Crystal lattice showing atoms and lattice planes with cubic symmetry.

Schematic diagram (lattice planes) showing an edge dislocation. Burgers vector in black, dislocation line in blue.

Plastic deformation in solids is a term used in metallurgy, materials science and solid state physics, and refers to an irreversible change in the internal molecular structure (or microstructure) of an object. This change may be due to either (or both) of the following factors:

• The action of an applied force (or work)
• A change in temperature (or heat).

In the former case, the applied force may be tensile (pulling) force, compressive (pushing) force, shear, bending or torsion (twisting) forces.

In the latter case, the most significant factor which is determined by the temperature is the mobility of the structural defects such as grain boundaries, point vacancies, line and screw dislocations, stacking faults and twins in both crystalline and non-crystalline solids. The movement or displacement of such mobile defects is thermally activated, and thus limited by the rate of atomic diffusion.

The change of an object's shape is called strain. The force causing it is called stress. Stress does not necessarily cause permanent change. As deformation occurs, internal forces oppose the applied force. If the applied stress is not too large these opposing forces may completely resist the applied force, allowing the object to assume a new equilibrium state and to return to its original shape when the force is removed. This is what is known in the literature as elastic deformation (or elasticity). Elastic deformation can be described by Hooke's Law for restoring forces, where the stress is linearly proportional to the strain. Larger forces in excess of the elastic limit may cause a permanent (irreversible) deformation of the object. This is what is known in the literature as plastic deformation in solids or plasticity.

Viscous flow near grain boundaries, for example, can give rise to internal slip, creep, fatigue in metals. It can also contribute to significant changes in the microstructure like grain growth and localized densification due to the elimination of intergranular porosity. Screw dislocations may slip in the direction of any lattice plane containing the dislocation, while the principal driving force for "dislocation climb" is the movement or diffusion of vacancies through a crystal lattice. One major difference between dislocation slip and climb is the temperature dependence. Climb occurs much more rapidly at high temperatures than low temperatures due to an increase in the rate of thermally activated diffusion of point vacancies. Slip, on the other hand, has only a small dependence on temperature.

Transmission electron micrograph of dislocations. The formation of the precipitate is due in part to the action of the dislocations in assisting atomic diffusion.

[edit] Introduction

The mechanical properties of the object help determine the type of deformation it will undergo. These include the breaking strength (or bond rupture strength) of the individual bonds (covalent, ionic, metallic, etc.) between the atoms and molecules composing the object. Less obvious factors are the object's capacity for internal friction and entropy production which both enhance an object's ability to resist plastic deformation and respond to a mechanical disturbance via reversible elastic deformation. Conversely a high capacity for stress relaxation will make an object more subject to irreversible plastic deformation and microscopic flow.

Dislocations are generally surrounded by relatively strained (and weaker) bonds than the primary bonds within the ordered crystal lattice. This explains why these bonds are the first to break under an applied load. The crystals thus lower their free energy through the re-formation of bonds within the crystalline structure. Thus interaction of dislocations with one another (and the atoms of the crystal) results in an energetically favorable energy conformation. Moreover, dislocations are a "negative-entity" in that they do not physically exist. They are merely aberrations in the ordered lattice—which does exist. As such, the detectable motion of the solid itself can be described as a change in the bonding pattern of largely stationary atoms.

Furthermore, the strained bonds around a dislocation are characterized by lattice strain fields. For example, there are bonds undergoing compressive strain which are adjacent to an edge dislocation, and bonds undergoing tensile strain bonds beyond the tip of an edge dislocation. These form compressive strain fields and tensile strain fields, respectively. These strain fields are also subject to the laws of attraction and repulsion. The visible (macroscopic) results of plastic deformation are the result of microscopic dislocation motion.

Factors such as these can be probed using acousto-optic techniques such as advanced forms of laser light (and sound) spectroscopy. For example, dynamic light scattering (or photon correlation spectroscopy, PCS) can reveal a great deal of information regarding the behavior of on object under stress. Such techniques are used to probe the concentration of microstructural defects, which often determine the degree of long-range order (as in crystalline solids) or short-range order (as in non-crystalline solids). Such methods are used to investigate and characterize the dynamic behavior over a range of length and time scales, from atomic or molecular vibrations to long-range atomic displacement (or density) fluctuations which are responsible for heat transfer (phonons) in solids at low temperatures. In this context, the observation of central phonon peaks and heterophase fluctuations provide details regarding structural changes which occur near the point of first-order phase transitions, phase transformations in solids, and/or a glass transition. By studying such properties and phenomena, material scientists hope to better understand the underlying processes involved when an object undergoes any type of irreversible microstructural change under the application of external forces or alterations in temperature.

