The 3D Kuramoto-Sivashinsky Equation for Nonequilibrium Defects Interacting through Self-Consisting Strain and Nanostructuring of Solids

Abstract

It is shown that the bulk defect-deformational (DD) nanostructuring of isotropic solids can be described by a closed three-dimensional (3D) nonlinear DD equation of the Kuramoto-Sivashinsry (KS) type for the nonequilibrium defect concentration, derived here in the framework of the nonlocal elasticity theory (NET). The solution to the linearized DDKS equation describes the threshold appearance of the periodic self-consistent strain modulation accompanied by the simultaneous formation of defect piles at extremes of the strain. The period and growth rate of DD nanostructure are determined. Based on the obtained results, a novel mechanism of nanostructuring of solids under the severe plastic deformation (SPD), stressing the role of defects generation and selforganization, described by the DDKS, is proposed. Theoretical dependencies of nanograin size on temperature and shear strain reproduce well corresponding critical dependencies obtained in experiments on nanostructuring of metals under the SPD, including the effect of saturation of nanofragmentation. The scaling parameter of the NET is estimated and shown to determine the limiting small grain size.

1. Introduction

In recent years, the spontaneous formation of periodic nanostructures upon the energy input into solids (laser or particle beams, severe plastic deformation, electrochemical etching, and others) has become the subject of intensive studies. An effective tool for the theoretical investigation of this phenomenon is the nonlinear Kuramoto-Sivashinsky (KS) equation [1, 2]. 2D KS equations with specific variables are used, for example, for the description of nanostructuring of solid surfaces using ion beams [3] and electrochemical etching [4]. Recently, a novel 2D KS equation has been derived for a defect-enriched solid layer formed at the surface by laser irradiation, and its numerical solutions have been used for the interpretation of the formation of surface nano- and microstructures under the action of laser pulses [5].

In [6], a model of the defect-deformational (DD) instability in the system of mobile defects (interstitials and vacancies), interacting in the bulk of an isotropic solid via quasistatic self-consistent strain, has been developed. The mechanism of the bulk DD instability consists in the following. An initial fluctuation of the defect concentration gives rise to self-consistent strain. This is accompanied by the strain-induced defect fluxes enhancing initial defect concentration fluctuation while the diffusion tends to smooth it out. At exceeding of a certain critical defect concentration, the positive feedback leads to the onset of DD instability with arising of nanometer scale bulk strain modulations and piling of defects at extrema of the strain. We note that, although the consideration of [6] was focused on point defects, starting DD equations possess a generality enabling description of the DD selforganization of any dilatation centers (including small dislocation loops) undergoing the diffusion. (The diffusion of nanometer size dislocation loops was observed in [7].)

In this work, we show that, invoking the nonlocal stress concept of the nonlocal elasticity theory (NET) [8, 9], one can, under some restrictions, reduce the equations of the bulk DD self-organization to a 3D nonlinear partial differential equation for the defect concentration of the KS equation type. In the mean field approximation (MFA), the obtained DDKS equation is reduced to the KS equation with critical behavior, which describes the threshold self-organization of bulk DD nanostructures. The period of self-organized DD structure is shown to be proportional to the scaling parameter of the NET.

In this work, we confine to investigation of linear solutions to the obtained nonlinear isotropic DDKS equation. The threshold defect concentration, the growth rate, and the period of the DD structure formed are determined.

Based on obtained results, a novel mechanism of nanostructuring under the SPD, stressing the role of defects generation and self-organization, described by the DDKS, is proposed. We show that solutions of the DDKS equation describe accurately experimental critical dependencies of the mean grain size on temperature and shear strain in metals, nanostructured by the severe plastic deformation. Comparing theoretical and experimental dependencies we regard the scaling parameter of the NET as a fitting parameter and, thus, estimate its value for considered cases. The effect of saturation of material nanofragmentation with increase of the shear strain finds description in the obtained dependencies, with the limiting small grain size being proportional to the scaling parameter of the NET.

