Review: Indentation size effect,indentation cracks and microhardness measurement of brittle crystalline solids – some basic concepts and trends

Indentation size effect, indentation cracks and microhardness measurement of some brittle crystals are reviewed against the background of the existing concepts of indentation deformation of crystalline solids. Several approaches reported in the literature devoted to relationships between applied indentation test load P and indentation diagonal length d are applied to analyze the experimentally observed normal and reverse indentation size effect (ISE) in brittle compounds. Using typical examples of normal and reverse ISE it is shown that the indentation induced cracking model does not give load‐independent hardness and the final expression describing the experimental data for various compounds is essentially another form of the Meyer law. Analysis of experiment data on crack lengths and indentation diagonals for different indentation loads suggests that the origin of ISE is associated with the processes of formation of indentation cracks following the general concepts of fracture mechanics. The load‐independent hardness H₀ may be determined reliably from plots of P /d against d of the proportional resistance model or of H_V against 1/d as predicted by strain gradient plasticity theories. It was found that the load‐independent hardness of depends on crystal orientation and state of the indented surface. Finally, some comments on determination of fracture toughness and brittle index of crystals are made. (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The cracking of Czochralski-grown crystals

Crystals of brittle materials are easily cracked during manufacture and processing. This paper examines the thermal strains existing in crystals during and after pulling and the strains which can be built-in by the facet effect. It is shown that during the growth of a crystal with a radius R, there is a maximum acceptable axial gradient roughly proportional to R^?1.5 and that to prevent cracking after growth there is a maximum rate of cooling proportional to R^?2. Results obtained with bismuth silicon oxide for which all the relevant parameters (breaking strain, expansion coefficient, thermal diffusivity and cooling constant) have been measured are in accord with these relations at least for 4 <R<23 mm. The analysis seems valid for other materials (e.g. lithium niobate) with small breaking strains and negligible plastic ranges. For bismuth silicon oxide, (110) and (100) facets have lattice constants larger than the bulk of the crystal by about 4 and 6 parts in 10⁵ respectively. The edges of these facets are thus regions which are strained by amounts corresponding to 20 or 30% of the breaking strain. Under normal growth conditions, this strain is not particularly significant, but it does affect post-growth handling and during rapid growth, interface instabilities develop first at the facet edges.     

Crystalloluminescence produced during the micro-fracture of growing crystallites

The present paper reports the theory of crystalloluminescence (CRL) produced during the micro-fracture of growing crystallites. Surface charges may be developed during the micre-fracture of crystallites due to several processes like piezoelectrification, movement of charged dislocations, barodiffusion of defects near the moving crack etc. The surface charges may be neutralized by the movement of charge carriers produced due to the dielectric breakdown near the surface of crystallites and consequently the recombination luminescence may occur. Considering the basic concepts of crystallization from the supersaturated solution, expressions are derived which are able to explain satisfactorily the temporal, thermal, spectral concentration and other characteristics of the CRL. It is shown that the induction period for crystal nucleation, lifetime of supersaturation, size of crystallites and time constant for the relaxation of charges on the newly created surfaces may be determined from the CRL measurements.     

Mechanoluminescence of halogenate crystals

Mechanoluminescence (ML) has been studied in twenty halogenate crystals. All piezoelectric crystals show ML similar to the emission from second positive group of molecular nitrogen, and all the crystals not exhibiting ML are non-piezoelectric. Some of the non-piezoelectric crystals also exhibit nitrogen emission ML which is comparable in intensity to that of piezoelectric crystals. The ML appears only during the different steps developing in the fracture region of the force versus compression curve of the crystals. No considerable change in the ML activity is found due to the number of crystallization or due to the addition of different impurities from 100 to 10.00 ppm. The ML activity per mole of the halogenate crystals varies around four orders of magnitude. It is concluded that in addition to the piezoelectrification of the newly created surfaces there exists some other process of electrification which may cause the ML exitation in halogenate crystals.     

