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Screened disclinations in solids
Materials Science and Engineering, A164 (1993) 58-68
58
Screened disclinations in solids Alexey E. Romanov* Max-Planck-lnstitut fiir Metallforschung, Institut fiir Physik, Heisenbergstrasse 1, 7000 Stuttgart 80 (Germany)
Abstract The properties of screened disclination configurations (loops, dipoles, defects in small particles, etc.) with relatively small energies are considered. The methods and results of calculations of elastic stresses and energies for screened disclinations are discussed. The following techniques of calculations are mentioned: linear and non-linear elasticity and computer simulation. The exact solutions recently obtained for disclinations in thin films and elastic spheres are briefly reported. On the basis of the properties of screened disclinations, a series of models for the processes of plastic deformation are proposed. The bands with misoriented crystal lattices in metals and other materials are described as a result of partial wedge disclination dipole motion. Other disclination models are applied to the description of work hardening at large strains and to the theory of grain boundaries in polycrystals. Disclination models for the structure and properties of amorphous solids and nanoparticles are also discussed.
1. Introduction At present, the concept of disclinations is recognized to be very helpful in condensed matter physics. It has been used for liquid crystals [1], magnetics [2], superconductors [3], superfluid helium [4], etc. Disclinations were initially introduced into the theory of solids together with dislocations as linear imperfections of a deformed body [5]. A unit disclination is a special and very strong source of internal elastic stress and possesses an energy which is very large in comparison with the energy of a lattice dislocation. For a long time, this feature of disclinations was considered to be an obstacle for their application in modelling the plasticity of solids [6]. However, in the case of screened disclination configurations (dipoles, loops, etc.), the energy may be relatively small and disclinations can exist in crystalline and amorphous materials. The present work deals with the properties of screened disclinations in solids. The geometry of disclinations, and their elastic fields and energies are considered for different types of screening. The applications of disclination models in the physics of strength and plasticity are also discussed.
the undeformed faces of the cut undergoing the displacement produced by mutual rotation around a fixed axis [5, 7, 8]. The elasticity of a body containing a disclination does not depend on the position and the shape of the surface of the cut, but is determined by the value, direction and the position of the axial vector of rotation to (Frank vector of the disclination), the shape of the defect line and the boundary conditions. One can investigate the geometry of isolated defects [8], e.g. individual disclinations, dipoles, multipoles, loops (Fig. 2), but special attention should be devoted to rectangular disclination loops (RDLs) (Fig. 2(b)) [9]. The significance of RDLs is clear, because they can be easily transformed into other disclination defects, e.g. an angular disclination, a disclination dipole, a Ldshaped defect or a linear disclination. In general, any disclination defect can be built up with given accuracy as a set of rectangular loops with a common rotation vector. This fact, together with the possibility of the extension of the shape of the loop defect, make it possible to regard the RDL as a fundamental structural
2. Some simple geometrical properties of disclinations The disclination is a linear defect (Fig. 1) which bounds the surface of a cut in a continuous body, with *Permanent address: Ioffe Physico-Technical Institute, Polytechnicheskaya 26, 194021 St. Petersburg, Russian Federation. 0921-5093/93/$6,00
(a) (b) (c) Fig. 1. Disclinations (Volterra rotational dislocations) in a cylinder: (a), (b) twist disclinations of Frank vector w; (c) wedge disclination. © 1993 - Elsevier Sequoia. All fights reserved
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Screened disclinations in sofids
unit when describing the geometry and elastic fields of complicated disclination networks. The presence in crystals of axes of discrete rotational symmetry leads to the existence of perfect and partial disclinations [8]. Partial disclinations are always related to plane defects (boundaries of misorientation) and have no lower limits to their power (modulus of the Frank vector). However, perfect disclinations do not restrict the surface of their plane defect and the symmetry axes are described by discrete values of the Frank vector to. Depending on the physical properties of the condensed media under consideration, the formation of disclinations of two types is possible: Frank disclinations, which are singularities of the field of spin (free) rotations, and Volterra disclinations, which are singularities of the field of orbital rotations connected with the non-uniform displacement field. Frank disclinations are peculiar to magnetics, liquid and "plastic" crystals [2]. In some cases, they have no upper limits to their power. They do not have any connection with plastic flow and do not have a translation dislocation analogue. However, in cholesterics and other mesophases with discrete translational symmetries, translational dislocations may be involved in a disclination. There exists a deep interrelationship between Volterra disclinations (rotational dislocations) and Volterra dislocations (translational dislocations). Any disclination can be represented in the form of a superposition of dislocations. Simultaneously, any dislocation defect admits an equivalent representation through a set of disclinations. As a characteristic example, a specific disclination dipole (Fig. 3(a)) may be considered. We call this configuration a single-line, two-rotation-axes disclination dipole [8]. In such a defect, the disclination lines coincide and the Frank vectors to and - t o have a mutual displacement by a vector t. As a result, the single-line dipole appears to be a natural equivalent of a dislocation of Burgers vector b = t Ato. The other important disclination dipole configurations are a two-line, two-rotation-axes dipole of wedge disclinations (Fig. 3(b)), which is similar to the terminated edge dislocation wall, and a two-line, one-rotational-axis dipole (Fig. 3(c)), which demonstrates the properties of a dislocation quadrupole. The presence of different types of disclination dipole is possibly due to the existence of three sets of parameters which are characteristic of a disclination defect: the value and the direction of the Frank vector to, the position r,~ of the Frank vector and the position of the disclination line. For disclination dipoles, we can have r_~, ¢ r.~ or r_,~ = r+,o. In the first case, one treats the dipole as a two-rotation-axes configuration and the lines of opposite-sign disclinations may or may not coincide. In the second case, it is a one-rotation-axis
/
/ - - - ed
(a)
/
~'
(b)
Fig. 2. Disclination loops: (a) circular pure twist disclination loop; (b) rectangular pure wedge disclination loop. For pure defects, the rotation vector is in the geometric center of the loop.