2. Derivation of the Defect-Deformational Kuramoto-Sivashinsky Equation

Let us consider an elastic solid with nonequilibrium mobile defects (dilatation centers). The energy of defect-strain interaction in the framework of the NET is 𝐻_𝑑(𝐫)=−𝜃_𝑑𝜉+𝑙²_𝑑Δ𝜉, (1) where 𝐫=(𝑥,𝑦,𝑧), 𝜃_𝑑 is the deformation potential of a defect (𝑑 for a dislocation loop, 𝑑=𝑣 for a vacancy, 𝜃_𝑣<0; 𝑑=𝑖 for an interstitial, 𝜃_𝑖>0, and 𝜃_𝑖≫|𝜃_𝑣|, see Appendix), Δ=𝜕²/𝜕𝑥²+𝜕²/𝜕𝑦²+𝜕²/𝜕𝑧², 𝜉(𝐫)=∇𝐮 is the strain, 𝐮=𝐮(𝐫) is the medium displacement vector in an isotropic material, and 𝑙_𝑑 is the scaling parameter of the NET. A short derivation of the formula (1) is given in the Appendix.

Using (1), we obtain the elastic force acting on a defect: 𝐅(𝐫)=−∇𝐻(𝐫)=𝜃_𝑑∇𝜉+𝑙²_𝑑Δ𝜉, (2) where ∇=𝐞_𝐱(𝜕/𝜕𝑥)+𝐞_𝐲(𝜕/𝜕𝑦)+𝐞_𝐳(𝜕/𝜕𝑧), 𝐞_𝐱, 𝐞_𝐲, and 𝐞_𝐳 are unit vectors along the 𝑥, 𝑦, and 𝑧 axes, respectively.

The defect flux consists of the diffusion and strain-induced parts: 𝐣_𝑑=−𝐷_𝑑∇𝑁_𝑑+𝐷_𝑑𝑘_𝐵𝑇𝜃_𝑑𝑁_𝑑∇𝜉+𝑙²_𝑑Δ𝜉, (3) where 𝐷_𝑑 is the diffusion coefficient of a defect, 𝑘_𝐵 is the Boltzman constant, and 𝑁_𝑑 (cm⁻³) is the concentration of defects.

From the continuity equation, with allowance for 𝐣_𝑑, (3), and neglecting the renormalization of the defect diffusivity by the strain, we obtain the equation for 𝑁_𝑑: 𝜕𝑁_𝑑1𝜕𝑡=−𝜏𝑁_𝑑+𝐷_𝑑Δ𝑁_𝑑−𝐷_𝑑𝜃_𝑑𝑘_𝐵𝑇𝑁_𝑑Δ𝜉+∇𝑁_𝑑−𝐷⋅∇𝜉_𝑑𝜃_𝑑𝑙²_𝑑𝑘_𝐵𝑇𝑁_𝑑Δ²𝜉+∇𝑁_𝑑⋅∇(Δ𝜉)+𝐺_𝑑, (4) where 𝜏 is the defect lifetime and 𝐺_𝑑 is the rate of defect generation.

To close this equation, we need to express the strain 𝜉 through the concentration of defects 𝑁_𝑑. For this end, we use the equation for the displacement vector 𝐮=𝐮(𝐫) in an isotropic medium from [10]. Taking into account the energy of the strain-defect interaction, (1), where we neglect the term proportional to 𝑙²_𝑑, we modify the equation for 𝐮 to the following form: 𝜕²𝐮𝜕𝑡²=𝑐²_𝑡Δ𝐮+𝑐²_𝑙−𝑐²_𝑡∇()+𝜃∇𝐮^𝑑𝜌∇𝑁_𝑑, (5) where 𝜌 is the medium density and 𝑐_𝑙 and 𝑐_𝑡 are the longitudinal and transverse sound velocities, respectively.