Strength of Crystals of Small Dimensions and their Triboluminescence

Stress-dependence of triboluminescence (TBL) in uranyl nitrate, tartaric acid and other nine crystals has been studied with the help of a new technique of crushing the crystals. As the appearance of TBL needs the creation of new surfaces in a crystal, the minimum stress at which TBL appears has been taken to be the fracture-strength of the crystal. The value of the fracture-strength and stress coefficient of binding energy determined from TBL measurements are found to be of correct order. It has been found that the rate of rise of TBL intensity with stress is higher for those crystals which have less value of the fracture-strength. The fracture strength (σ_f) is found to be higher for those crystals which have less value of stress coefficient of binding energy (β) and the product of σ_f and β is higher for those crystals which have higher melting point. It has been shown that the rate of rise of TBL intensity with stress decreases at higher values of stress due to strain hardening in the crystals.     

Brittle fracture of single crystals and active structure of cleavage surfaces

By the method of vacuum decoration at an electronmicroscopic scale with the use of lowangle electron diffraction the dependence of brittle fracture of alkali halide crystals and of the active structure of cleavage surfaces on the character of ordered distribution of point defects in crystals has been studied. The occurrence of cleavage steps reflects the presence in crystal volume, where brittle fracture took place, of a high degree of periodicity of point defects mainly in one crystallographic direction. The smooth areas of the cleavage surface reflect the existence in these places of a periodic distribution of point defects in two mutually perpendicular crystallographic directions. Coincidence of periods between the cleavage steps with the periods in the arrangement of point defects on smooth areas of the cleavage surface allows the steps to be regarded not only as elements of the geometric microrelief but primarily as electrically active linear elements of the crystal surface, reflecting the linear arrangement of point defects. Together with cleavage steps exhibiting a stable in time activity there arise cleavage steps with an extremely high, but rather shortlived activity being the consequence of the brittle fracture process proper. The electrically active linear elements of the point defect lattice – linear elements of the potential relief of the surface – play a most important role in heterogeneous processes, in particular crystallization processes.     

Cleavage Mechanoluminescence in Crystals

When a crystal is cleaved, initially the mechanoluminescence (ML) intensity increases linearly with time, attains an optimum-value I_m at a particular value of timet_m, and then decays exponentially with time. Cleavage ML provides a new tool to determine the velocity, v of cracks in crystals, and it may be given by v = H/t_m, where H is the thickness of the crystal. Both, the peak ML intensity I_m and total ML intensity I_T increase linearly with the area of newly created surfaces A as well as with the surface charge density γ. The ML intensity decreases with temperature primarily due to the decrease in the surface charge density. Beyond a particular temperature, the surface charge density may decrease to such a value where the breakdown of gases and solids may not be possible and thereby the ML may not appear. Depending on the prevailing conditions either the ML emission resembling gas discharge or other types of the luminescence of solids, or that having these two characters may be obtained. There exists a good correlation between the theoretical and experimental results obtained for cleavage ML in crystals.     

Fracture in precursor-derived Si–C–N ceramics – Analysis of crack roughness and damage mechanisms

Roughness analysis of fracture in precursor-derived amorphous and phase segregated Si–C–N ceramics using fractal methods is reported, towards examining the possible correlations between fractal scaling of roughness and fracture properties as well as fracture damage mechanisms. Topography of the fracture surfaces created at a crack velocity of ～10^?4 m/s was recorded using atomic force microscopy, and analyzed using RMS roughness and second order height–height correlation functions. The evolution of roughness was well correlated with the evolution of structural and compositional inhomogeneities in the amorphous materials, and the formation of second phases in the phase segregated materials. All the investigated fracture surfaces displayed self-affine scaling with a correlation length of ～50–100 nm and a roughness exponent of 0.8 ± 0.1, commensurate with the universal exponent conjectured by Bouchaud et al. corresponding to dynamic damage regime. No correlation was observed between the roughness exponents and the fracture toughness of the corresponding materials. Examination of the crack opening near the tip region revealed no persistent damage cavities assignable to ‘plastic deformation’ preceding fracture, suggesting that the fracture in the Si–C–N ceramics proceeds in a brittle manner at the employed crack velocities.     

Designing Crystal Morphology by a Simple Approach

In separation processes containing solids, mostly crystals, the interest in shape modifications of the solids has increased. The idea in it is the necessity to save energy, to achieve specific product requirements, e.g. purities, or to provide downstream processes with better feed material. For an organic crystal a simple way to select a growth inhibitor is presented here, in contrast to the concept of “tailor-made additives”. The additive selected is structurally different from the host molecule and modifies the habit in a desired manner. The procedure of additive selection involves the analysis of the molecular structure of the crystal to be modified and the orientation of the molecules at the surfaces of the morphological most important faces. Furthermore the problem is discussed if the chosen additive is incorporated into the crystal or not. Results are shown for the system caprolactam/ethanol.     