Je
(a)
\
~
t
Y/~./
33/ _L ..L 3_ -!=
.L 3_ 2_ 2_ 2_ 3_ (b)
(c)
Fig. 3. Disclination dipoles and their dislocation analogues: (a) single-line, two-rotation-axes dipole of twist disclinations; (b) two-lines, two-axes wedge disclination dipole; (c) one-axis dipole.
configuration, which is the limited case of an extremely elongated disclination loop.
3. Disclinations and structure levels of plastic flow
There are four main structure levels which are dealt with in the physics of plasticity [8, 10]: a microscopic level--the scale of the lattice constant; a mesoscopic level--the scale of dislocation substructures;
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A. E. Romanov
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Screened disclinations in solids
a structural level--the scale of grains in polycrystals; a macroscopic level--the scale of a sample size (Fig. 4). Each level of structure requires the employment of special physical models and mathematical languages in compliance with its deformational characteristics. The mesoscopic level is conditioned by collective effects in the ensemble of defects at the microscopic level. Its scale is
bG
Z
frles
1)
m
Oi
where b is the lattice dislocation Burgers vector, a~ is the resistance to the motion of an individual dislocation and G is the elastic modulus. Usually, /m~=O.1 -
1.0/~m.
,/~7_~ (a)
i
--
l
e"
~LL~L~L B -
=
For each structure level, the elementary linear defects, i.e. dislocations and disclinations, may be introduced. For a crystal lattice, these are the usual translational lattice dislocations: defects of the first order. For the mesoscopic level, these are partial disclinations and dislocation pile-ups: defects of the second order. For the grain structural level, these are defects of the third order and so on. At a critical concentration of the defects of any level, the strong interaction between them becomes very important and the cooperative effects in this level come into play. As a result, the next level joins in and the defects of the next order appear. The rotations to observed in crystals lie within a wide range of angles, regardless of the crystal lattice type. Therefore, it is not crystallographic considerations that determine the nature of the rotations. From this, it follows that disclinations in crystals are partial, i.e. imperfect; they inevitably entail surface distortions, such as subgrain, grain or twin boundaries. The disclination power to can vary over a wide range. The disclination line characteristic scale also has a spectrum of length values, i.e. from /r = b (for loops enveloping twisted polymeric chains (Fig. 4(a)) to lr = 106-10Sb for macrorations during necking. However, the most typical scale for disclinations in crystals is the scale of the mesoscopic level with lr = 103-104 b.
Nb
¢M
(b)
(c)
-/ (d)
Fig. 4. Scale levels of plastic deformation in solids: (a) microscopic level (typical lengths being the size of a kink on a dislocation or the diameter of a disclination loop in twisted chains of a polymer); (b) mesoscopic level connected with dislocation substructures--a dislocation wall (wedge disclination dipole) or a pile-up (superdislocation); (c) structural level with translational or rotational motion of grains in polycrystals; (d) macroscopic level of the order of a sample size.
4. Different approaches to the calculation of energetic properties of disclinations To evaluate the elastic stresses and energies of screened disclinations, one can use linear or non-linear theory of elasticity, or apply the technique of computer simulation. The last two methods are more precise but more complex problems may be investigated exactly in the framework of linear elasticity. Therefore, it is necessary to determine the limits and the accuracy of the above three methods when analysing the properties of disclinations. If we take into account geometrical non-linearity (tensor of finite strains instead of tensor of small strains) and materials non-linearity (Murnaghane's law instead of Hooke's law), we can obtain the following expression for the elastic energy of a wedge disclination per unit length of a cylinder with a free surface [11]:
E~a
X/l(~. + ~)
2(2+2/~) /c
2R 2
[
l,]i
+x/~ [ 2(/].~~,u) 2
+ ~+/'t ( a 2 - Otl) + -3V2- - Vl Ct3] ~3R2 /~ 3~
(2)
A. E. Romanov
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Screened disclinations in solids
where ~c: ~o/2n characterizes the power of the disclination, R is the radius of the cylinder, and a, 51 and c~ are combinations of the linear elastic moduli ~ and /~ (Lame constants) and the second-order elastic constants v~, w and v3- In eqn. (2), the first term gives the well-known solution of linear elasticity. Usually, it is written in the form
(3) where D = G/2~(1 - v ) , with G the shear modulus and v being Poisson's ratio. The analysis shows that, for a not very large power of defects (for example, typically for partial disclinations of mesoscopic level ~o~<20°) and the secondorder constants typical for real solids, the stresses and energy of isolated disclinations are described by linear elasticity with an accuracy 10% or better. However, there is a set of qualitatively new non-linear effects for disclinations. As follows from eqn. (2), there is a difference in energies between positive and negative disclinations. Unlike the case in the linear theory, the mean dilatation of the body containing a wedge disclination is not equal to zero. This typical second-order
61
effect is evidently the main motive for the application of non-linear elasticity theory in elastostatic problems involving disclinations. It turns out that, in the secondorder theory, the dilatation 6 does not depend on the sign of o); a typical value for ~o= 10 ° is 6 = 10-3 and, for perfect disclinations with ~o = ~/3, the dilatation is d = 10-2-3 x 10 2. According to current ideas about the structure of amorphous materials, this result can be taken as being a rough estimate of the difference in densities between the amorphous and crystalline states of the material. There is another way to take into account nonlinearity when investigating the properties of disclinations, namely computer simulation [12, 13]. The idea of the molecular dynamics technique consists of the construction of an initial atomic configuration for the disclination, the introduction of the interaction potential between atoms and the performing of the relaxation procedure (Fig. 5). It has been shown [12, 13] that, for potentials characterizing metals, computer simulation of disclinations gives good agreement with the linear theory of elasticity (for defect energies and stresses). At the same time, the results of the non-linear approach are also confirmed. The energies of positive and negative disclinations are distinguished and non-zero total dilatation exists.
