent heat and a peak in specific heat is observed instead. 7 transitions, due to the shape of their specific heat capacity near the phase transition M ost of the so called displacive transitions within the solid crystal phase are second order. Displacive transitions are caused by the displacement of atoms or ions to a new equilibrium position within the crystal lattice As a result of the displacement, the symmetry of the crystal structure changes, while the long range order is p reserved. 7 In principle, t ransitions to any n th order may exist original definition of an n n 1 )th derivative of Gibbs free energy is continuous, while the n th derivative is not continuo us at the

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9 transition temperature. 7 Gibbs free energy is a measure of the work that can be extracted from a closed system. This definition of n th order phase transitions is considered outdated, however, and transitions above the second order are rarely co nsidered. 1.3 BaTiO 3 Structure and Phase Transition s Barium titanate (BaTiO 3 ) is a ferroelectric. This inorganic compound is a nonmetallic crystalline solid BaTiO 3 exists primarily as a crystalline white powder or a larger, transparent crystal with perov skite structure Perovskite is a type of crystal structure first determined for CaTiO 3 by Russian mineralogist Perovski. This structure is pictured in Figure 1. 4. Figure 1. 4 C ubic unit cell with Perovskite structure ABO 3 In this idealized depiction of the cubic unit cell, a B cation six face centered oxygen atoms and eight A cations at the cube corners. Barium titanate, as its chemical formula suggests, consists of a titanium ion (B) surrounded by oxygen (O) and barium (A) ions The barium ions occupy the corners of the cube 8

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10 Cubic BaTiO 3 is paraelectric and exists above the Curie temperature As the temperature decreases bulk barium titanate undergoes successive structural transitions into t hree distinct ferroelectric phases. Around 393 K, cubic BaTiO 3 becomes tetragonal (T) ; around 278 K, it becomes orthorhombic (O) ; and below 183 K, it is rhombohedral (R) 8 These transitions are distortions of the paraelectric cubic symmetry involving el ongation along the edges and diagonals of the unit cell Such distortions result in displacement of the Ti cations with respect to the oxygen octahedra, yielding net polarization within the material. various crystalline phases found by neutron scattering by G. H. Kwei et al : 8 Figure 1. 5 Phases of BaTiO 3 in descending order of temperature The deformations exaggerated for clarity, illustrate the directions of polariza tion for ea ch phase. P stated that structural

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11 instabilities in the ferroelectric phases might be due to the titanium ion moving around in the oxygen octahedron with titanium displacement on the order of 0.01 Kwei however, argues that the instabilities are caused by competition between ionic and covalent forces on the titanium and oxygen ions. 8 ferroelectric to paraelectric transition is usually classified as displacive, but shows elemen ts of both displacive and order disorder transitions. 7 According to the displacive theory the lowest frequency BaTiO 3 vibrational mode (called the soft phonon mode) approaches zero near the Curie transition temperature, leading to instabilities in the cr ystal structure. A new, more stable crystal structure is established a f t er the transition 7 Another theory speculates that above the Curie temperature, the Ti ions are disordered in eight off centered s ites with in the oxygen octahedra 9 Therefore, the f erroelectric to paraelectric transition has elements of a first order (order disorder) phase transition, with the paraelectric phase being the disordered phase. Barium t heory, but both phonon vibrations and ionic movements contribute. 1.4 Synthesis of BaTiO 3 Barium titanate may be synthesiz ed either from solid powders or from solutions containing barium and titanium. Several possible methods are discussed below. One way of synthesiz ing BaTiO 3 is through precipitation of a solution For example, barium chloride (BaCl 2 ) or barium carbonate (BaCO 3 ) can be combined with a solution of titanium oxychloride (TiOCl 2 ) and oxalic acid (H 2 C 2 O 4 ) In the correct ratios, BaTiO 3 precipit ates out of the solution. 10 Similar results can be obtained by adding solid

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12 barium salts to a concentrated solution of oxalic acid in hydrous titanium tetrachloride (TiCl 4 ). 10 Barium and titanium ions may also be dispersed in a colloidal suspension call ed sol, consequently dried and transformed to gel, and then heate d at high temperatures (anywhere from 300C to 1350C) 11 This process is known as the sol gel method and is used for producing solid materials from smaller molecules in liquid solutions L arger b arium titanate samples may be created from powders in a process called sintering. P owdered material is held in a mold while heated to a high temperature (800C 1000C) The heating process does not melt the material, but fuses the preexisting pa rticles by speed ing up atomic diffusion across particle boundaries. Sometimes sintered samples are annealed after an initial cooling, meaning that they are heated again at slightly lower temperatures ( 700 C ) and allowed to cool slowly to remove any intern al stress. 12 Already prepared samples may be crushed or ground into a powder if desired; the powder size is mechanically controlled by techniques such as milling. A ball mill is a type of grinder used to grind materials into extremely fine powder Small balls inside the mill (for reference, one experiment milling BaTiO 3 used 5 mm Yttrium stabilized zirconia balls) w ork to crush the sample into a fin er powder. 10 1.5 Size Effects B phase transitions depend on the particle size. With decreasing decreases, the temperature of the tetragonal cubic (T C) transition decreases, and the