[edit] Internal friction near defects

SEM micrograph of surface of colloidal solid. Structure and morphology consists of ordered domains with both interdomain and intradomain lattice defects.(Amorphous colloidal silica particles of average particle diameter 600 nm).

Highlighted image of surface of colloidal solid. Emphasis on microstructural defects to illustrate the defect/domain morphology typical of a simple one-component glass.

Originally considered by Clarence Zener in his classic studies of plasticity in metals, internal friction was defined as the capacity of a solid to convert its mechanical energy of vibration into internal energy of various forms. This unavoidable degradation of ordered energy into disordered energy prevents the amplitude of a solid's vibrations from becoming infinite when the frequency of an applied force approaches a natural frequency of the specimen (the resonant condition). It can be measured by plotting the amplitude of vibration against the frequency of the exciting force near a resonant frequency. ^{[1] [2]}

The amplitude can become large when the driving frequency ω is near the natural atomic vibrational frequency, or resonant frequency ω = ω_o . When the damping is small, the increase in amplitude near ω = ω_o  is very large. In an ideal frictionless system, resonance occurs at ω = ω_o  and the resonant peak becomes infinite. In such a case, energy is being continuously transferred into the system and none is dissipated. For real dissipative systems, however, the presence of friction limits the amplitude of the resonant peak to a finite value.

Small amplitudes are generally used for the investigation of elastic phenomena, while internal friction of large amplitude is primarily a measure of the capacity of a solid to undergo irreversible plastic flow and deformation under large stresses. Zener proposed that internal friction depends in general on thermal currents which occur owing both to: 1) the inhomogeneous strains set up by the vibrations in a solid, and to 2) the variations in elastic constants from point to point in a polycrystalline solid. ^{[3] [4]}

The mechanism of the absorption of sound in solids which is responsible for the damping of elastic lattice waves was considered by Akheiser, who regarded the absorption as arising partly from heat flow and partly from viscous damping. Akheiser's calculation of the viscous damping contribution to sound absorption is based on the notion that the sound wave modulates the elastic properties and hence the thermal phonon frequencies of the medium through which it propagates. The modulated phonons relax towards local thermal equilibrium via phonon-phonon collisions caused by an anharmonic interaction. This relaxation is dissipative (or entropy producing) and thus attenuates the sound wave driving the process. ^[5]

It has since become possible to predict the acoustic loss of non-crystalline solids from the known thermal and elastic properties. Results indicate that infrared optical vibrational modes can contribute to such viscous phenomena, which is not surprising in light of the conclusions of Slack that optic phonons can transport heat in crystalline solids if the acoustic-optic energy gap is small enough, and if the optic phonon velocity Vg (Opt.) is large. ^{[6] [7] [8] [9]} In addition, mechanisms of attenuation of high-frequency shear and longitudinal waves have been considered in viscous liquids, polymers and glasses, with subsequent work leading to a fresh interpretation of the glass transition in viscous liquids in terms of a spectrum of relaxation phenomena occurring over a range of time and length scales. ^{[10] [11]}

[edit] Stress relaxation

The recognition of relaxation as a cooperative phenomenon resulting in structural rearrangement within glass-forming liquids resulted in the formulation of a molecular-kinetic theory to explain the temperature dependence of this process. Such network contributions to the anelasticity of glasses have been measured using internal friction techniques. ^{[12] [13]}

In developing a stress relaxation function for the interpretation of such data, Majumdar assumed that the stress field can be decomposed into elementary modes of stress relaxation of a range of wavelengths, λ. The smallest wavelength is of the order of the dimensions of the tetrahedral unit in oxides (or the domain size in elemental melts) whereas the largest wavelength is of the order of the sample size—or determined by the domain size, free of mechanical defects, etc. The wavelength λ scales with the degree of atomic or molecular order. ^[14]