{
AO~Oa_
o]exoO-~o~o~.eo~,~o;..~ ° g ~ i O _ q P X i O ~ 0 i I N I g I ~IIP~
o°° °co ----
oo o
Fig. 5. Disclinations in two-dimensional crystallites. Results of computer simulation: (a) initial configuration for a negative perfect disclination of power ~o= -s~/3; (b) relaxed configuration with the additional defects generated at the surface of a crystallite; (c) relaxed configuration with partial disclinations of ~o~-20°; (d) amorphization of disclination core.
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62
With the help of computer simulations, some new effects of disclination screening have been discovered [13]. These are the amorphization of the disclination core when new disclinations are generated near the initial defect (Fig. 5(d)) and the appearance of a dislocation cloud formed by defects going from the surface of crystallite to the disclination (Fig. 5(b)). Both these processes lead to an increase in the number of defects and to a decrease in the energy of the crystallite under consideration.
5. Screened straight disclinations in an infinite continuum
other dipole (Fig. 6). For small values of a I and a 2 ( a l, a2"~.x2+y2), eqn. (5) transforms into the interaction law for edge dislocations which are macroscopically equivalent to these dipoles, i.e. 2 _y2 Fxb = 4Dtol tOzalaEx (xx +y2)Z
(6)
The physical consequences of the difference between eqns. (5) and (6) include the peculiarities of rotational structure development during plastic deformation.
6. Disclination loops in an infinite continuum
The elastic fields of straight disclinations in an infinite isotropic elastic continuum may be found in analytical form from the general relations of the theory of defects [14]. Typical for disclinations is the In divergence of stresses in the defect core and at large distances r from the defect line. Twist disclinations possess also the linear divergence along their lines. The In divergence of the long-range stresses can be disposed of by joining straight disclinations in dipole and other multipole configurations. One-axis dipoles and quadrupoles of wedge disclinations are selfscreened defect configurations. Their energies depend only on the distance between the disclinations inside the configuration. For example, for a dipole with axes of rotation shifted by the distances t 1 and t2 for each disclination, the energy is given by
Disclination loops are defect configurations with maximal screening. It is convenient to investigate the properties of circular disclination loops [8] and RDLs [9]. The elastic fields and energies of RDLs may be expressed through elementary functions. The analysis is easy in comparison with the case of circular loops, where special functions must be involved. The general expressions for the energy of RDLs with Frank vector components (.Oy and ~oz are .
+lul
+xxo In
-
ln(p+v)
V2
I-
E d = Do9313a 2 + 2a(t2 - / l ) + {2a(a +/2 - t t ) - t2tt}
+P('9218
) / X =x2-Rc Xo=X2 y = y 2 - R c yo=Y2 ~"2XXo____ x=x,+Rcxo=xlY=yl+Rcyo=Y,
(7)
xln(~a)+t22+t121n(R-~)12
(4)
13p3
E'°'=Dc°z24 2#(xx°+ yY°)+ 18 where 2a is the dipole arm, R c is the core radius of the dipole disclinations and R~ is an external screening parameter, e.g. the body size. From eqn. (4), for t2 = t I = 0 we obtain the energy of a two-axes dipole with unshifted axes of rotation and, for t 1 = - t: = a, the energy of the one-axis dipole. It is obvious that, in the last case, the energy has no dependence on R s. When analysing the interaction of disclination dipoles it is convenient to use their dislocation representation. The force of interaction between two dipoles with arms 2al and 2a 2 was determined using [8] Dtot to2 Fxa -
2
- v xxo
in(p + v) + v{(2yoVX +yo2U -v2x)
x ln(p + u) +(2XoUy+Xo2V-
u2y)
[{x 2 +(y+ax + a2)2}{x2 +(y-a1 -a2)2}]
xln[{x2+(y+al-a2)2}{x~e~~a2)2}] x In(/) + v)}]. =,:& .o-x=y=y=-g~yo=y~ (5)
where x and y are the coordinates of the center of
one of the dipoles with respect to the center of the
-I IX = X I + R¢lx 0 =x I
(8)
lY=Yl + R~lYo=Yl
where it is supposed that the loop is placed in the plane z = 0; x 2 and Xl, and Y2 and Yl are given by the projec-
A. E. Romanov
\
\,
/
63
Screened disclinations in solids
/
\,/
5
I--d-.. (a)
, / H / /
/
/
/
,,
,,(•
Y
|- -
(b)
............ .n ,o, :-~z_
~
~
m'
//
7~,(=)- mo~
~'~'n azc
/ ma~ ,-,o,
i
i
I
I
t
2
3
~
I
5 x/a
Fig. 6. The map and schematic diagram of interaction between two disclination dipoles. I and II are asymptotes (obtained from eqn. (6)) for q~0 and ~P2(the functions determining zero-value and minimal value of the force Fxd). The curves are given for the case a~ = 2a~. The region of attraction is shaded.