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13 temperature of the orthorhombic tetragonal (O T) transition increases 11 The te mperature of the T C transition drops to 0 K below a certain critical particle size, which has been estimated at anywhere from 30 nm to 300 nm 13 Below this size, BaTiO 3 appears to remain in the cubic phase, with little evidence of ferroelectricity even a t low temperatures. Sun et al. actually attempts to model the Curie temperature as a function of particle size. 13 For a BaTiO 3 particle of diameter d he assumes the Curie point T c to be (7) where T c is the value for bulk Ba TiO 3 and d c0 is the critical size at which T c is 0 K. The most cited value of T c is 403 K and the value of d c 0 is predicted at 44 nm 14 P arameter A assumes the va lue of 0.22 to fit with experimental data obtained by Wang et al. 14 Eq. 7 is assumed valid only when d > d c 0 Factors such as sample preparation method may impact critical size findings for BaTiO 3 Lattice defects affected by heating time and temperatur e, may alter the phase transition properties. Mechanical distortion like milling may also lower t he tetragonality of the lattice 15 ferroelectric to paraelectric transition constitutes two separate events: a T C phase transition and a cell volume expansion. While the T C transition temperature remains at 135C regardless of particle size the volume expansion occurs at lower temperatures with decreasing particle size 15 A nother study suggests that tetragonal a nd cubic phases can coexist in the same material. The relative amount of the cubic phase increases with decreasing particle size. 3 Th e s e interesting result s imply that structural transitions occur over a

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14 range of temperatures and sizes and that these transitions are difficult to explain quantitatively 1.6 Heat Capacity heat capacity is the amount of heat (thermal energy) require d to increase the temperature of that substance by one degree, measured in units of energy per deg ree Since the amount of heat varies with the amount of mass in the substance, a more useful quantity is specific heat capacity, defined as the heat required per unit mass for the temperature to be changed by one degree. It has units of J/gK or J/kgK. H eat capacit y at constant volume ( C v ) and at constant pressure ( C p ) are defined by : ( 8a ) and (8b) w here U is the energy in the substance H is the enthalpy o f the system, Q is the heat transferred to the system, and T is the temperature. The enthalpy of the system is the total energy of the system, and is given by : (8c) where U int is the internal energy of the system and p and V are t he pressure and volume, respectively. The contribution of phonons to the heat capacity of a crystal is called the lattice heat capacity, denoted by C lat 6 The p honon is the quantum unit of a crystal lattice vibration. This means that the vibrational ener gy of the lattice when quantized, is represented as an oscillating particle. Lattice vibrations can be longitudinal (in which the particle displacement is parallel to the

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15 direction of wave propagation), transverse (particle displacement is perpendicular to the direction of propagation), or some combination of the two. The energy of any phonon vibrating with frequency is equal to (9a) where n is an integer > 0 and is the reduced Planck constant equal to 1.054 571726 10 34 J s. T he total energy U of all lattice phonons at temperature (where k B is the Boltzmann constant, equal to 1.3806488 10 23 J/K) may be written as the sum of the energies over all phonon modes ( ) indexed over all wavevectors k and polarizations P : (9 b ) where is the average thermal equilibrium occupancy of phonons. 6 The value of d epends on temperature and is given by the Planck distribution function: (10) The Debye approximation, based off of a classical model of the density of states, yields a total phonon energy of (11) w here V is volume, v is the constant velocity of sound, = vk and cutoff frequency D is determined by where N is the total number of acoustic phonon modes. 6 In the substitution, and Debye temperat ure T D is defined by (12) The heat capacity is found by differentiating Eq. 11 with respect to temperature, giving: 6 (13)

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16 The Einstein heat capacity f o r s o l i d s which differs from the Debye model mainly in the assumption that a l l o f t h e p h o n o n s v i b r a t e a t t h e s a m e f r e q u e n c y i s g i v e n b y : 6 (14) Fig. 1.6 compares the two models. Figure 1. 6 Debye vs. Einstein models. 16 The shapes are similar but diverge slightly at lower temperatures. In this study, b as a function of temperature i s fitted by a third order polynomial, see Table 3.1. Debye temperature s for BaTiO 3 range from around 415 K (for rhombohedral phase) to 618 K (cubic phase) 17 T h e s e v a l u e s p u t t h i s s t u d y i n t h e m i d d l e r a n g e o f t h e c u r v e i n F i g 1 6 g i v i n g v a l u e s o f T / T D f r o m a b o u t 0 4 t o 0 7

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17 1.7 Experimental Methods for Studying BaTiO 3 Phase Transitions As previously mentioned, the phase tra nsitions in barium titanate are studied in multiple ways including differential scanning calorimetry, Raman spectroscopy, X ray diffraction, and neutron diffraction. 1.7.1 Differential Scanning Calorimetry A differential scanning calorimeter (DSC) measures the amount of heat needed to increase the temperature of a sample compared to th e heat needed to raise the temperature of a reference by the same amount. In practice, the DSC heats two sealed aluminum sample pans simultaneously a reference (empty) pan and a sample pan filled with the studied material The re are two possible methods of obtaining heat flow differences. In a power compensation DSC, the pans are heated separately but are kept at the same temperature set to increase at a steady rate. The sampl e pan requires more heat to maintain the temperature a nd the difference is recorded Depending on the transition type an endothermic or exothermic peak, or a rise or fall in sample heat capacity value may occur. By comparing the difference in heat flo w between sample and empty pan, the calorimeter calculates values for the sample heat capacity and transition enthalpy. A heat flux DSC uses a single heating source for both samples meaning that the same amount of energy is put into both pans T he temper ature difference between the two samples is measured and converted into an energy differential in mW Fig. 1. 7 displays a s chematic for this type of DSC which is the kind that this study uses

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18 Figure 1. 7 A heat flux DSC sche matic. The heat flow to both pans is constant. 1 8 A typical BaTiO 3 heating curve with well pronounced phase transitions is presented in the figure below Figure 1. 8 Differential heat ing flow curve for BaTiO 3 300 nm powder 0.14 0.13 0.12 0.11 0.10 0.09 0.08 0.07 0.06 0.05 0.04 100 150 200 250 300 350 400 450 500 Heat Flow, W/g T, K