Acting with the operator ∇ on both sides of the equation (5), we obtain the equation for the strain: 𝜕²𝜉𝜕𝑡²=𝑐²_𝑙𝜃Δ𝜉−^𝑑𝜌Δ𝑁_𝑑. (6) In the adiabatic approximation, (𝜕²𝜉/𝜕𝑡²=0), from (6), we obtain formula for the self-consistent quasistatic strain: 𝜃𝜉=^𝑑𝜌𝑐²_𝑙𝑁_𝑑. (7) Using (7) in (4), we obtain the following closed equation for defect concentration: 𝜕𝑁_𝑑1𝜕𝑡=−𝜏𝑁_𝑑+𝐷_𝑑𝑁1−^𝑑𝑁_dcΔ𝑁_𝑑−𝐷_𝑑𝑙²_𝑑𝑁_dc𝑁_𝑑Δ²𝑁_𝑑+∇𝑁_𝑑⋅∇Δ𝑁_𝑑−𝐷_𝑑∇𝑁_𝑑²𝑁_dc+𝐺_𝑑, (8) where the critical defect concentration is introduced: 𝑁_dc=𝜌𝑐²_𝑙𝑘_𝐵𝑇𝜃_𝑑². (9) For point defects, at 𝜌𝑐²_𝑙=10¹²erg∗cm⁻³, 𝑇=300K, 𝜃_𝑑=10²eV (see Appendix), we have the estimate 𝑁_𝑑𝑐∼10¹⁸cm⁻³.

The 3D nonlinear equation (8) is similar to the KS equation [1, 2] with the following essential differences. The expression in the brackets in the second term on the right-hand side of (8) depends on the dynamic variable 𝑁_𝑑 and can change sign with an increase in this variable, while in the KS equation this coefficient is a constant. The coefficient of the dispersion term (∼Δ²𝑁_𝑑) also depends on the dynamic variable and is not constant as in the KS equation. Besides, (8) contains an extra term proportional to (∇𝑁_𝑑•∇(Δ𝑁_𝑑)).

In the mean field approximation (MFA), the concentration of defects is represented as 𝑁_𝑑(𝐫,𝑡)=𝑁_𝑑0+𝑁_𝑑1(𝐫,𝑡), where 𝑁_𝑑0 is the spatially uniform concentration and 𝑁_𝑑1(𝐫,𝑡) is the spatially nonuniform component that spontaneously arises above the instability threshold. Under the condition 𝑁_𝑑1(𝐫,𝑡)<𝑁_𝑑0, we can neglect the relatively small term (∇𝑁_𝑑•∇(Δ𝑁_𝑑)) in the square brackets in (8). Then, introducing the parameter 𝜀=𝑁_𝑑0/𝑁_dc, we obtain from (8) the following equation: 𝜕𝑁_𝑑11𝜕𝑡=−𝜏𝑁_𝑑1−𝐷_𝑑(𝜀−1)Δ𝑁_𝑑1−𝐷_𝑑𝜀𝑙²_𝑑Δ²𝑁_𝑑1−𝐷_𝑑𝑁_𝑑𝑐∇𝑁_𝑑1². (10) The spatially uniform defect concentration 𝑁_𝑑0 can be found as a steady state solution of the spatially uniform equation (8): 𝑁_𝑑0=𝜏𝐺_𝑑. (11) The spatially uniform defect concentration 𝑁_𝑑0 (i.e., the parameter 𝜀) plays the role of a control parameter of the DD instability.

The DD equation (10) is similar to the stabilized KS equation, which takes into account the term proportional to 𝜏⁻¹. However, a significant difference lies in the fact that (10) contains an additional free parameter 𝜀, which characterizes the excess over DD instability threshold. Note the critical dependence on 𝜀 and the fact that (10) corresponds to the KS equation [1, 2] only when 𝜀>1.

3. The Linear Regime of the DD Instability

We employ the Fourier-series expansion of the defect concentration: 𝑁_𝑑(𝐫,𝑡)=_𝐪𝑁_𝑑(𝐪,𝑡)exp(𝑖𝐪𝐫). (12) Here, 𝑁_𝑑1(𝐪) is an initial fluctuation amplitudes.

According to (7), the Fourier-series expansion of the strain is 𝜉()=()=𝜃𝐫,𝑡div𝐮^𝑑𝜌𝑐²_𝑙𝐪𝑁_𝑑(𝐪,𝑡)exp(𝑖𝐪𝐫). (13) Each pair of Fourier amplitudes 𝑁_𝑑(𝐪,𝑡) and 𝜉(𝐪,𝑡)=(𝜃_𝑑/𝜌𝑐²_𝑙)𝑁_𝑑(𝐪,𝑡) describes the DD grating with the wave vector 𝐪.