(c)
(d)
Fig. 8. External screening for wedge disclinations.
example, the stresses of an RDL at large distances r behave as r 4p or r-4p ~ for p~>l or p ~ 1, respectively, where p = c/d is the parameter of the loop shape (ratio of rectangular sides). In general, all the characteristics and properties of RDLs are strongly affected by their shape, in contrast to those of dislocation loops. If the loop shape is changed, the sign of the energy of interaction between a disclination loop and a source of internal stresses may even be reversed. 7. External screening for disclination defects
Fig. 7. RDL in the coordinate system used in calculation of the loop energy.
tion of the loop sides on the coordinate axes; u = x0 - x, v =Y0-Y, P = u2 +v2 (Fig. 7). Therefore, ~oz is responsible for, the twist RDL (~ozez is normal to the loop plane) and ~Oyfor the wedge R D L (Wyey is in the loop plane). As follows from the analysis, square loops have larger elastic energies than do circular loops of the same area, as is obvious from general physical considerations. Nonetheless, in most applications this difference (which does not exceed 25%) is insignificant. What is important is the similar dependence on the loop size, i.e. proportional to c 3 or a 3, where c is the side of the square and a is the circular loop radius. The asymptotes of stresses for circular disclination loops coincide with those for square loops within a factor of 3/~. RDLs permit one to study the effect of the shape on the elastic fields and energies of disclination loops. For
At present, there are solutions [8] for a set of problems concerning the calculation of stresses and energies of disclinations near free surfaces or interfaces in solids: for disclinations in uniform (Fig. 8(a)) and nonuniform [15] cylinders; for straight disclinations whose lines are parallel to a free surface (Fig. 8(b)); for straight wedge disclinations perpendicular to the surface of a half-space; for disclination loops and dipoles in a plate of finite thickness (Fig. 8(c)); for a straight wedge disclination in an elastic sphere [16]; for disclinations near grain boundaries (Fig. 8(d)). For example, for the wedge disclination, with o) displaced by a distance R~ with respect to the cylinder axis the Airy stress function, which permits one to find the stresses in a state of plane strain, is given by Do
'
, [ R2{(x_R~)2 + yZ}
x=-~ - {(x- Rl)Z+ fi} ,n [(x~, ~ q
(x 2 +y2)(R,2 - R 2 ) / R2 ]
+--~2 (9)
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Screened disclinations in solids
The introduction of any type of surface means the appearance of external screening for disclinations. It is obvious that different surfaces affect differently the elastic properties of disclinations, depending on their geometry and physical properties. Less trivial is the fact that external screening for disclinations is stronger than in the case of dislocations. If for dislocations, instead of the stress dependence o = 1/r at large distances r,> d (d is the distance from a defect line to the surface of a half-space), there appears o = 1 / r 2 - 1 / r 3, disclinations are characterized, instead of the stress divergence 0-~ In r, by the relationship 0 -~ 1/r 2 - 1 / r 3, depending on the type of disclination and the stress tensor component considered. The energy per unit length of a wedge disclination is proportional to the square of the typical screening parameter in the problem, e.g. to the distance d to a plane free surface, to the half-thickness of a plate h or to the radius R of a cylinder. In dependence on the body shape, the degree of screening diminishes in the following sequence: a cylinder (see also eqn. (2)) 1 Dof Ec= ~
(R2 ~ Rl2)2 _
(10)
a plate E h ~ 0.182Da)2h 2
(11)
8. Screened disclinations in rotational structures in crystalline solids The appearance and the development of rotational defect structures is characteristic for practically all crystalline materials, for different temperature ranges and for various regimes of mechanical loading. Block, cell, fragment and grain structures are related to rotational structures [8]. The main feature of all the above structures is the presence of misorientations between neighboring cells, fragments or grains. Very often, rotational structures realize themselves in the form of misorientation bands, which are regions of the crystal lattice misoriented with respect to the surrounding volume, with one dimension much smaller than the other two dimensions. One of the most important problems in the theoretical investigation of rotational structures is the description of their geometry and calculation of the internal stresses and latent energy in such structures. To find the solutions of these problems, it is natural to introduce a three-dimensional network of partial disclinations [10, 17, 18]. This network must be formed from screened disclinations at joints of grains (fragments, subgrains) (Fig. 9). In this way, fragment boundaries and subboundaries are identified with surfaces of partial disclinations and come into existence by the motion of the partial disclinations in the crystal interior.
a half-space 1 Ed=~ Dof d 2
1
(12)
?3
The energy of the disclination in a sphere is given by [16] E~ a = D~o2A ~
X
- , , =j
-.-4
--
(4m +3){ 8v2m 2 + 2m(5v 2 + 3 v - 1)+ (1 + v)( 1 + 2 v)}]
,j
(a)
I*-
ly'tO r,
(b)
(13)
where A is the sphere radius, v is Poisson's ratio and the bulky summation term is responsible for only small corrections (of the order of 10%) to the disclination energy. The external screening of defect elastic fields in small particles, thin films or in subsurface layers leads to decreases in the disclination energy down to values which are comparable with the energy of an individual lattice dislocation. Grain and phase boundaries have very low screening ability for individual disclinations but they can cause additional configurational forces acting on disclination dipoles (Fig. 8(d)) and loops.