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19 A heating curve including the melting point (first order phase transition) of indium is shown At around 430 K, the heat flow jumps very sharply, as is characteristic of first order discontinuities. Figure 1. 9 Differential heat ing curve for Indium sample Todd and Lorenson defin e heat capacity values for the tetragonal cubic peak at at 285 K) 1 9 Lee et al. identify a low point for the transition enthalpy of 0.8 J/g and Hoshina et al. put th e enthalpy at 1.0 J/g. 20 ,15 1.7.2 Raman Spectroscopy Raman scattering is defined as the inelastic scattering of photons with the term inelastic indicating that the energy and frequency of the scattered light is not conserved This effect is named after C.V. R aman, who in 1930 discovered that some of the light deflected off of a transparent material changes in wavelength. Th e change in wavelength 3.50 3.00 2.50 2.00 1.50 1.00 0.50 0.00 400 410 420 430 440 450 Heat Flow, W/g T, K

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20 and therefore in photon energy occurs when the photons interact with molecular vibrations and phonons in the ma terial they are striking. The inverse relation between energy, E and wavelength, is called the Planck relation and is given by: ( 15 ) w here the Planck constant Js, the speed of light m/s, is in units of meters, and E is in units of Joules. Some of the in order to excite them to a higher vibrational state and the light that is scattered back is of a lower energy than the original light. Occasionally, the outgoing photons absorb some of the bond energy and are emitted at a higher energy, but this happens much less frequently. Raman spectroscopy is used to measur e the se shifts in energy which can provide information on the vibration al modes of the particular system. Typical ly in Raman spectrosco py, a laser of known wavelength illuminates a sample Outgoing light is collected with a lens and filtered so that the Rayleigh scattered light (light that has been elastically scattered, which comprises most of the reflected light) is removed from the fi nal signal. A notch filter, which blocks a small wavelength range of light, may be used for separating the Rayleigh and Raman scattered light. The remaining light is collected on a detector which measures the wavelength of each striking photon, as well as recording the number of photons of each wavelength Results are often collected on a graph of relative intensity vs. Raman shift ( cm 1 ). The Raman shift is the inverse wavelength of the collected light, and is therefore proportional to its energy.

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21 To study temperature dependent phase transitions, Raman spectra of barium titanate samples were taken at five degree intervals o ver a large temperature range. Fitting software was used to identify the most prominent peaks and track their changes in position over different temperatures. 1.7.3 X ray Diffraction X ray diffraction (XRD) techniques are used to study periodic structures, like those of crystals. This type of diffraction, which only works if the wavelength of the incident radiation is smaller than the l attice constant, is also termed Bragg diffraction. The theory behind it, proposed by W.L. Bragg, is as follows. Rays striking parallel equidistant atomic crystal planes at an incident angle may be constructively scattered reflected off of the planes. In accordance with the law of reflection, the angle of reflection is equal to the angle of incidence. For parallel planes spaced distance d apart, th e condition for constr uctive interference is called Bragg Law: 6 ( 16 ) where n is a positive integer and is the angle of incidence In XRD, the angle of incidence is measured from the surface of the interface. Fig. 1.10 demonstrates this relation.

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22 Fig ure 1. 10 Bragg's law geometry In x ray diffraction of a single crystal, a monochromatic beam of known wavelength is incident on a crystal angled towards the beam. The intensity of the diffract ed beam order to obtain the intensity ov er a large angle range. Peaks in the recorded intensity indicate the angles satisfying the Bragg equation, which can be used to calculate lattice spacing d Two dimensional images of the diffraction pattern, as opposed to intensity measurements at specif ic locations, may be used to create a three dimensional model of the electron density. Because the observed diffraction pattern represents the reciprocal 6

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23 Figure 1. 11 Schematic for a basic x ray diffractometer. In the above schematic, t he detector could be flat or curved (the curvature allows for greater incident angles) and is used to determine I vs. Whereas a single crystal produces a simple, dot ted diffraction pattern from a single beam of light, a powdered sample creates a ringed diffraction pattern. The rings are caused by randomly oriented crystals Because every angle is represented, the entire intensity spectrum is shown at once. The brig ht rings still represent areas of constructive interference. Figure 1. 12 Comparison of x ray diffraction signal maxima for single crystal and powder diffraction.

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24 1.7.4 Neutron Diffraction Neutron diffraction i s similar to x ray dif fraction. The main differences arise from the ways that neutrons and x rays interact with atoms. Whereas x rays are deflected off of the electron cloud, neutrons interact directly with the atomic nuclei. Diffracted x ray intensity is larger for atoms wi th larger atomic number, Z; by contrast, intensity of diffracted neutrons depends on the atomic isotope. Light atoms (with low Z numbers ) often contribute noticeably to the diffract ogram in the presence of heavier atoms. Because neutrons scatter primarily from the atomic nuclei, which are much smaller than electron clouds, neutron diffraction proves capable of resolving crystalline structures to a high level of precision. There are a few downsides to this technique, however. Neutron diffraction is used m ore often with powders than it is with single crystals. Crystals studied with this technique must be large on the order of 1 mm 3 2 1 Additionally, the neutrons for this technique are produced in nuclear reactor s, which are less accessible than synchrotr ons or x ray generators

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25 Chapter 2: Experiment 2.1 Materials and Apparatus Differential scanning calorimetry was used to study barium titanate BaTiO 3 nanoparticles 50 nm to 300 nm in size were analyzed. The masses of powder samples with alumin um Tzero holding pans were measured using a Mettler Toledo XP2U Ultra microbalance, accurate to TA Instruments Q20 DSC, with a liquid nitrogen cooling system (LNCS). Universal Analysis software was used to view heating curves and band s. Figure 2. 13 DSC and LNCS DSC LNCS