On a short time scale, a glass should respond as a rigid disordered solid. For a relaxation mode of wavelength λ, the linear dimension of material sustaining such a mode must be of the order of l. If l is much larger than the short-range order parameter d, then the mode involves motion of parts of the glass only incoherently linked. Such modes will be subject to viscous friction and may be strongly damped. The probability of excitation of a relaxation mode of wavelength λ is reduced to the exponential function [exp (-l/d)] which should therefore contain information on the microstructure at various length scales. A large body of data has accumulated relating to the relaxation behavior of viscous glass-forming liquids, obtained largely by the use of acoustic resonance and ultrasonic propagation studies. It should be noted that the highest frequency available in such an experiment is on the order of 10⁹  Hz, so that this technique is useful only for the study of phenomena on time scales longer than 10⁻¹⁰ seconds. ^[15]

Critical insight into the nature of the heat flow associated with thermal relaxation may be gained by consideration of Frenkel's observation on thermal conductivity that the quantity of heat flowing to a given volume element is used partly to heat it and partly for performing external work. It should be noted that in the Debye theory of the heat motion in condensed matter, the elastic vibrations describing this motion are treated without any a priori reference to the temperature, the latter being introduced merely as a measure of the average intensity of these vibrations. Such a treatment implies a strict validity of the principle of superposition of normal longitudinal and transverse vibrational modes; that is a neglect of their deviations from a linear law of force (anharmonicity). ^{[16] [17]} Owing to these deviations, the energy of a vibrational mode is transferred to a number of other modes. Nonlinear effects become irrelevant at high frequencies or short wavelengths, where the vibrations are propagated in a practically "isothermal" way; i.e., no energy exchange with vibrations of still higher frequency. ^[18]

[edit] Entropy production

The preceding considerations show that Debye's "acoustic" theory of the heat motion gives an incomplete picture of reality. The longitudinal fluctuations in density (or pressure) and the transverse fluctuations in shearing stresses are not treated as the effects of the heat motion, but as its mechanical constituents. Temperature variations associated with such fluctuations cannot be fitted into the frame of Debye's theory as long as it aims at a purely mechanical description of the heat motion. An accurate description of this kind requires the introduction of anharmonicity effects due to the nonlinearity of the equations of motion.

Thus, processes which result in entropy production must be treated by the introduction of thermal elements into the picture of heat motion. The motion should be treated as a superposition of 1) mechanical (elastic) fluctuations in the form of acoustic waves, and 2) thermal fluctuations, associated with local variations of the temperature T or the specific entropy S, the latter of which cannot be propagated in the form of waves. This introduces the possibility of representing ΔT or ΔS in the form of a Fourier series of standing waves. If the density of the body ρ is considered as a function of the (local) pressure fluctuation p + dp and of local temperature fluctuation T + dT (or specific entropy fluctuation S + dS), then structural fluctuations of the density ρ can be represented as

or

The first (mechanical) part is propagated in the form of progressive acoustic waves vibrating harmonically, while the second (thermal) is non-vibrating, displaying a relatively slow and irregular variation with time.

[edit] Brillouin scattering

Thus, as a result of thermal agitation, fluctuations occur in both density and orientation. Both kinds of fluctuations cause an optical inhomogeneity of the medium, which result in light scattering. The optical inhomogeneity is characterized by the fact that the dielectric constant in the volume of the fluctuation differs from the mean value of n²  where n is the index of refraction. The index of refraction, however, is a function of the density alone. The scattering is therefore a direct result of density fluctuations, and the variation of density is due both to a variation of the pressure and of the temperature or entropy. Thus, the light scattered in a given direction must consist of two parts: a "mechanical" one and a "thermal" one.

Light scattering makes possible the study of molecular processes from time intervals as short as 10⁻¹¹  seconds. This is equivalent to extending the available frequency range from 10⁹  Hz to greater than 10¹⁰  Hz. The scattering, according to Brillouin, is caused by the diffraction of the incident plane monochromatic light waves by spontaneous, sinusoidal density fluctuations, that is, by standing thermal sound waves or acoustic phonons. The light wave is considered to be scattered by the density maximum or amplitude of the acoustic phonon, in the same way that X-rays are scattered by the crystal planes in a solid, the role of the crystal planes being played by the planes of the sound waves. ^[19]

The increased frequency range associated with light scattering is a critical one for the study of thermal relaxation since estimates indicate that many liquids have a vibrational relaxation time of the order of 10⁻¹¹  seconds. Brillouin scattering by thermally driven density fluctuations has been used to measure structural relaxation and viscoelasticity in liquids as well as compressibilities, phase separation and vitrification in glasses. ^[20] In addition, the introduction of digital correlation spectroscopy has made possible the measurement of the time dependence of spatial correlations in liquids and glasses in the relaxation time gap between 10⁻⁶ and 10² seconds. ^{[21] [22]}

