Alternative Approaches to Measure a Dislocation

Dmitrii Vasilev Physical Metallurgy December 17, 2012

Alternative Approaches to Measure a Dislocation Density A dislocation is a crystallographic defect, or irregularity, within a crystal structure. The presence of dislocations strongly influences many of the properties of materials. The theory was originally developed by Vito Volterra in 1905, but the term 'dislocation' was not coined until later by the Professor Sir Frederick Charles Frank of the Physics Department at the University of Bristol. In 1934, Egon Orowan, Michael Polanyi and G. I. Taylor, roughly simultaneously, realized that plastic deformation could be explained in terms of the theory of dislocations. Dislocations can move if the atoms from one of the surrounding planes break their bonds and rebond with the atoms at the terminating edge. In effect, a half plane of atoms is moved in response to shear stress by breaking and reforming a line of bonds, one (or a few) at a time. The energy required to break a single bond is far less than that required to break all the bonds on an entire plane of atoms at once. Even this simple model of the force required to move a dislocation shows that plasticity is possible at much lower stresses than in a perfect crystal. In many materials, particularly ductile materials, dislocations are the "carrier" of plastic deformation, and the energy required to move them is less than the energy required to fracture the material. Dislocations give rise to the characteristic malleability of metals [2]. It is important to study dislocations to explain different phenomena, which cannot be explained in another way. For example, dislocations are responsible for hardening effect: In FCC metals the yield stress and the strain hardening are governed by the forest interactions, i.e., the interactions with the dislocations intersecting the glide planes of moving dislocations. Dislocations also affect the electrical and optical properties of polycrystals. The studies of dislocations are mostly focused on their nucleation, geometry, on their role in plastic deformation and on their kinetics. In this paper I am aimed to investigate different ways of measuring dislocation density. In particular, I am interested to analyze the recently developed method of measuring dislocation density using an ultrasound [1]. Most of the experimental dislocation observation methods are not based on the identification of the dislocation line itself, which is practically difficult to implement, but on the registration of stresses or distortions of the lattice due to dislocation. In essence, dislocations were first directly observed by transmission electron microscopy (TEM) by the group of Peter B. Hirsch at Oxford [3]. The main methods of observing dislocations today are: 





The method of selective chemical etching - the warped interatomic bonds are more vulnerable to chemical and physical effects than non-deformed. Consequently, if one etch the crystal (usually in an acid solution), the place where dislocations emerge on the surface etches more intensely than the surrounding material. The ends of dislocations, observed in microscope are usually called "pits" or "bumps". The method of photoelasticity - own stress fields of the dislocations adds up and create a noticeable tension within the crystal, which can be observed through piezo-optical effect (photoelasticity). This method is particularly suitable for dislocations observation in cubic crystals, which are optically isotropic in a relaxed state. The dislocations’ stress fields are recorded via birefringence. X-ray structural analysis - provides an opportunity to explore the thick and large enough samples. It uses X-ray Bragg reflection while the diffraction contrast is obtained due to the fact that the local lattice deformation associated with the defect changes the conditions of reflection and scattering of X-rays. The intensity of the diffracted X-ray beam near the defect is reduced, so that the defect is visible as a dark line on the common light background. X-ray diffraction topography is usually used in crystals, where the defect density does not exceed per . This is due to the fact that the magnification in X-ray topography method is almost equal to one, but the width of the diffraction image and, therefore, the resolution is of a few microns order. Therefore, for large dislocation density the image can overlap and the opportunity to explore the characteristics of individual defects will be lost. On the other hand, the ability to obtain a picture of the dislocations distribution over a significant area of the crystal is a significant advantage. Modern cameras can obtain topograms of crystals having a diameter of 150 mm.

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Electron microscopy - dislocations observation using electron microscope. The method is based on the effect of dislocation contrast arising from the interaction of the electron beam with the displaced atoms in the stress field around the dislocation. Due to the interference between transmitted and diffracted electron beam, one can see a picture of dislocation lines, series and dislocation walls in the electron microscope.