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26 2.1.1 DSC Calibration and Maintenance To ensure accurate data collection, the DSC required weekly conditioning. This conditioning was accomplished with the preset Cell/Cooler Conditioning routine, which heated the DSC cell to 75C and held it at that temperature for 60 minutes. Nitrogen was used as the purge gas. T his procedure burns away and cleanses the cell of any accumulated small particles. It also proved necessary to run occasional temperature calibration checks on the DSC. Indium, whic h has a 156.6C melting point, was used for this purpose. 2 2 Approximately 10 to 15 mg of indium was heated above the melting point and then appeared, as expected, as a s harp negative band near 156.6C ( 430 K ) on the differential heat flow vs. temperature graph, Fig. 1.9 2.2 Experimental Procedure The procedure consisted of three parts: sample preparation (selecting, taking the mass and sealing the material and pans), data acqui sition ( putting materials in DSC for set method including three heating and two cooling runs ), and data analysis (finding the location and intensity the transition bands and calculating the specific heat capacity over a large temperature range). 2.2.1 Sa mple Preparation BaTiO 3 powders consisting of spherical particles with diameters of 50 nm, 100 nm, 200 nm, and 300 nm were purchased from US Research Nanomaterials, Inc. After

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27 ensuring that the microbalance was zeroed, an empty sample pan and lid were m e asur ed individually, their weights recorded. The pan was then loaded with the desired sample (usually BaTiO 3 powder taken from a small plastic container labeled with the particle size) using a small metal tool for scooping. Care was taken to remove the p an from the microbalance before filling and to en sure that the pan exterior was clean before measuring its mass again. These procedures kept the scale clean of stray material, ensur ing the accuracy of mass measurements As previously mentioned, the micro g Usually this process of adding sample to pan and taking the combined mass was repeated several times. Once the desired amount of sample was added to the pan (sample mass ranged from about 20 to 35 mg), the combined mass of p an and sample was recorded. A press specifically designed for DSC pans was then used to push the lid into the pan, sealing the sample inside. The pressing procedure helps samples remain in good thermal contact with the bottom and sides of the pan at all times. Random movement in the powder particles during the measurement may result in fluctuations in the thermal contact with the walls of the pan and the thermal contact between particles. 2.2.2 Data Acquisition For any given DSC run, the sealed sample pan was placed inside the DSC on t he right thermocouple and the various lids to the heating/cooling chamber were replaced. A n empty hermetically closed pan was placed on the other thermocouple to serve as a reference (Fig. 2.2) The LNCS was checked for its liquid nitrogen levels and refilled if

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28 they were too low to complete the run ( below 40% ) T he necessary purge flow gases were also turned on: nitrogen at 50 mL/minute and helium at 25 mL/minute Figure 2. 2 DSC cell containing empty reference pan Information about the sample was entered into the computer connected to the calorimeter, in the data acquisition software ( TA Instruments ) Sample name, notes, and most importantly, mass (used for calculating the differential heat flow) were recorded for each sample run. A typical data acquisition run consists of twenty steps and five thermal cycles, performed following the algorithm below, in which Gas 2 refers to the helium purge flow : 1: Select Gas: 2

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29 2: Mass flow 25.0 mL/min 3: Equilibrate at 15 0.00C 4: Isothermal for 5.00 min 5: Ramp 20.00C/min to 180.00C 6: Isothermal for 5.00 min 7: Mark end of cycle 0 8: Ramp 20.00C/min to 150.00C 9: Isothermal for 5.00 min 10: Mark end of cycle 1 11: Ramp 20.00C/min to 180.00C 12: Isothermal for 5.00 min 13: Mark end of cycle 2 14: Ramp 20.00C/min to 150.00C 15: Isothermal for 5.00 min 16: Mark end of cycle 3 17: Ramp 20.00C/min to 180.00C 18: Isothermal for 5.00 min 19: Mark end of cycle 4 20: End of method Essentially, the sample was heated an d cooled several times between 150C and 180C. This method resulted in measurements for three heating and two cooling c ycle s completed in two hours The data was viewed and analyzed in the aforementioned

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30 Universal Analysis software, which displays a g raph of differential heat flow in units of W/g v s. temperature. Endotherms are shown going down ; exotherms, up 2 .3 Specific Heat Measurement The specific heat of BaTiO 3 was calculated using the differential heat flow measurements. A pressed, empty sam ple pan and a sapphire sample were measured following a similar calibration algorithm The empty pan was used to correct for pan contribution to the barium titanate measurements. The sapphire also sealed inside a sample pan, served as a calibration stan dard for heat capacity. s heat capacity is known accurately over a wide range of temperatures 2.3.1 Specific Heat Calculation The basic equation for heat capacity of a given DSC sample, taken from a 1981 Application Notes, is : 2 3 ( 16 ) where C p (s) = C p of the sample, J/gK C p (st) = C p of the sapphire standard, J/gK W s = sample mass W st = reference mass D s = sample d i f f e r e n t i a l h e a t f l o w empty pan d i f f e r e n t i a l h e a t f l o w mW a n d D st = reference d i f f e r e n t i a l h e a t f l o w empty pan d i f f e r e n t i a l h e a t f l o w mW

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31 More precise specific heat capacity measurements modify this equation to account for the differences in mass of the holding pans. Below is an equation for calorimetric sensitivity, E taken from a test method issued b y ANSI in 2001: 2 4 ( 17 ) where b = heating rate C p (c) = specific heat capacity of the aluminum specimen holder, 0.9 J/gK st = sapphire pan mass empty pan mass and all other constants are the sam e as in Eq. 16 Th e equation for specific heat capacity of a sample, C p (s) is: 2 4 (1 8 ) where s = sample pan mass empty pan mass a nd all other constants are the same as in Eqs. 16 and 17 Combining Eqs. 17 and 1 8 gives the following equation with pan mass corrections for DSC determined specific heat capacity: (1 9 ) which includes the original heat capacity calculation of Eq. 16 along with an additional term. All constants in Eq. 1 9 have been explained in Eqs. 16 1 8