[edit] Doppler shifts

Thus far, we have not considered the frequency distribution of the spectrum of light scattered from a viscoelastic body. Frequency shifts of incident light rays occur as a result of their interaction with normal vibrational modes. Thus, a standing acoustic wave may be decomposed into two oppositely directed waves of equal intensity and frequency. Density maxima will therefore be traveling with the velocity of sound, and the frequency of the light wave will be Doppler shifted, according to

ν = ± (ν - dν)

yielding two infinitely sharp lines in the resulting spectrum at (ν + dν) and (ν - dν). This frequency shift, first introduced by Brillouin, is commonly referred to as the Brillouin doublet and is observed in the spectrum of light scattered from all liquids. This doublet arises from the interaction of the light rays with the "mechanical" component of the thermal motion. Furthermore, the spectral width of the Brillouin doublets gives the lifetime of the sound wave, and thus the sound absorption coefficient. ^[23]

Alternatively, the light scattered at constant pressure (the "thermal" component) is not shifted in frequency but is instead broadened somewhat as a result of the dissipative processes which damp out these fluctuations. ^{[24] [25]} This feature of the spectrum of light scattered quasi-elastically from thermal fluctuations in a liquid was first observed by Gross and subsequently by Benedek, et al. at MIT with their "self-beat" method of spectrometry near the critical point. ^{[26] [27]} Debye gave the theory for the width of the central component in binary liquid mixtures, and Alpert, Yeh and Lipworth gave the first measurements of the width of this central component in a liquid mixture near the critical mixing temperature. ^{[28] [29]} Landau and Plazcek suggested that the width of the central line in the spectrum, which is broadened due to thermal dissipative processes, is determined by the size and lifetime of the density fluctuation which is responsible for the line. ^[30]

[edit] Critical phenomena

Until the study of strontium titanate, pre-transition phenomena associated with second-order structural phase transitions appeared satisfactorily described by the "soft-mode" concept (see Phase transformations in solids). In this description, a force constant for a lattice wave decreases with temperature, with the frequency of the normal vibrational mode approaching zero at the critical point. High-resolution scattering, however, has shown that pre-transition phenomena in solids is more complex. A central phonon peak occurs between the Brillouin doublet and diverges as the temperature approaches a transition point, T_c. A variety of mechanisms have now been proposed for this result. ^[31]

One emerging view of the central peak phenomenon is that it is associated with scattering by domains of the low temperature phase which appear as fluctuations above T_c having a relatively long lifetime t compared to the inverse vibrational frequency. These domains are not unlike the heterophase fluctuations proposed by Yakov Frenkel. The width of the central phonon peak resulting from such fluctuations should be proportional to the time required for the formation and collapse of such thermally excited embryos.

Such fluctuations are always present, but at temperatures far above (or pressures far below) the critical point, they are short lived and small in amplitude, usually involving only a few atoms or molecules. In addition, for a discussion of the angular distribution of scattered light it is not enough to know what the amplitudes of the density fluctuations are. We must also know how much of a spatial correlation exists between the fluctuations in two points a distance apart. This correlation is measured by a correlation function C(r) which represents the average product of the fluctuations in two points a distance r apart, divided by the average square of the fluctuation. The correlation function can then be used to define a correlation length L.

In approaching a transition point, not only the amplitude of the fluctuations, but also their correlation length increases, which means that the distance over which a certain fluctuation is maintained gets larger and larger, the nearer the temperature comes to the critical point. Moreover, the processes by which the fluctuations "relax", or fade into the background fluid, slow down. At the critical point itself, the correlation length which defines the size of the fluctuation becomes infinite and the rate of relaxation becomes infinitely slow. This "critical slowing down", yielding long-lived, long wavelength correlations yields the light scattering phenomenon of critical opalescence, which frequently is apparent by a "milky" appearance.

One such process is the solidification of a supercooled liquid, the rate of which is limited by the rejection of the heat of solidification. Quasi-elastic light scattering experiments at the ice-water interface by Bilgram, et al. indicate that heterophase fluctuations at the solid-liquid interface may be responsible for this thermal transport. In these experiments, scattering in excess of the bulk liquid was only observed if a critical growth velocity V_t = 1.5 mm/sec was exceeded. The intensity of the Rayleigh component is found to be dependent on this growth rate. It is suggested that the width of the Rayleigh component is a function of the amplitude of the fluctuations and depends upon the rate of decay of such fluctuations.