Satatistical Model of Mechanoluminescence in Crtstals

The present paper reports on a statistical model of the mechanoluminescence (ML) excitation in crystals where attempts have been made to explain stress, temperature-, strain rate-, activator concentration dependence and several other aspects of ML. It is found that ML emission should take place only during the period at which stress will change with time. The total ML intensity should be directly proportional to the square of the stress and there should be phase difference between the ML pulse and the applied stress pulse. It is found that the ML intensity should decrease faster with temperature as compared to the corresponding photoluminescence intensity. within the limit of concentration quenching the ML intensity should increase directly with the defect centre concentration. The ML intensity should increase with the stress rate or strain rate of the solids. Furthermore, the concept of mechanoluminescent and non-mechanoluminescent materials is explored.     

Präparation und Charakterisierung von Granateinkristalloberflächen

The preparation of single crystal substrates of gadolinium gallium garnet as a typical example of a brittle material is described. The mechanical polishing with iron oxide and diamond as well as a mechano-chemical polishing procedure basing on SiO₂/H₃PO₄ are investigated. Working damage is investigated by X-ray topography, double crystal spectrometry and selective etching. Results are discussed in the frame of a brittle fracture model of the abrasion process.     

Zum Problem der Zweikristall-Topographie von kleinen Defekten mit Bragg-Bragg-Reflexion

Double crystal back reflection topography was applied to study the contrast of small defects as they are created near silicon surfaces during oxidation. The contrast varies stronger with the extinction depth than with the orientation between reflecting lattice planes and possible Burgersvectors. This is obviously due to surface relaxation of the strain field. Since the extinction distance varies strongly for different reflections reliable determination of the nature of the defects appears impossible without comparison to contrast simulation.     

脆性材料切削温度理论模型及实验研究

通过将单次断裂扩展成连续断裂,并利用被加工面表面形貌面积和投影面积之比k1,计算断裂次数,根据脆性材料微观断裂机理的能量守恒原理,建立了脆性材料切削温度理论模型;使用硬质合金刀具切削氟金云母陶瓷,开展车削实验,验证模型可靠性;结果表明,建立的脆性材料切削温度理论模型,可以在一定范围内,对温度随工艺参数变化趋势进行预测.     

Twin laws explained by partitions of space

Twin laws of monoclinic potassium feldspars, potassium sulphate, cerussite and hexabromobenzene can be explained by partitions of space. The face indices of the crystals denote points of a lattice in the reciprocal space. A Fourier transform leads to the morphological lattice. The reciprocal crystal is the Dirichlet domain of the morphological lattice and the structural content consisting of groups of atoms or molecules is the morphological unit. The bond strength between neighbouring morphological units is proportional to the area of the common face of the reciprocal crystals. The reciprocal crystal represents the energetical structure of a crystal. Twins can be explained by continuation of the energetical structures across the twin boundaries. The symmetry operations result from pseudosymmetries of the reciprocal crystals. It is possible to predict planes favoured for the formation of twins or parallel grown individuals.     

Die Bedeutung mechano-elektrischer Erscheinungen für die Bestimmung der Oberflächenenergie von Kristallen

Using the method of crystal cleaving a calculation of the surface free energy of mica is given. For this an effective elastic stiffness constant is introduced, this taking into consideration the electrostatic charging of the crystal surface. For measuring the charges arising from the cleaving of the crystal a special experimental equipment is used. Microgeometry and mechano-exoelectronic flow are determined. With sufficient high vacuum the latter becomes the main process of stabilizing instable electric crystal surfaces. The electrostatic charging only little affects the values of measured surface energies. The deformation, however, of layers near the surface gives a larger effect. Considering this deformation a surface free energy of 380 erg · cm⁻¹ for mica is given.     

Emergence of regular meandered step structure in simulated growth of GaN(0001) surface

Step meandering during the growth of gallium nitride crystal is studied on using kinetic Monte Carlo method. Cause of instability is identified to be the particle advection caused by the step flow. Growth process is conducted in N-rich conditions and GaN(0001) surface kinetics is modeled by setting jump probabilities for Ga atoms adsorbed at the surface. We show that at low enough temperatures and relatively high external particle fluxes periodic regular pattern of meanders is created with its wavelength inversely proportional to the particle flux. An increase of the meander amplitude saturates after some period and further crystal grow is stationary, creating “finger-like” structure. For medium fluxes regular structure of meanders builds up for low or zero value of Schwoebel barrier. For higher fluxes wavelengths of meanders become shorter than the terrace width and they start to grow independently and finally transfer the surface to a rough structure. For very low fluxes or at relatively high temperatures steps move steadily remaining their initial shapes of straight, parallel lines.     