WtO
(c)
(d)
Fig. 9. Disclinations in rotational structures: (a) disclination in the node of three grain boundaries of misorientations go1,go, and go3; (b) disclination separating the portions of a subboundary with different misorientations; (c) disclination dipole associated with a jog in the misorientation boundary; (d) disclination representing the termination of fragment or subgrain boundaries.
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Screeneddisclinations in solids"
The concept of screened disclinations permits one to describe the properties of misorientation boundaries themselves. The most promising use here is the application of the disclination model to large-angle grain boundaries, where the dislocation description does not work [19, 20]. Within the framework of this model and using a limited number of experimentally measured parameters (the energies 71 and 72 of the boundary at two favored misorientations c#1 and q~2), the dependence of the equilibrium grain boundary energy on the misorientation ep was evaluated as [20] [[
,,-,
,,
E'g- 16Jr: [{nf('~2)+ Z
Z
i=l / = i + 1
65
orientation may be found from computer models or experimentally. With the aid of eqn. (14), the dependence Eg(Cp)(Fig. 10) was calculated for symmetrical equilibrium tilt boundaries in AI [20]; these calculations are in good agreement with experimental results. Moreover, the disclination model permits one to investigate the socalled non-equilibrium state of grain boundaries. For this purpose, a random distribution of disclination dipoles should be considered. Owing to the nonuniformity of the dipole positions, the grain boundary contains an additional energy which may be comparable with the grain boundary energy in the lowest state.
{f(Hi-H,+2.
9. Disclination mechanisms of plastic deformation and hardening +
mdl~'l + nd2y2
Db2~o2n
H
JrH
(14)
where H = m d l +nd 2 is the period of the boundary under consideration, dl and d2 (X2 = ~d2/H) are the periods of favored boundaries (the dimensions of the structural elements in favored boundaries) and Hi=yi:~/H are coordinates of disclination dipoles which appear in junctions of structure elements of two different types. These coordinates and the number of structure elements (m and n) are determined in the framework of the model and are connected with the misorientation q). In eqn. (14), the last term is caused by the contribution of the disclination cores and the special function
When moving, disclinations contribute to plastic deformation. To calculate the amount of strain e, the geometry of disclination motion must be known in detail. To obtain an elementary estimate of strain rate, one can apply the following relationship: ,~= O¢olrV
(15)
where 0 is the density of screened disclination systems (dipoles) with the power ~o, V is their velocity and lr is the characteristic size (the dipole arm 2a). The motion of two-axes dipoles of wedge partial disclinations is the main mechanism of rotational plastic deformation (Fig. 11). The terminated band of misorientation is modelled as a disclination dipole [21]. The dipole movement occurs when edge dislocations
/
f(t) =16 f ( u - t ) ln(]2 sin ul) du [I
was introduced and investigated in the work [19]. The set of faw)red boundaries for the given axis of misRg
E36.87° /i,
>7.~ %
[~o-~,,j ~
(a)
@
•. '. v
0--
)0
Fig. 10. Energy of symmetrical tilt boundaries in A1 as function of misorientation angle q~ around the [001] direction: disclination model [20]; e, experimental data [20]; o, computer simulation.
(b) Fig. 1 1. Disclination mechanism of plastic deformation: (a) contribution of disclination dipole motion to the change in shape of a sample; (b) micromechanism of wedge partial disclination dipole motion.
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Screened disclinations in solids
travel from the volume to the planes where partial disclinations are located (Fig. 11(b)). The disclination approach permits one to describe all the details of the development of misorientation bands in crystals and other solids. It permits one to connect the parameters of a band (the thickness 2a and misorientation q~= to) with the density p of mobile dislocations and the applied stress a e. For example, for the dislocation density, the following limitations exist:
p~/z < ~ < a p b
(16)
where b again is the value of the lattice dislocation Burgers vector. The following estimates for the parameters of a dipole have been found [21]: 2 a = 0 . 1 - 1 . 0 / ~ m and to=0.01-0.1 for p = 1 0 9 - 1 0 m cm -2. These values correspond to experimental observations [17] of misorientation bands in crystals. The evolution of the dislocation-disclination ensemble may be investigated in the framework of the kinetic approach [22, 23]. The equations of twoelement, reaction-diffusion kinetics look as follows: a2___P ~ t = F(p) - L( O) Bp 2 - mpO + D Ox2
O0 - - - - Q ( O)+ ttMpO at
(17)
Q(0), L(0) and F(p). Two cases are of particular interest. The first case, with L ( 0 ) = I , F(p)=Ap and Q(0) = QO, gives time-periodic uniform solutions for densities of dislocations and disclinations. This explains periodic variations of material properties (for example, microhardness) with plastic deformation under active loading [22]. The second case, i.e. L(O)=O, F(p)=P and Q(O)=QO 1/2, leads to the formation of spatially periodic, stationary dislocation and disclination patterns [23]. For a small deviation in the defect densities from the stationary uniform solutions Ps and 0s of eqns. (17) and (18), the characteristic pattern period is
Iu(D/M) 1/2 x , = 2 : r / 2 p / Q _ QB/M 2
Numerical estimation of XT again leads to the values typical for the mesoscopic level; i.e. XT= 0.1-1.0/~m. Hardening when rotational structures begin to form is connected with the resistance to dislocation motion in the elastic fields of screened disclinations. At the stage of active rotational deformation, the hardening appears to be due to the interdisclination interaction (Fig. 12). Typical for disclination hardening is the dependence of deforming stresses
a = flGto (18)
Here p and 0 are the respective densities of dislocations and disclinations. The function F(p) is the source term for dislocations; in the case of Frank-Read sources, F(p)=P=const., and in the case of the double-cross slip mechanism of dislocation multiplication, F(p)=Ap. The second term on the right-hand side of eqn. (17) describes the annihilation of mobile dislocations. The function L(O) reflects the possible catalytic role of disclinations in the process of dislocation annihilation: L(O)= 1 in the absence of catalytic properties and L(0) = 0 in their presence. The interaction of disclinations with mobile dislocations leads to a decrease in the dislocation density p, with the rate MpO, and to the formation of disclination defects. The coefficient/a must be much less than unity, since, for the generation of a disclination defect, it is necessary to have a large number of dislocations [22]. The function Q(0) depends on microscopical mechanisms of disclination disappearance in the absence of other defects. It is also supposed that the mobility of dislocations can be described with the aid of a diffusion-like term. The parameters of the model A or P, B, M, D and/~ must be determined by microscopical considerations. The eqns. (17) and (18) model various physical situations when one chooses different dependences
(19)
(20)
where fl is the geometry factor determined by the actual type of disclinations interacting. Considering eqn. (20) helps to explain deviations from the well-
(a)
(b)
-
(c)
Io1
(d)
Fig. 12. Hardening mechanisms owing to disclination interactions: (a)-(d) possible arrangements of disclination interactions.