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32 The heat capacities of different BaTiO 3 powders were calculated at 0.5 in tervals between 134 K and 451.5 K. The results were plotted in Microsoft Excel. For each of the 50 nm 100 nm and 200 nm samples, the second heating curve (out of three) was used to gen erate the plot of heat capacity. The first heating curve of each ru n was usually more noisy than the other two, which tended to be very similar For the 300 nm powder, the heat capacity was calculated from the average of two signals from the same sample, due to variation in heating curves. Below is a table of values and parameters used for specific heat calculations in this experiment: Parameter Sample Pan Sapphire mass 26.131 mg 50.736 mg 300 nm BaTiO 3 powder 29.974 mg 50.664 mg 200 nm BaTiO 3 powder 29.851 mg 50.889 mg 100 nm BaTiO 3 powder 29.891 mg 51.020 mg 50 nm BaTiO 3 powder 24.290 mg 51.039 mg Empty pan mass 50.730 mg Specific heat of aluminum sample pan 0.9 J/gK Table 2.1 Experimental values for the masses and the specific heat capacity of the aluminum The specific heat of aluminum is quoted at va lues surrounding 0.9 J/gK. No information value over the entire experimental temperature range was assumed. 2.3.2 Specific Heat of Sapphire Although the specific heat ca pacity of sapphire (J/gK) is known to four decimal places over a wide temperature range, it is only quoted at 10 degree intervals. 2 5 A much

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33 obtain an estimate for heat values in the range of interest (90 to 460 K) were fitted in Fityk to a second degree polynomial. The specific heat capacity of sapphire, C p (st) as a function of temperature T is given below: ( 20 ) Below is the plot of known heat capacity values displayed with Eq. 20 Figure 2. 3 Plot of literature values for sapphire heat capacity from 90 K 460 K fitted with Eq. 20 2.3.3 Differential H eat Flow for Sapphire Below is the differential heat flow for the sapphire standard that was used in heat capacity calculations. The sapphire gave a highly consistent signal throughout its entire heating cycle, so no averaging of different heating cycles was necessary. y = 5.27 10 6 T 2 + 0.00539T 0.36754 R = 0.9995 0 0.2 0.4 0.6 0.8 1 1.2 0 100 200 300 400 500 Cp, J/gK T, K

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34 Figure 2. 4 Sapphire heating curve The values from the above plot were used as the reference curve values in Eq. 19 2.3.4 Differential Heat Flow of an Empty Sample Pan The heat flow data for an empty sample pan was taken over the sa me temperature range as the powder samples. Results proved somewhat inconsistent between runs, however. Many of the heating curves appeared nois y and displayed erratic leaps and dips 9 we re taken from the two most stable runs Because the signals were slightly offset from one another, they were averaged in the heat capacity calculation. Below is a plot of the two signals, superimposed: 0.14 0.13 0.12 0.11 0.1 0.09 0.08 0.07 0.06 0.05 0.04 100 150 200 250 300 350 400 450 500 Heat Flow, W/g T, K

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35 Figure 2. 5 Heating curves for the empty pan data used in heat capacity calculations. Heat flow units are mW instead of W/g because the sample mass was entered into the DSC computer as 0.0 mg. Both signals have a similar shape, but the first empty pan signal is approximately 0.02 mW lower than the secon d one. Sample mass was entered as 0.0 mg because that 2.4 Phase Transition Band Analysis After the specific heat capacities of BaTiO 3 samples were calculated, the data points for all of the powders were input into Fityk, a computer program used for curve fitting. The baseline subtraction tool proved especially useful for revealing the lower temperature and less resolved transition bands not always apparent in the raw heat capacit y plots. The fitting proce e ded as follows. After the heat capacity data for a particular sample was loaded into Fityk, a baseline curve was defined with a spline of multiple 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 100 150 200 250 300 350 400 450 500 Heat Flow, mW T, K

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36 points along the baseline. For the 50 nm and 100 nm powders, six points were use d to define the baseline; the 200 nm and 300 nm powders required eight points. The background was then subtracted from the heat capacity plot so that the bands themselves were easier to locate and fit with different distribution functions. All transition bands were analyzed using both Lorentzian and Split Pearson fits. The Lorentzian function, L is a single peak distribution given by: (21 ) where x = x 0 is the center of the peak and is a parameter representing the peak width. ( 22 ) and the function is equ al to half of its maximum at ( 23 ) 2 6 Figure 2. 6 An example of the Lorentzian function 2 6

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37 The Pearson VII function, developed by Karl Pearson in 1895, is an intermediate bet ween the Gaussian and Lorentzian functions, frequently used for X ray diffraction curve fitting. The function is given by 2 7 ( 24 ) where H is the peak height at the center x 0 of the peak, and x is the independent variable The parameter is the full width half max (FWHM), while parameter represents the proximity of the function to a Gaussian ( or Lorentzian ( = 1) function. For ~ 50, the Pearson VII function is essentially Gaussian. 2 8 Fig. 2.7 compares Pearson band shapes that result from varying the value of Figure 2. 7 Pearson VII peak shapes. 2 7

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38 The Split Pearson V II function, SP VII (x) consists of two Pearson VII functions with the same values of H and x 0 but different values for and This function is piecewise, and is represented by the following equation. 2 9 ( 25 ) In order to decrease the experimenter error, each heat capacity plot was loaded into Fityk ten times, with independent background choice f ollowed by subtraction and band fitting performed each time. This procedure proved especially important for resolving the broad, low intensity bands. Small differences in baseline subtraction could potentially result in different peak parameters. Releva nt parameters from each fitting were recorded and averaged over the ten trials for each powder sample. These parameters included information like peak location, height, and full width half max ( F W H M ) For the Split Pearson VII functions, half width half max for both sides of each peak function were included. In general, the Split Pearson functions fit the bands more closely, due to their flexible shape and ability to accommodate asymmetry. Half width half max (HWHM) parameters obtained from the Split Pearson fi ts were used to set integration bounds for the latent heat calculations. The latent heat is the area under the heat flow curve for a given transition. For each band, the lower integration bound was three times the HWHM 1 subtracted from the peak temperat ure. The upper integration bound was obtained by adding three times the HWHM 2 to the peak position. HWHM parameters were used similarly in conjunction with the Raman DSC comparison; the peak bounds were used to estimate the approximate onset and offset of each peak. The lower bound was obtained by subtracting two times the HWHM 1 from