Macrosteps dynamics and the growth of crystals and epitaxial layers

Step pattern stability of the vicinal surfaces during growth was analyzed using various surface kinetics models. It was shown that standard analysis of the vicinal surfaces provides no indication on the possible step coalescence and therefore could not be used to elucidate macrostep creation during growth. A scenario of the instability, leading go macrostep creation, was based on the dynamics of the step train, i.e. the step structure consisting of the high (train) and low (inter-train) density of the steps. The critical is step motion at the rear of the train which potentially leads to the step coalescence i.e. creation of the double and multiple step. The result of the analysis shows that the decisive factor for the step coalescence is the step density ratio in and out of the train. The ratio lower than 2 prevents double step formation irrespective of the kinetics. For higher ratio the coalesce depends on step kinetics: fast incorporation from lower terrace stabilizes the single steps, fast incorporation from upper leads to step coalescence. The double step is slower than the single steps, so the single steps behind catch up creating multistep and finally macrostep structure. The final surface structure consists of the macrosteps and superterraces, i.e. relatively flat vicinal segments. The macrostep alimentation from lower superterrace leads to emission of the single steps which move forward. Thus the single step motion is dominant crystal growth mode in the presence of the macrosteps. These steps finally are absorbed by the next macrostep. The absorption and emission of single steps sustain the macrostep existence, i.e. the macrostep fate is determined the single step dynamics. The condition for single step emission was derived. In addition, the macrosteps are prone to creation of the overhangs which results from surface dynamics coupling to impingement from the mother phase. The angular preferential access of the bulk material to the macrostep edge, leads to the overhang instability and creation of inclusions and dislocations.     

Explanation of some peculiarities of crystal morphology deduced from the BFDH law

The morphological variety of crystal habits is due to differences in relative growth rates of faces of which the crystal is composed. For equilibrium, the growth rates of faces are proportional to the distances from the centre of the crystal to the respective hkl faces. According to the BFDH law, such distances are inversely proportional to the interplanar distances, therefore the observed crystal faces are those with the largest interplanar distances. This paper tries to explain some peculiarities of the crystal morphology deduced from the BFDH law and shows that the crystal geometry influences morphological importance of faces and because of crystal geometry, the faces of the largest interplanar distances are not necessarily the largest faces in the BFDH morphology.     

Effect of precipitation procedure and detection technique on particle size distribution of CaCO3

The deposition of inorganic salts (“scales”) such as calcium carbonate is an important flow assurance problem during crude oil production. The knowledge of the features of the precipitated solids, mainly the particle size and morphology, is crucial to understand the nature of the solids and to avoid or reduce the effect of their deposition. For instance, the use of additives is one of the most usual procedures to mitigate this problem. Additives interact with scale-forming substances either by increasing the induction time, or by inhibiting crystal growth, changing the morphology of solids.     

Nucleation and growth kinetics for the crystallization of ice from dextrose solutions in a continuous stirred tank crystallizer with supercooled feed

The crystallization of ice in aqueous dextrose solutions is studied in an adiabatic continuous stirred tank crystallizer with a supercooled feed stream. The effective diameter of the ice crystals was determined for various values of mean crystal residence time, feed supercooling, magma density, stirring rate, and dextrose concentration. For all process conditions the supercooling was measured at 9-12 different locations in the crystallizer. These local supercoolings were averaged algebraically to yield the bulksupercooling. From the experimental results growth and nucleation rates have been calculated. By comparing the experimental growth rates to growth rates calculated by means of a mathematical model kinetics for the inbuilding of water molecules into the ice lattice have been determined. The growth rate appears to be directly proportional to the interface supercooling. The rate constant decreases exponentially with increasing weight percentage of dextrose in the solution. The nucleation rate was found to be directly proportional to total crystal surface per unit volume of suspension and proportional to the bulksupercooling to the power 2.1. Nucleation is believed to occur by breakage of dendrites from the surface of parent crystals.