A. E. Romanov
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Screened disclinations in sofids
known dependence a = a G b p 1/2. For the disclination mechanisms of hardening, one can obtain a = p " with ~
67
pentagonal axis (motion of the disclination to the particle surface) and splitting of the pentagonal axis (formation of new partial disclinations) (Fig. 14). This conclusion is in good agreement with the available experimental results.
10. Disclinations in non-crystalline materials 11. Conclusions
If a misorientation band moves in composite or in polymer materials, its front can be modelled as a dipole of wedge partial disclinations, similar to that in homogeneous crystals. The peculiarities of these materials are connected with their plastic anisotropy. Dipole motion in the matrix of the composite occurs by the redistribution of dislocations. However, screened disclination defects are left behind by the dipole when it passes plastically undeformed inclusions, fibers or plates. The stresses and energies of these defects, the origin of which is similar to the origin of Orowan dislocation loops near obstacles to dislocation motion, may be easily found on the basis of the properties of screened disclinations. Transformations of the defects may lead to the destruction of strengthening inclusions or fibers [8]. In amorphous materials, a three-dimensional disclination network is an essential part of their structure. This network appears as a result of the formation of non-crystallographic groups of atoms, which are stable at small sizes, i.e. decahedrons and icosahedrons (Fig. 13) [24]. Structural disclinations in an amorphous body are defects of the first order at the microscopic scale level. The following model of the amorphous state may be proposed. There is a "sea" of positive disclination of small power w+ = 2 : r - 10 arcsin(3-1/2) ~ 7020 ' (Fig. 13(b)) with negative disclinations of power w_ = 2 arcsin(3 1/2),~2:r/5 (Fig. 13(c)) sprinkled in it. The total disclination charge is equal to zero. In such a disclination system, effective screening takes place, which results in the finite density of the latent energy of amorphous solids 5 e = ~ . Dw 2 ~5
G 120(l-v)
(21)
When compared with experimental results, this latent energy proves to be equal to the energy released during the crystallization of metallic glasses. Topological disclinations of power w+ also may be observed in pentagonal nanoparticles. Owing to the presence of a disclination, non-uniform elastic stresses are generated in the particle interior. These stresses may then relax in different ways. It was shown in ref. 25 that the principal relaxation channels in pentagonal particles are dislocation creation, displacement of the
Disclinations in solids exist only in the form of screened configurations with lower self-energies than for isolated disclinations. Partial disclinations, which may be observed in defect structures of deformed solids, manifest themselves on the mesoscopic level, i.e. 0.1-1.0/zm. Their generation and motion are the results of collective effects in dislocation ensembles. The type of boundary conditions strongly affects the behavior of isolated disclinations in solids. The energy of wedge disclinations is determined by the screening parameter, e.g. the thickness of a plate, the size of a small particle, the diameter of a cylinder, etc. In self-screened disclination systems, the energy decreases, owing to the interaction with defects of opposite sign (disclination loops, dipoles or multipoles) or owing to the interaction with defects of different type (dislocation ensembles). In contrast to dislocation loops, the elastic fields and energies of disclination loops depend essentially on the loop shape. Models based on the properties of screened disclinations are important when considering deformation, work hardening and fracture of polycrystals under high degrees of plastic strain. Screened disclinations strongly contribute to the properties of small particles, amorphous solids and grain boundaries.
(a)
(b)
(c) Fig. 1 3. Disclinations in packing of tetrahedra: (a) deficit of solid angle in packing five tetrahedra with a common edge; (b) compact decahedron with disclination of power 09+; (c) disclination of power ~o_ introduced into the decahedron.
A. E. Romanov
68
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Screened discfinations in solids
( (a)
(b)
(c) Fig. 14. Mechanisms of screening of disclination elastic fields in pentagonal nanoparticles: (a) generation of lattice dislocations; (b) motion of the axis of pentagonal symmetry to the surface; (c) splitting of the pentagonal axis and formation of two disclinations.