The dislocation density is of particular interest if we want to compare properties of different materials or explain/predict the material behavior. The dislocation density is a measure of how many dislocations are present in a quantity of a material. Since a dislocation is a line defect, this is defined as the total length of dislocation per unit volume. Consequently the units are m/m3 = − . Equivalently, it is the number of dislocation lines intersecting a unit area. Dislocation density is usually of the order of 10 − in a metal, increasing to ~ 15 − after work hardening. In the first method, the sequence of actions to calculate the dislocations density is the following: 1) Calculate , the number of "pits" and "bumps" 2) Calculate , the area of observation − 3) Calculate dislocations density . It is the only method where a human interaction with specimen during experiment can strongly affect the result (over etching causes wrong results, human eye is not perfect, etc). Besides, counting by hand is laborious, local and intrusive. More often dislocation density is determined by transmission electron microscopy (TEM) direct observation, but this method is rather complex and time consuming. Moreover, the small size of the regions observed, associated to the heterogeneity of the dislocation distribution in plastically deformed metals, may lead to ρ values that are not representative and are inconsistent with the plastic behavior of the material. There is one more method, not included to the list above, the essence of which is described in work [1]. According to [1], dislocations in a material will, when present in enough numbers, change the speed of propagation of elastic waves. Consequently, two material samples, differing only in dislocation density, will have different elastic constants, a quantity that can be measured using Resonant Ultrasound Spectroscopy (RUS). To check this assumption, an experiment was conducted, using 1100 pure aluminum. From the same bar, five samples were taken classified as original, annealed and laminated material: two samples were annealed at 400◦C, one for 5 h and another for 10 h. Other two were coldrolled, one at 33% and another at 43%. The measurements, made via RUS allow to calculate the dislocation density in different samples. First, let us consider the principle of operation of RUS [4]. RUS uses normal modes of elastic bodies to infer material properties such as elastic moduli. In principle, the complete elastic tensor can be inferred from a single measurement. The scheme utilizes the fact that the mechanical vibration resonance spectrum depends on the geometry, mass density, and elastic tensor of the sample. Samples (generally parallelepipeds) are polished to a mirror finish on all six sides, and accurate dimensions are obtained. The sample is then lightly sandwiched between transducers so that a resonance spectrum can be obtained (see Fig.1 ). The spectrum (center frequencies of the resonance peaks) is then used as input data for a program which adjusts the initially guessed elastic tensor until the calculated spectrum most closely matches the measured spectrum [5]. In present experiment the wave propagation speed was measured through the measurement of the resonant frequencies of the samples for stress-free boundary conditions. These frequencies, in turn, provide the elastic Figure 1 Schematic of the twotransducer resonant ultrasonic constants (in Voigt notation) after the mass and linear dimensions spectroscopy set up. [Resonant of the samples have been independently measured. In order to satisfy ultrasound spectroscopy: applications, this boundary condition, samples are placed in between an emitter and current status and limitations. R.B a receiver, held by two opposite corners, the receiver being held by a Schwarz, , J.F Vuorinen1.] set of springs mounted on a linear air bearing. The weight of the receiver part of the set-up is such that, at equilibrium with the spring force, the distance between the emitter and receiver surfaces is slightly larger than the sample’s diagonal. In this way, the sample can only be held by its corners in the setup if a small mass is added to the receiver part, typically of

10 g. Thus, the sample–apparatus contact force is small, with an upper bound of 0.1 N, corresponding to about one-half of the sample weight Mg = 0.22 N. The estimated contact area is 0.1–0.2 [6]. The theory of the waves propagation gives us that the medium of a crystal allows for the propagation of longitudinal (acoustic) and transverse (shear) waves with propagation velocity and , respectively. Work [7], however, reports that the RUS is known to give √ more precise measurements of transverse (or shear) waves propagation speed, which means that theoretical transverse wave propagation speed should be accompanied with transverse wave propagation speed from experiment (and not with longitudinal waves!), which is, in essence, done in the experimental work. Dislocations are modeled as one-dimensional objects (“strings”, [8]) X(s, t), where s is a Lagrangean parameter that labels points along the line of length L, pinned at the ends, whose equilibrium position is a straight line, and t is time. They are characterized by a Burgers vector b, perpendicular to the equilibrium line. Their unforced motion is described by a conventional vibrating string equation: (1) where m - mass per unit length, - line tension, B is a phenomenological term that describes the internal losses of the string due to, for example, interactions with phonons and electrons. The wave propagation equation is: (2) where

is the tensor of elastic constants, and i, j, k, l = 1, 2, 3.