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39 the peak position and the upper bound was obtained by adding two times the HWHM 2 to the peak position. The baseline subtracted graphs in Chapter 3, created to clearly dis play the bands, were prepared in Microsoft Excel rather than Fityk. Third order polynomial functions were fitted to the heat capacity plots with band points deleted. An example fitting (for the 200 nm sample) is shown below: Figure 2. 8 Baseline fitt ing for 200 nm BaTiO 3 powder. The Excel obtained baseline functions and their R 2 values for each powder were recorded. y = 1 10 9 x 3 3 10 6 x 2 + 0.0021x + 0.0424 R = 0.9997 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 100 150 200 250 300 350 400 450 500 Cp, J/gK T, K

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40 Chapter 3: Results and Discussion 3.1 Raw Heat Flow Data for BaTiO 3 Particles The DSC measures differential heat flow needed to main tain the same sample and reference temperature. Data are collected in a wide temperature range 134 K to 451.5 K. The following plots show raw data taken from heating curves for 50, 100, 200, and 300 nm particles at every 0.5. All of the following plots display signals from the second heating cycle for each respective BaTiO 3 powder, except for the 300 nm plot. The 300 nm plot averages the signals from the second and third heating cycles for the same sample. Peak location appeared to vary the most betwe en different heating cycles for the 300 nm powder. This phenomenon could have several possible explanations: all samples had small variation (less than 0.005 W/mg) noticeable because its phase transition bands were the na rrowest. For the Curie transition, for instance, FWHM values decreased significantly with increasing particle size from 34.3 for the 50 nm powder to 5.01 for the 300 nm powder. The 300 nm sample may have had more random error perhaps the powder moved in the pan while it was in the DSC.

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41 Figure 3. 1 Raw h eat flow data for 300 nm (a) 200 nm (b) 100 nm (c) and 50 nm (d) BaTiO 3 powders.

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42 The heating scan for the 50 nm powder (Fig. 3.1d) appears to be featureless. The 100 nm powder (Fig. 3.1c) heatin g scan displays one weak band near 265 K, but otherwise appears featureless. Both 50 nm and 100 nm temperature scans appear to have weak, broad phase transition bands near 160 K. These bands are difficult to evaluate without fitting software, and were no t examined in their raw forms. They are considered later with the analysis of the heat capacity. A lack of bands for the smaller BaTiO 3 powders indicates that these powders may not exhibit the phase transitions that larger powders exhibit in this tempera ture range. By contrast, the BaTiO 3 traces for 300 nm and 200 nm BaTiO 3 powders (Figs. 3.1 a and b) display well pronounced phase transition bands. The band parameters, such as peak position, height, FWHM, and integral area are summarized in Table 3.2. T hey each exhibit peaks corresponding to the orthorhombic tetragonal (around 290 K) and tetragonal cubic (around 400 K) structural transitions. They also appear to have weak broad bands around 210 K, which could be peaks due to the rhombohedral orthorhombi c phase transition that should occur around 183 K. Again, fitting software proves necessary for analyzing these weak bands 3.2 Heat Capacity of BaTiO 3 and Indium The data discussed in S ection 3.1 were used to calculate (Eq. 1 9 ) the specific heat for th e BaTiO 3 powders over the entire temperature range. The sample and pan masses were measured and recorded prior to data collection. The mass data is summarized in Table 2.1. The plots of specific heat vs. temperature plots are presented both as is and wi th the baseline subtracted, for a better view of the bands (Fig. 3.2). As previously

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43 noted, these baseline subtraction plots were prepared with Microsoft Excel rather than Fityk, and are mostly for visual rather than analytical purposes.

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44 Figure 3. 2 H eat capacity plots for 300 nm (a) 200 nm (b) 100 nm (c) and 50 nm (d) BaTiO 3 powders with ( lower trace r i g h t s c a l e ) and without ( upper trace l e f t s c a l e ) background subtraction.

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45 The coefficients of the background polynomial, obtained with Microsoft Excel fitting, are summarize d in Table 3.1. The lowest order polynomial fitting all temperature cycles well turned out to be a third order one. Sample a 10 9 b 10 6 c 10 4 d 10 4 R 2 value 300 nm 0. 5 3 21 451 0.9998 200 nm 1 3 21 424 0.9997 100 nm 2 4 25 133 0.9999 50 nm 4 6 31 367 0.9999 Table 3.1 Background polynomial coefficients for BaTiO 3 powder samples, in the form of f(T) = aT 3 + bT 2 + cT + d. The linear term appears to be the most significant one, and tends to increase with decreasing particle size Smalle r particles appear to be more temperature sensitive. The shapes of these heat capacity plots can be compared with that of the 15 mg indium sample, shown below. Figure 3.2e Heat capacity for indium near the band corresponding to its melting point. 0.00 2.00 4.00 6.00 8.00 10.00 12.00 14.00 400 410 420 430 440 450 Cp, J/gK T, K