Acknowledgments The present article was prepared in Max-PlanckInstitut fiir Metallforschung (Stuttgart) during my tenure of a fellowship from the Alexander von Humboldt Foundation (Bonn). The support of both these organizations is gratefully acknowledged.
References 1 E C. Frank,,Disc. Farad. Soc., 25 (1958) 19. 2 M. Kleman, Points, Lines and Walls, Wiley, New York, 1983.
3 K. Anthony, U. Essmann, A. Seeger and H. Trfiuble, in E. Kr6ner (ed.), Mechanics of Generalized Continua, Springer, Berlin, 1968, p. 355. 4 M. N. Salomaa and G. E. Volovik, Rev. Mod. Phys., 59 (1987)533. 5 V. Volterra, Ann. Ecole Norm. Sup. Paris, 23 (3)(1907) 401. 6 J. Friedel, Dislocations, Pergamon, London, 1964. 7 E R. N. Nabarro, Theory of Crystal Dislocations, Clarendon Press, Oxford, 1967. 8 A. E. Romanov and V. I. Vladimirov, in E R. N. Nabarro (ed.), Dislocations in Solids 9, North-Holland, Amsterdam, 1992. 9 N. A. Pertsev, A. E. Romanov, V. I. Vladimirov, Philos. Mag. A, 49(1984) 591. 10 V. E. Panin, V. A. Likhachev and Yu. V. Grinyaev, Structure Levels of Deformation in Solids, Nauka, Novosibirsk, 1985 (in Russian). i 1 V. I. Vladimirov, I. A. Polonsky and A. E. Romanov, Soy. Phys. Tech. Phys., 58 (1988) 882. 12 M. Doyama and R. M. J. Cotterill, Philos. Mag. A, 50 (1984) L7. 13 L. V. Zigilei, A. 1. Mikhailin and A. E. Romanov, Phys. Met. Metall., 66 (1988) 56. 14 R. de Wit, J. Res. Nat. Bur. Stand. A, 77 (1973) 607. 15 V. G. Gryaznov, A. M. Kaprelov, I. A. Polonsky and A. E. Romanov, Phys. Status Solidi B, 167 ( 1991 ) 29. 16 I. A. Polonsky, A. E. Romanov, V. G. Gryaznov and A. M. Kaprelov, Philos. Mag. A, 64 ( 1991 ) 281. 17 V. V. Rybin, Large Plastic Deformation and Fracture of Metals, Metallurgia, Moscow, 1986 (in Russian). 18 W. Bollmann, Philos. Mag. A, 57(1988) 181. 19 J. C. M. Li, Surf. Sci., 31 (1972) 12. 20 V. Yu. Gertsman, A. A. Nazarov, A. E. Romanov, R. Z. Valiev and V. I. Vladimirov, Philos. Mag. A, 59 (1989) 1113. 21 V. I. Vladimirov and A. E. Romanov, Sov. Phys. Solid State, 20 (1978) 1795. 22 B. K. Barakhtin, S. A. lvanov, I. A. Ovid'ko, A. E. Romanov and V. I. Vladimirov, J. Phys. D, 22 (1989) 519. 23 A. E. Romanov and E. C. Aifantis, Scr. Metall. Mater., to be published. 24 D. R. Nelson, Phys. Rev. B, 28 (1983) 5515. 25 V. G. Gryaznov, A. M. Kaprelov, A. E. Romanov and I. A. Polonsky, Phys. Status Solidi B, 167 (1991) 441.
58
Screened disclinations in solids Alexey E. Romanov* Max-Planck-lnstitut fiir Metallforschung, Institut fiir Physik, Heisenbergstrasse 1, 7000 Stuttgart 80 (Germany)
Abstract The properties of screened disclination configurations (loops, dipoles, defects in small particles, etc.) with relatively small energies are considered. The methods and results of calculations of elastic stresses and energies for screened disclinations are discussed. The following techniques of calculations are mentioned: linear and non-linear elasticity and computer simulation. The exact solutions recently obtained for disclinations in thin films and elastic spheres are briefly reported. On the basis of the properties of screened disclinations, a series of models for the processes of plastic deformation are proposed. The bands with misoriented crystal lattices in metals and other materials are described as a result of partial wedge disclination dipole motion. Other disclination models are applied to the description of work hardening at large strains and to the theory of grain boundaries in polycrystals. Disclination models for the structure and properties of amorphous solids and nanoparticles are also discussed.
1. Introduction At present, the concept of disclinations is recognized to be very helpful in condensed matter physics. It has been used for liquid crystals [1], magnetics [2], superconductors [3], superfluid helium [4], etc. Disclinations were initially introduced into the theory of solids together with dislocations as linear imperfections of a deformed body [5]. A unit disclination is a special and very strong source of internal elastic stress and possesses an energy which is very large in comparison with the energy of a lattice dislocation. For a long time, this feature of disclinations was considered to be an obstacle for their application in modelling the plasticity of solids [6]. However, in the case of screened disclination configurations (dipoles, loops, etc.), the energy may be relatively small and disclinations can exist in crystalline and amorphous materials. The present work deals with the properties of screened disclinations in solids. The geometry of disclinations, and their elastic fields and energies are considered for different types of screening. The applications of disclination models in the physics of strength and plasticity are also discussed.