Both equations (1) and (2) acquire the right-hand side source—terms, whose structure has been discussed in detail in [9] and [10]. The string equation (1) is written for the component of motion along the glide direction, and, loaded by a Peach-Koehler force it becomes: ̈

̇

(3)

The wave propagation in equation (2) is best described not in terms of particle displacement u but in terms of particle velocity v = ∂u/∂t and the wave equation (2) becomes: (4) Thus, the elastic wave – dislocation interaction is described using equations (3) and (4). Without deep dive derivations, which can be obtained from [1] and [10], the dislocation density can be obtained through: (5)

5

However, it is not explicitly shown in the work, how to calculate the dislocation density directly from equation (5). I suggest that in order to make it more explicit, we to transform it to another form: −

(6)

5

In equation (6) the last term, , is a unitless quantity (n is a number of dislocations per unit volume). In order to calculate the dislocation density we impose the right-hand side of this term: (7) Where we can define

as a dislocation density.

Thus, the theoretical wave propagation speed ( ) is known, is measured by RUS, all other quantities are known (one can impose the dislocation line length), it is now possible to calculate the

dislocation density. The authors, however, offers only the results of measuring the wave propagations speed, without providing quantities for dislocation density. The results of the experiment are given in the table below: Table 1 Rectangular parallelepiped dimensions, mass, mass density, and increasing expected dislocation density.

for the five samples. Columns are ordered for

The dislocation density should be increasing along with increasing rate of deformation in Rolled samples because of work hardening effect [11]. The dislocation density should be decreasing along with increasing annealing time in annealed samples due to removal of crystal defects during the first stage of annealing. [12]. Note from (6), with increasing transverse wave propagation speed, the dislocation density should be decreasing, what we observe in the table. However, the study does not provide a quantified assessment of the result in terms of dislocation density. Let us now calculate from the given data the dislocation density and analyze the results. Table 2 Measured dislocation density for 5 different samples

−

Sample 1 3202 9285714,3 1E-8 2,6E+10 2,86E-10 2800 1,21473E+16

Sample 2 3192 9285714,3 1E-8 2,6E+10 2,86E-10 2800 1,21473E+16

Sample 3 3196 9285714,3 1E-8 2,6E+10 2,86E-10 2800 1,21473E+16

As it can be seen from results, is so large so even after 5 significant digits the variation is not observable. Generally, we can assume that, to some extent, dislocation density is constant for all 5 samples. The results compare well with the predictions of the theory and practice though, because the calculated makes sense for aluminum [13]. It does make sense now to compare the results with those, obtained via another independent instrument. The work [6] reveals the results of XRD analysis of the same 5 samples. An XRD pattern obtained from the aluminum samples is shown in Fig. 2. The characteristic (111), (200), (311) and (222) signals are observed, but the (220) reflection is absent. For all peaks present in the diffractograms, the larger the matrix deformation (from the annealed to the extreme laminated

Sample 4 3181 9285714,3 1E-8 2,6E+10 2,86E-10 2800 1,21473E+16

Sample 5 3170 9285714,3 1E-8 2,6E+10 2,86E-10 2800 1,21473E+16

Figure 2 XRD pattern for Sample 2. The characteristic (111), (200), (311) and (222) signals are observed, but the (220) reflection is absent due to texturing induced by the deformation of the sample. The inset shows a zoom of the (111) peak of the XRD patterns for the five samples.

conditions), the larger the peak width. The inset in Fig. 2 shows this effect for the Al (111) reflection. Current theory [14] accounts for the broadening of XRD peaks as arising from two effects: finite crystallite size and presence of dislocations. If the two effects are uncorrelated, they will act additively: ∆𝐾