For the analysis of the linear regime, we put 𝑁_𝑑1(𝐪,𝑡)=𝑁_𝑑1(𝐪)exp(𝜆_𝑞𝑡) in (12). The Fourier transform of the linearized DDKS equation (10) then yields the growth rate of the DD grating: 𝜆_𝑞=𝐷_𝑑𝑞²𝜀1−𝑙²_𝑑𝑞²−1−1𝜏. (14)

To graphically represent the dependence of the growth rate of DD Fourier amplitudes on the wavenumber, we introduce the dimensionless wavenumber 𝑄=𝑙_𝑑𝑞, the growth rate 𝜆(𝑄)=𝜆_𝑞/(𝐷_𝑑𝑙_𝑑⁻²), and the scaling parameter 𝑆=𝑙_𝑑/𝜏𝐷_𝑑. Then, formula (14) acquires the form 𝜆(𝑄)=𝑄²𝜀1−𝑄²𝑆−1−²𝑄². (15) This dependence is shown in Figure 1 at two values of the scaling parameter: 𝑆=0 and 𝑆=0.2 (the former corresponds to the case 𝜏→∞). It is seen that, when the condition of instability is met (see below), a whole band of DD modes with wavenumbers lying around the dominant (with the maximum growth rate) DD harmonic with 𝑄=𝑄_𝑚 (i.e.,𝑞=𝑞_𝑚) becomes unstable. Their amplitudes grow exponentially in time; this leads to the generation of a band of DD Fourier harmonics with 𝑞∼𝑞_𝑚.

Figure 1 

Dependence of the dimensionless growth rate of DD Fourier amplitudes, 𝜆(𝑄), on the dimensionless wavenumber 𝑄 calculated with formula (15) at 𝜀=2, 𝑆=0 (solid), and 𝑆=0.2 (dashed).

The growth rate 𝜆_𝑞 reaches a maximum value at 𝑞=𝑞_𝑚=1−𝜀^−11/2√2𝑙_𝑑. (16) The corresponding period of the dominant DD grating with the wave number 𝑞_𝑚 is given by 𝑑_𝑚=2𝜋𝑞_𝑚=2𝜋√2𝑙_𝑑1−𝜀^−11/2, (17) where the control parameter 𝜀=𝑁_𝑑0/𝑁_dc.

The period 𝑑_𝑚 is proportional to the scaling parameter of the NET 𝑙_𝑑, critically increases at 𝑁_𝑑0→𝑁_dc, and reaches a constant minimum level at 𝑁_𝑑0/𝑁_dc≫1. For the DD grating with 𝑞=𝑞_𝑚, the maximum growth rate is given by 𝜆_𝑚=𝐷_𝑑𝜀1−𝜀⁻¹²4𝑙²_𝑑−1𝜏. (18)

The growth rate becomes positive (𝜆_𝑚≥0), when the condition 𝜀≥𝜀_𝑐≡𝑙1+2^2𝑑𝜏𝐷_𝑑+𝑙1+2^2𝑑𝜏𝐷_𝑑²−1 (19) is fulfilled. On the other hand, formula (16) shows that a real value of 𝑞_𝑚 emerges at the condition 𝜀>1. Thus, the critical value of control parameter 𝜀_𝑐 is given by the equality in (19). Under the condition 2𝑙²_𝑑/𝜏𝐷_𝑑≪1, the critical value is 𝜀_𝑐≈1 and the periodic DD grating is created when the critical defect concentration is exceeded: 𝑁_𝑑0>𝑁_dc (𝜀 > 1).

At a constant concentration of defects (𝑁_𝑑0=const), the period depends on the temperature 𝑇 at which the DD instability takes place: 𝑑_𝑚=√8𝜋𝑙_𝑑𝑇_𝑐𝑇_𝑐−𝑇^1/2, (20)

where the critical temperature is 𝑇_𝑐=𝑁_𝑑0𝜃²_𝑑𝑘_𝐵𝜌𝑐²_𝑙. (21) For point defects, at 𝜌𝑐²_𝑙=10¹²erg∗cm⁻³, 𝑇=300K, 𝜃_𝑑=10²eV, 𝑁_𝑑0∼4∗10¹⁸cm⁻³, we have the estimate 𝑇_𝑐∼740K. Dependencies (17) and (20) are two main results of the linear theory of the DD instability. Comparison of these results with experiment is done in Section 4.