The relationship between the growth and decay of the fluctuations responsible for the scattering and the motion of the interface is addressed in a subsequent work, where Cahn's original interpretation of interfacial structure as a function of driving force is re-explored. Below the critical velocity V_c, anisotropic light scattering yields a periodically oscillating correlation function which does not decay. Above V_c however, a transition occurs in which the oscillations in the correlation function become damped and disappear finally while an exponentially decaying one emerges. At the same time, the scattering of the light becomes more and more isotropic, a condition referred to as diffusive scattering. It should be noted that the process which leads to the formation of the non-faceted diffuse layer is very poorly understood. The explanation of such a non-equilibrium structure would be difficult to rationalize in terms of screw dislocations or any other equilibrium defect structures which are stable in the crystalline state. ^[32]

[edit] Non-crystalline solids

While there is clearly a thermodynamic driving force toward crystallization, there is very little evidence for the physical mechanism(s) responsible for the tendencies of most glasses towards long-range order formation or crystallization -- not even on most geological time scales. Many glasses (e.g. traditional silicate glasses more than a few degrees below their glass transition temperature) in fact show no evidence of crystallization whatsoever! Five degrees below T_g it takes over a year at high stress to see any evidence of structural or stress relaxation in the glass, let alone the amount of rearrangement necessary for actual crystallization to occur. This is at energies near that required for the liquid state to form. At hundreds of degrees below this temperature, the rate of atomic rearrangement towards the thermodynamically stable crystalline state would be so low as to question at the age of the universe, let alone geological time scales.

Most arguments to the contrary are largely academic—yet highly respected within the majority of the materials science community. Some empirical "proof" in the literature is that relating to recent work on quasi-elastic light scattering in glasses (referenced above). The consensus of these works is a clear indication of dynamic non-equilibrium behavior of non-crystalline silica (the most basic glass former known) at the molecular level. This would indicate some degree of irreversible (plastic) deformation on the smallest length scales. The net result of this over time would most likely be irreversible (plastic) deformation on continuously larger spatial scales.

For example, a rheid is a solid that deforms slowly by microscopic viscous flow. Almost any type of rock can behave as a rheid under appropriate conditions of temperature and pressure. The outer crust of the Earth, which is composed largely of silica and its compounds, is believed to undergo convection over long time scales. Since the mantle is believed to be solid (i.e. it supports the propagation of shear waves) it must be behaving as a rheid. Microcrystalline granite has a measured viscosity at standard temperature and pressure of ~ 4.5 • 10¹⁹ Pa·s, so it should also be considered a rheid.

It should further be emphasized that plastic deformation in crystalline solids (e.g. most metals) is highly localized due to the extremely finite and localized distribution of point and line lattice defects, and their limited mobility at most experimental temperatures. Amorphous solids, on the other hand, consist largely of defects (by definition). Thus, such irreversible deformation is much more likely to occur on a massive scale throughout the entire microstructure, and potentially on a much larger length scale.

This type of massive structural rearrangement resulting from the continuous interaction of compositional or density fluctuations (remnants of the initial liquid state) is what constitutes the basic mechanisms of the glass transition at the molecular level. Furthermore, observation of the inelastic scattering of density fluctuations at internal surfaces (or "defects") constitutes one of the more recent microscopic approaches to the measurement of this phenomenon in the materials science laboratory.

Thus, as metallurgists or material scientists, we investigate the mechanisms of plastic deformation (aka structural "relaxation" or internal friction) using techniques such as dynamic light scattering (aka PCS: photon correlation spectroscopy or quasi-elastic light scattering). Using PCS, scientists investigate such contributing factors as central phonon peaks (relevant primarily near critical points in more traditional phase transitions), entropy production (typical of any "plastic" or irreversible phenomena), and thermally arrested density fluctuations (or "heterophase" fluctuations, as originally described in detail by Frenkel in his classical text on the theory of liquids and the vitrification process).

It is through this type of experimental work that we will be begin to shed additional light on the highly controversial and much debated physical phenomenon known commonly as the glass transition. To dismiss this work is to avoid the reality of highly sophisticated experimental results continuing to be published around the world by our most highly educated and inquisitive minds, using what we commonly refer to in the scientific community as the "scientific method".

[edit] See also

[edit] References

[edit] Further reading