∆𝐾 𝐷

∆𝐾

(8)

where ∆𝐾 is a full width at half maximum (FWHM) of the diffraction peak at wave vector K. ∆𝐾 is the contribution of the crystallite’s finite size, and ∆𝐾 𝐷 is the contribution of the dislocations. On dimensional grounds ∆𝐾 = 1 where D (dimensions length) is the crystallite size, and 1 is a dimensionless constant. This quantity is independent of the X-ray wavelength. After some derivations, authors offer the following equation: ∆𝐾

0.9 𝐷

𝑀

𝛬

̅

ℎ00

𝑞𝐻 𝐾

(9)

In equation (9) we take = 0.1 [15], [16]. Consequently, a plot of ∆𝐾 vs. 𝐾 will yield a straight line: the slope determines the dislocation density , and the intercept with the vertical determines the crystallite size D. The results are shown in Fig. 3 and Table 3. Table 3 Dislocation density K and crystallite size D, obtained by XRD.

First, the results obtained by me in RUS experiment differ from those obtained by XRD by one order. However, I believe, the results obtained via RUS and XRD are difficult to trust, because the difference in dislocation density between highly annealed sample and highly rolled sample is less than one order, but which, according to literature [17] should be in average of 4 orders. The annealed at 673K sample №1 had to pass all 3 stages of annealing, including recrystallization phase (recommended full annealing temperature for 1100 aluminum is 610K [18], Figure 3 ∆𝑲 𝟐 vs. 𝑲𝟐 𝑪 plot for the five different aluminum [19]), thus, it definitely should have way less samples, obtained through the modified Williamson–Hall dislocation density than original sample and method. all the more so than rolled samples. Also, we can think about the material choice. Aluminum is definitely much easier to deform and anneal, it is a ductile material, thus, it has to has larger dislocation density than, for instance, an average steel (which holds as we can see from results). The author does not explain the material choice however. The ultrasound found itself as a good method of nondestructive analysis in extreme cases also. For instance, work [20] reports about successful utilization of ultrasound waves to monitor changes of dislocation density in reactor pressure vessel (RPV) components of nuclear reactor due to irradiation. Premature shutdowns may be successfully prevented. I believe, there are still many open questions for the latter method. There is a chance that the dislocation structure can evolve in an ultrasonic field [21] so that the computed dislocation density is not representative anymore. It is still not clear how the ultrasound waves interact with other defects i.e.

impurities, vacancies, etc. Moreover, some works [26] report about temperature rising near the dislocations due to ultrasonic field. In this paper [26] the result of investigations of the ultrasonically stimulated phenomenon of the temperature rise around dislocations in semiconductor mercury–cadmium telluride (MCT) crystals is presented. The physical origin of the ultrasonic (US) effects on a crystal with extended defects is connected with interaction between acoustic waves and dislocations in the frame of the vibrating string model of Granato and Luecke [27]. Such interaction results in an effective transformation of the absorbed US energy into the internal vibration states of the crystal stimulating numerous defect reactions. Previous study of the ultrasonically induced transformation of the crystal defect structure and the modification of charge-carrier scattering conditions in MCT crystals has shown a sensitivity of this material to sonic vibrations [28]. Longitudinal US vibrations with frequency = 5– 7MHz and intensity 0.5W/ were generated by a LiNb transducer ( Y-cut). It was used an in situ pre-threshold intensity regime of US influence, i.e. switching off the ultrasound led to a relaxation of all sample parameters to their original values. During the experiment, several thermocouples were placed along the samples investigated and the nonuniform macroscopic temperature distribution in their surface during in situ US loading was detected. The value of the sonic-stimulated deviation of the temperature from the average value in the crystal was within ∆ 10–20K at US intensity 0.5W/ . The chemical selective etching of samples was performed and an inhomogeneous distribution of extended defects such as dislocations was observed using optical microscope. It was determined that the sonic-stimulated heating of imperfect regions takes place. It is thought that the sonicstimulated nonuniform heating of the crystal is connected with a selective absorption of the US energy at dislocations. In conformity with the Granato-Luecke model the dislocation moves in an ultrasonically loaded crystal as a vibrating string. Through the damping of the dislocation motion by electrons and phonons its kinetic energy is dissipated into the heat. A vast amount of acoustic energy is put into the material during period of the ultrasonic wave which was − s in the experiment. Since is less than the time of the heating relaxation, the heat may not have sufficient time to dissipate throughout the sample or to radiate into the environment during one period of the external influence. Thus, the temperature around dislocations may become considerably raised. The dislocation moving under the US action was considered as a linear thermal source. According to [30] the temperature distribution around the dislocation line L can be written as:

0

0

∫

∫

1

In the above expression, G(R, t) is Green’s function of a point thermal source, 1 is a dislocation line coordinate, 0 is the energy dissipated by a dislocation per unit time and length, ρ is a crystal density, C is a crystal specific heat, and 0 is an average equilibrium temperature in a crystal. It is assumed that the dislocation line is placed along the OY direction ( 1 = {0,y, 0}), is normal to plane xz and is the infinity. The temperature rises around the dislocation until an equilibrium temperature distribution is reached during a time which can be determined as { , 1 , }, where 1 is a characteristic time for thermal equilibrium attainment between individual dislocations, and is a characteristic time of the thermal equilibrium attainment between the sample and the environment. The initial conditions and the results of the experiment are the following: equilibrium crystal temperature is 0 = 78K, sonic-stimulated temperature increase is ∆ = 0.4W/cm2. 0 = 25K, intensity of the ultrasonic loading is Work [22] is another interesting approach to measure dislocation density in materials, the theory of which was discussed in more details in [23]. In essence, that method utilizes the indentation size effect (ISE) and can be a good competitor for RUS-based method. It is a well-known fact, that, after plastic deformation the dislocation density increases, to accommodate the strain gradients. If a deformation region is small and is under indenter pyramid or conic form, the so called geometrically necessary dislocations (GND) nucleate. The smaller the deformation scale the larger the density of GND. consequently, the hardness of the material, increase when the size of the deformed region decreases: this is known as the indentation size effect (ISE, “Smaller is stronger”). According to Nix and Gao [23], the ISE results from an increase in the density of the geometrically necessary dislocations required to accommodate the plastic deformation gradient around the indentation. The authors developed a model and deduced a simple expression to relate hardness with indentation depth:

𝐻 𝐻0

ℎ∗

1 ℎ

(10)

where H is the hardness for a given depth of indentation, h; 𝐻0 is the hardness in the limit of infinite depth and h* is a characteristic length that depends on the shape of the indenter, the shear modulus and 𝐻0 . Basically, after obtaining experimental hardness results (in present work a Ni film was used for experiments), one can get the Fig. 4. From which the ℎ∗ can be explicitly obtained (In the present work ℎ∗ 4 . m). Then, using Eq. (11): 1 𝑎 𝜗 𝑓3 ℎ∗

(11)

where is the density of dislocations statistically stored in the lattice (SSDs), θ the angle between the surface of the material and the surface of the indenter, b the Burgers vector of the dislocations and f a correction factor for the size of the plastic zone. In the present work θ=20°, f=1.9 [24], and b=0.25 nm [25]. 1 − Introducing these values in Eq. (11) a dislocation density of 4. is obtained, which correlates good enough with TEM-based experiment of dislocation density measurements of the same Ni film. I believe the model, suggested in [23] is reliable and the method proved to be as accurate as TEM based method, but without providing additional information, i.e. structural analysis. It can be used in the labs where the equipment doesn’t include TEM, providing fast and efficient dislocation density evaluation. However, the main disadvantage of that method is that it is destructive, i.e. the sample is not necessarily represents quite well the bulk material and additional errors may be caused during the sample preparation. At the same time, ultrasound probe is nondestructive but the model lying inside that method is weaker to my mind.

Figure 4 Regression curve of the experimental results fitted on the Eq. (10)

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