In the linear regime, besides the dominant DD grating with 𝑞=𝑞_𝑚 and the growth rate 𝜆_𝑚, a whole continuum of independent DD gratings with wave vectors 𝐪 and growth rates 𝜆_𝑞>0 grows with time and forms the superpositional DD structure (12), (13).

4. Comparison with Experimental Results

4.1. The Critical Temperature Dependence of the Grain Size in Metals and Semiconductors, Nanostructured by the SPD

Figure 2 shows the experimental temperature dependence of the grain size in titanium nanostructured by the severe plastic deformation from [11] in comparison with the critical theoretical dependence given by formula (20). The experimental curve is accurately reproduced at 𝑇_𝑐=450°C and the scaling parameter 𝑙_𝑑=11 nm (for the introduction of this parameter, see Appendix).

Figure 2 

Experimental temperature dependence of the grain size in titanium nanostructured by the severe plastic deformation [11] (stars). The curve is drawn with the use of formula (20) at 𝑙𝑑=11 nm, 𝑇𝑐=450°C.

It is interesting to note that, in semiconductors, nanostructured by severe plastic deformation, the mean grain size is of an order of magnitude less than that in metals (24 nm in Ge and 17 nm in Si [11]). Based on results of the present work one may suppose that the scaling parameter of the NET-𝑙_𝑑 in semiconductors is of order of 1 nm.

4.2. The Critical Shear Stress Dependence of the Grain Size in Metals and Semiconductors, Nanostructured by the SPD and the Saturation of Nanofragmentation

Figure 3 shows the experimental shear strain dependence of the grain size in nickel nanostructured by the SPD at room temperature [12]. To model this dependency, we used formula (17), where we set 𝜀=𝛽⋅𝛾, (22) where 𝛾 is shear strain in the material induced during the SPD and 𝛽 is the proportionality constant. We note that the linear dependence of the defect (vacancy) concentration generated by the SPD on the shear strain is confirmed experimentally (see, e.g., [13]).

Figure 3 

Experimental dependence of the grain size in nickel nanostructured by the SPD [12] (squares) on the shear strain 𝛾. The curve is drawn according to formula (17) at 𝑙𝑑=43 nm and 𝜀=0.435𝛾.

A good fit of the experimental dependence is obtained at 𝛽=0.435 and the value of the scaling parameter 𝑙_𝑑=43 nm (Figure 3).

It is seen that the theoretical curve describes two salient features of the experimental dependence: critical increase of the grain size with approaching the critical value of the shear strain (𝛾→𝛾_𝑐=2.3) and the saturation of the grain size decrease at 𝛾→∞.

One more correspondence between the theory and experiment is obtained if one compares the point defect concentration 𝑁_𝑑0, taking part in the DD instability, with the experimentally determined defect concentration. Since 𝜀=𝑁_𝑑0/𝑁_dc and 𝑁_dc∼10¹⁸cm⁻³ (see (9), from (22) one has the estimate of point defect (vacancy) concentration: 𝑁_𝑑0∼0.435𝛾∗10¹⁸cm⁻³∼1.7∗10¹⁸−1.3∗10¹⁹cm⁻³ within the interval 4≤𝛾≤30, shown in Figure 3. This estimate of vacancy concentration is in accordance with characteristic experimentally measured values for Cu and Ni achieved with SPD (see [13, 14]).