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46 Table 3.2 summarizes the results of the Cp band fitting for all of the different powder sizes and transitions. The parameters from the Split Pearson fits, discussed in Section 2.4, typically fit the bands better than the Lorentzian functions. The enthalpy of some of the transitions is also included. The enthalpy was calculated by the method described in Section 2.4. The indium heat capacity was not able to be fit with the software used, due to the sharpness of its increase near 130 K. 50 nm 100 nm 200 nm 300 nm R O Position, K 211.1 + 0.7 209.3 + 0.6 Height, J/gK 0.0030 + 0.0004 0.0043 + 0.0005 FWHM, K 26 + 4 23 + 3 HWHM 1, K 14 + 4 9 + 1 HWHM 2, K 11.7 + 2.0 14 + 3 Enthalpy, J/g 0.3138 0.2196 O T Position, K 155.3 + 0 .4 285.6 + 0.1 287.2 + 0.2 Height, J/gK 0.0091 + 0.0008 0.0188 + 0.0006 0.0219 + 0.0006 FWHM, K 34 + 3 19.1 + 0.5 14.8 + 0.2 HWHM 1, K 11.7 + 0.7 5.5 + 0.2 4.1 + 0.1 HWHM 2, K 22 + 3 13.6 + 0.4 10.7 + 0.3 Enthalpy, J/g 0.2746 0.2120 T C Position, K 158.2 + 0.4 267.2 + 0.3 398.31 + 0.02 401.12 + 0.04 Height, J/gK 0.0147 + 0.0003 0.0093 + 0.0004 0.093 + 0.001 0.079 + 0.001 FWHM, K 34.3 + 0.7 19.3 + 0.7 5.52 + 0.03 5.01 + 0.08 HWHM 1, K 13.3 + 0.5 9.3 + 0.7 3.89 + 0.04 3.20 + 0.09 HWHM 2, K 21 + 1 10.0 + 0.4 1.63 + 0.03 1.81 + 0.04 Enthalpy, K 0.2039 0.3782 1 0.3085 Table 3.2 Phase transition band parameters for BaTiO 3 powders. Expected peak positions for the 200 and 300 nm particles would occur near 2 7 8 K (O T), and 39 8 K ( T C). 11 The transitions corresponding to the different bands in the 200 nm and 300 nm samples were easy to identify; there were three transitions to look for, and three bands corresponding to phase transitions. Transitions in the 50 and 100 nm proved les s obvious. For example, the transition centered around 267.2 K in the 100 nm sample 1 The method used for determining the bounds of integration for latent heat was questionable for this band, as the right bound appeared to be only slightly outside of the peak offset.

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47 could correspond either to an orthorhombic tetragonal (O T) or tetragonal cubic (T C) phase transition. I chose to interpret it as T C phase transition, for multiple reas ons. First, the temperature at which the Curie transition occurs has been shown to decrease with decreasing particle size, whereas the O T temperature has been shown to increase. 11 Second, in both the 200 nm and 300 nm particles, the transition at the Cu rie temperature has the most narrow and intense band. If it exists for the 100 nm sample, then it can be expected to be the most prominent band for that sample. It is difficult to determine if the lowest temperature bands in the 50 nm and 100 nm samples r epresent a phase transition The y appear at the low end of the temperature range, meaning that there is very little baseline to the left of them. This lack of baseline makes them difficult to fit The low temperature bands were included because averagin g over ten trials gave consistent results for their locations and parameters, but it proves hard to know how accurate they are. T he bands, which are very broad and weak, could be part of the background function. In the DSC Raman comparison for 50 nm powd er (Fig. 3.4) the band does not line up with any interesting Raman features, furthering the idea that it may not be significant. As Figs. 3.2a and 3.2b demonstrate, the heat capacity plots for 300 nm and 200 nm BaTiO 3 powders appear relatively similar. T heir peak locations differ only by a few degrees and the corresponding band shapes are comparable. Interestingly, the 200 nm powder has an overall lower specific heat than the 300 nm powder, but its band at the Curie transition appears more intense More experiments need to be conducted to explain peak heat capacity is higher than that of the 100 nm powder. Both the 50 nm and 100 nm peak heat capacities

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48 are also, on average, higher than t he baselines of the 200 nm and 300 nm powders. Perhaps this effect has to do with the fact that the 50 nm and 100 nm samples exist in the cubic phase at lower temperatures than the 200 nm and 300 nm samples. If the cubic phase of BaTiO 3 has a higher specific heat than other phases, then cubic 50 nm and 100 nm samples may have a higher specific heat than powders with larger particle size that are tetragonal or orthorhombic at the same temperature. The tetragonal phase may have a higher s pecific heat than the orthorhombic phase, and so forth. The baselines of specific heat may also be affected by the packing efficiency of the powders. The smaller powder samples probably contain less empty space, which could impact the overall thermal con ductivity and skew the specific heat results. Eq. 7 predicts T C transition temperatures of 80 K for 50 nm, 339 K for 100 nm, 378 K for 200 nm, and 388 K for 300 nm BaTiO 3 powders. These predicted values of the y well with experimental results (158 K, 267 K, 398 K, and 401 K, respectively) particularly for the smaller particles This result seems to indicate that the model is flawed somehow, or that the methods used in its study. 3.3 DSC Raman Comparison Figs. 3.3 and 3.4 compare the DSC data with Raman scattering data for the 50 and 300 nm BaTiO 3 powders. The Raman spectra for these powders were taken with a 785 nm laser at every 5 between 83.15 K and 573.15 K. Fittin g software was used to manually track Raman shifts of individual modes over this temperature range. The temperature dependency of the Raman shift (cm 1 ) of each individual mode was plotted

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49 in order to view the vibrational mode behavior at the phase transi tions. The plots below were taken from the 260 cm 1 mode of the Raman scattering. This mode shows the most distinct phase transition data of all the modes considered. Figure 3. 3 Comparison of 300 nm BaTiO 3 Raman data (upper trace) with DSC baseline subtraction (lower trace) Thicker d ashed lines represent peak locations as calculated from the DSC data, and thinner dashed lines surrounding the bands estimate peak width as outlined in Section 2.4 0 0.05 0.1 0.15 0.2 0.25 0.3 210 220 230 240 250 260 270 100 150 200 250 300 350 400 450 500 Cp, J/gK Raman Shift T, K