5. The Outline of a New DD Mechanism of Nanostructuring of Solids under the Severe Plastic Deformation and Conclusion

Based on results obtained in this work, we propose the following novel scenario of nanostructuring of solids upon the severe plastic deformation. Let us consider a polycrystalline solid with the initial grain size 𝐿. Consider one of such grains. The SPD process leads to the generation of the dense field of defects with a mean concentration 𝑁_𝑑0 in the bulk of this grain. If 𝑁_𝑑0≥𝑁_dc, then the threshold DD self-organization of defects occurs in the volume of the grain leading to the formation of the 3D DD structure with the period 𝑑, which is the function of 𝑁_𝑑0 (see (17)). The DD self-organization process is governed by 3D DDKS equation (10). Inspecting numerical solutions of similar 2D DDKS [5], one may expect the formation in the considered 3D isotropic solid of a 3D cellular hexagonal DD structure. After the DD self-organization, the initial large grain of size 𝐿 is fragmented into small grains of size 𝑑<𝐿. Boundaries of small grains are formed by piles of defects with the concentration 𝑁_𝑑1. At this stage, the defect field consists of two parts: defects with concentration 𝑁_𝑑0 in the bulk of the small grains and defects with concentration 𝑁_𝑑1 accumulated in the boundaries of small grains (𝑁_𝑑1<𝑁_𝑑0). The DD walls (small grain boundaries) are in a dynamic equilibrium with the defect concentration field 𝑁_𝑑0. An increase of 𝑁_𝑑0 leads, according to (17), to decrease of 𝑑 (see Figure 2). Thus, the continuation of the SPD process, accompanied by the shear strain 𝛾 (and defect concentration) increase, leads to further fragmentation of grains occurring via grain boundaries rearrangement. From Figure 2 it is seen that, at large 𝛾, the saturation of fragmentation occurs at the minimal grain size 𝑑_min=2𝜋√2𝑙_𝑑. This picture of successive fragmentation of an initial large grain into small ones is observed in experiment together with the nanofragmentation saturation effect (see [12]).

When 𝑁_𝑑1>𝑁_dc1, where 𝑁_dc1 is the threshold defect concentration of dislocation nucleation, the formation of dislocation loops begins at DD grain boundaries. At this stage of the DD self-organization, grain boundaries decouple from the bulk, spatially uniform defect concentration field 𝑁_𝑑0, and the structure of nanofragments with a small size 𝑑, achieved at this point, is written permanently and irreversibly inside the initial large grain of size 𝐿. The grain boundaries in this structure consist of dislocation pileups. If 𝑑<𝑑_min, we return to the initial point of our consideration with a grain of size 𝑑 playing the role of starting large grain 𝐿. If the process of the SPD is resumed, it will lead to further fragmentation of the grain with size 𝑑 until the saturation of fragmentation occurs.

In conclusion, the novel DDKS equation for nonequilibrium defect concentration is derived, and its linear solutions are shown to reproduce salient experimental dependencies describing the nanofragmentation of solids during the SPD. The new DD mechanism of nanofragmentation based on the notion of nonequilibrium defects generation and self-organization during the SPD is proposed.

Appendix

The Derivation of Energy of Defect-Strain Interaction in the Framework of the Nonlocal Elasticity Theory

We define the energy of the defect-strain interaction as a work done under creation of a defect (dilatation center) in the stressed material. In an isotropic solid, this work is 𝐻_𝑑(𝐫)=−𝜎(𝐫)Δ𝑉_𝑑, (A.1) where 𝜎(𝐫) is the nonlocal stress at point 𝐫 and Δ𝑉_𝑑 is the change of the sample volume under creation of the defect in its bulk (Δ𝑉_𝑣<0 for vacancies, Δ𝑉_𝑖>0 for interstitials, and Δ𝑉_𝑖≫|Δ𝑉_𝑣|).

The equation for the nonlocal stress in the considered case is 1−𝑙²_𝑑Δ𝜎=𝐾𝜉, (A.2) where 𝐾 is the bulk elasticity modulus. The (A.2) follows from (4) of [15] in the case of isotropic solid.

Solving (A.2) with the use of perturbation theory, we obtain the nonlocal stress: 𝜎(𝐫)=𝐾𝜉(𝐫)+𝑙²_𝑑𝐾Δ𝜉(𝐫). (A.3) Substituting (A.3) in (A.1), we obtain formula for defect-strain interaction: 𝐻_𝑑(𝐫)=−𝜎(𝐫)Δ𝑉_𝑑𝜎(𝐫)=−𝜃_𝑑𝜉(𝐫)+𝑙²_𝑑Δ𝜉(𝐫), (A.4) where 𝜃_𝑑=𝐾Δ𝑉_𝑑 is the deformation potential for a defect. For point defects: 𝜃_𝑣<0, 𝜃_𝑖>0 and 𝜃_𝑖≫|𝜃_𝑣|. At 𝐾=10¹²erg∗cm⁻³, |Δ𝑉_𝑑|=𝑎³∼10⁻²²cm³ (𝑎 is the crystal lattice parameter), we have the estimate 𝜃_𝑑∼10²eV (𝑑=𝑣,𝑖).

The case of small (nanometer size) dislocation loops as strongly anisotropic dilatation centers needs separate consideration and is not addressed here.