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50 Figure 3. 4 Comparison of 50 nm BaTiO 3 Raman data (upper trace) with DSC baseline subtraction (lower trace) Thicker d ashed lines represent peak locations as calculated from the DSC data, and thinner dashed lines surrounding the bands estimate peak width as outlined in Section 2.4 The T C band in the 300 nm DSC (Fig. 3.3) lines up with what appears to be a narrow upward slope in the Raman data. The O T band in the 300 nm DSC lines up very well with a wider upward slope; the R O transition lines up with a local minimum, but its ny interesting features on the Raman trace. The 50 nm the single band found in the DSC data 3.4 Size Dependent Parameter C omparison Fig. 3.5 is a s et of plots comparing the transition temperatures, FWHM values, and transition enthalpies for the different transitions and powder sizes. 0.001 0.004 0.009 0.014 0.019 0.024 0.029 0.034 0.039 0.044 217.00 222.00 227.00 232.00 237.00 242.00 247.00 252.00 257.00 262.00 100 150 200 250 300 350 400 450 500 Cp, J/gK Raman Shift T, K

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51 Figure 3.5 Particle size dependent plots of FWHM, peak location, and latent heat for the three phase transitions. Unsurprisingly, the 200 and 300 nm BaTiO 3 powders display similar peak positions and FWHM (Figs. 3.5 a f) especially for the two higher temperature transitions. The latent heat of the 200 nm powder appears to be greater than the latent heat for the 300 nm powder for all three transitions (Figs. 3.5 g i) which is unexpected. More experiments

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52 need to be conducted in order to determine if this is a property of the material or a difference between the two samples used.

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53 Chapter 4: Concl usions and Future Experiments 4.1 Conclusions The results of this experiment confirmed that size differences in BaTiO 3 nanoparticles considerably alter their heat capacity and phase transition behaviors. Overall, the 200 nm and 300 nm particles displayed similar transition behaviors. All three expected structural transitions appeared as bands in the heat capacity plots, and peak locations for each transition of the 300 nm powder were within a few degrees of the corresponding 200 nm peaks (Table 3.2). Fo r the T C transition, the transition temperature dropped approximately 3 from the 300 nm to the 200 nm sample. For the O T transition, the transition temperature dropped approximately 2 from the 300 nm to the 200 nm sample. Both sets of transition temp eratures were near the transition temperatures found by Frey and Payne, although the O T transition in particular was about 7 9 higher than expected. The transition enthalpies for the 200 nm sample appeared to be larger than those for the 300 nm sample, an effect that is currently unexplained. DSC scans for 300 nm BaTiO 3 powders complemented Raman data taken on powder of that size. In particular, the bands corresponding to O T and T C phase transitions appeared to line up with sections of the data in wh ich the Raman shift near 260 cm 1 was increasing. The 100 nm and 50 nm particles also displayed similar transition behaviors, although the 100 nm particles exhibited a transition band around 267 K that the 50 nm particles did not. Both powders showed a v ery faint, broad transition band centered between 155 and 160 K. The presence of these transition bands, particularly in the 50 nm powder, was somewhat unexpected. 50 nm BaTiO 3 is often below the critical size values

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54 quoted in literature, although some a rticles do argue for a critical particle size lower than 50 nm. 13 It seems likely that the low temperature band found in the 50 nm powder shows the Curie transition happening at a very low temperature because the transition temperature decreases with dec reasing particle size In all four powder samples, the clearest transition was the T C transition, followed by the O T transition. The R O transition was difficult to resolve even for the larger particle sizes. Essentially, the structural phase transit ions become harder to view with findings the T C transition is the last to disappear but the cause for this phenomenon has not been investigated in this paper. The D SC results given by these four samples are by no means comprehensive, but they do clearly demonstrate the existence of size and temperature dependent phase transitions in barium titanate nanoparticles. 4.2 Future Experiments Although there was a great de al of variation in the DSC data between the four different samples, it could be informative in future experiments to include even more particle sizes. In particular, larger particles would give information about the behavior of the transition temperatures in powders well above the critical size. The FWHM and peak locations for the 200 nm and 300 nm samples were very similar, implying that these parameters might not change much above a certain particle size. Data points for 400 nm, 500 nm, and larger powd ers would help to confirm or refute this relation. Data could be taken on multiple samples of the same powder size in order to control for possible random error in sample preparation or issues with thermal contact

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55 during data collection. Sometimes DSC si gnals display random blips or squiggles that are due to powder shifting inside the pan or other factors that are difficult to prevent. Analyzing more trials and including more samples to average over would ensure that the resulting data is not dependent o n a single sample. It could also be worthwhile, in future experiments, to see if the reported parameters vary at all with changing sample mass. More emphasis in this experiment was placed on standardizing the various sample masses (all but the 50 nm BaTiO 3 are approximately 30 mg of powder) than on investigating mass effects. Given more time and resources, however, it could be a valuable question for consideration. Some authors report aging effects for barium titanate and other ferroelectrics. Aging effe cts occur when a ferroelectric material is held in the ferroelectric state for an extended period of time, and result in an increase in the Curie transition temperature. For BaTiO 3 the longer aging times appear to correspond with proportional increases i n T c as compared with a non aged or de aged sample. A de aged sample is one that is held above the ferroelectric to paraelectric phase transition temperature prior to analysis. 30 Figure 3.6 Shift of DSC peaks of ferroelectric paraelectric phase trans ition of a BaTiO 3 single crystal sample. Aging temperature is 80 C. 30

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56 This aging effect was not explored at all or taken into account in this experiment, but it would be useful to consider in future experiments. Perhaps the same sample could be cycled repeatedly, or DSC running methods could be created to age or de age the samples before heating them in order to control for different levels of ferroelectric aging. Finally, some preliminary attempts were made to look at pressed powder samples (at 20 ft lbs and 60 ft lbs) with the DSC. The results were uninteresting and have not been included in the current study, as pressing the powders mainly appeared to lessen the intensity of the phase transition bands. Given more time, however, pressed BaTiO 3 powde rs could be more thoroughly studied and analyzed.

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