the undeformed faces of the cut undergoing the displacement produced by mutual rotation around a fixed axis [5, 7, 8]. The elasticity of a body containing a disclination does not depend on the position and the shape of the surface of the cut, but is determined by the value, direction and the position of the axial vector of rotation to (Frank vector of the disclination), the shape of the defect line and the boundary conditions. One can investigate the geometry of isolated defects [8], e.g. individual disclinations, dipoles, multipoles, loops (Fig. 2), but special attention should be devoted to rectangular disclination loops (RDLs) (Fig. 2(b)) [9]. The significance of RDLs is clear, because they can be easily transformed into other disclination defects, e.g. an angular disclination, a disclination dipole, a Ldshaped defect or a linear disclination. In general, any disclination defect can be built up with given accuracy as a set of rectangular loops with a common rotation vector. This fact, together with the possibility of the extension of the shape of the loop defect, make it possible to regard the RDL as a fundamental structural
2. Some simple geometrical properties of disclinations The disclination is a linear defect (Fig. 1) which bounds the surface of a cut in a continuous body, with *Permanent address: Ioffe Physico-Technical Institute, Polytechnicheskaya 26, 194021 St. Petersburg, Russian Federation. 0921-5093/93/$6,00
(a) (b) (c) Fig. 1. Disclinations (Volterra rotational dislocations) in a cylinder: (a), (b) twist disclinations of Frank vector w; (c) wedge disclination. © 1993 - Elsevier Sequoia. All fights reserved
A. E. Romanov
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59
Screened disclinations in sofids
unit when describing the geometry and elastic fields of complicated disclination networks. The presence in crystals of axes of discrete rotational symmetry leads to the existence of perfect and partial disclinations [8]. Partial disclinations are always related to plane defects (boundaries of misorientation) and have no lower limits to their power (modulus of the Frank vector). However, perfect disclinations do not restrict the surface of their plane defect and the symmetry axes are described by discrete values of the Frank vector to. Depending on the physical properties of the condensed media under consideration, the formation of disclinations of two types is possible: Frank disclinations, which are singularities of the field of spin (free) rotations, and Volterra disclinations, which are singularities of the field of orbital rotations connected with the non-uniform displacement field. Frank disclinations are peculiar to magnetics, liquid and "plastic" crystals [2]. In some cases, they have no upper limits to their power. They do not have any connection with plastic flow and do not have a translation dislocation analogue. However, in cholesterics and other mesophases with discrete translational symmetries, translational dislocations may be involved in a disclination. There exists a deep interrelationship between Volterra disclinations (rotational dislocations) and Volterra dislocations (translational dislocations). Any disclination can be represented in the form of a superposition of dislocations. Simultaneously, any dislocation defect admits an equivalent representation through a set of disclinations. As a characteristic example, a specific disclination dipole (Fig. 3(a)) may be considered. We call this configuration a single-line, two-rotation-axes disclination dipole [8]. In such a defect, the disclination lines coincide and the Frank vectors to and - t o have a mutual displacement by a vector t. As a result, the single-line dipole appears to be a natural equivalent of a dislocation of Burgers vector b = t Ato. The other important disclination dipole configurations are a two-line, two-rotation-axes dipole of wedge disclinations (Fig. 3(b)), which is similar to the terminated edge dislocation wall, and a two-line, one-rotational-axis dipole (Fig. 3(c)), which demonstrates the properties of a dislocation quadrupole. The presence of different types of disclination dipole is possibly due to the existence of three sets of parameters which are characteristic of a disclination defect: the value and the direction of the Frank vector to, the position r,~ of the Frank vector and the position of the disclination line. For disclination dipoles, we can have r_~, ¢ r.~ or r_,~ = r+,o. In the first case, one treats the dipole as a two-rotation-axes configuration and the lines of opposite-sign disclinations may or may not coincide. In the second case, it is a one-rotation-axis
/
/ - - - ed
(a)
/
~'
(b)
Fig. 2. Disclination loops: (a) circular pure twist disclination loop; (b) rectangular pure wedge disclination loop. For pure defects, the rotation vector is in the geometric center of the loop.
Je
(a)
\
~
t
Y/~./
33/ _L ..L 3_ -!=
.L 3_ 2_ 2_ 2_ 3_ (b)
(c)
Fig. 3. Disclination dipoles and their dislocation analogues: (a) single-line, two-rotation-axes dipole of twist disclinations; (b) two-lines, two-axes wedge disclination dipole; (c) one-axis dipole.
configuration, which is the limited case of an extremely elongated disclination loop.
3. Disclinations and structure levels of plastic flow
There are four main structure levels which are dealt with in the physics of plasticity [8, 10]: a microscopic level--the scale of the lattice constant; a mesoscopic level--the scale of dislocation substructures;
60
A. E. Romanov
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Screened disclinations in solids
a structural level--the scale of grains in polycrystals; a macroscopic level--the scale of a sample size (Fig. 4). Each level of structure requires the employment of special physical models and mathematical languages in compliance with its deformational characteristics. The mesoscopic level is conditioned by collective effects in the ensemble of defects at the microscopic level. Its scale is
bG
Z
frles
1)
m
Oi
where b is the lattice dislocation Burgers vector, a~ is the resistance to the motion of an individual dislocation and G is the elastic modulus. Usually, /m~=O.1 -
1.0/~m.
,/~7_~ (a)
i
--
l
e"
~LL~L~L B -
=
For each structure level, the elementary linear defects, i.e. dislocations and disclinations, may be introduced. For a crystal lattice, these are the usual translational lattice dislocations: defects of the first order. For the mesoscopic level, these are partial disclinations and dislocation pile-ups: defects of the second order. For the grain structural level, these are defects of the third order and so on. At a critical concentration of the defects of any level, the strong interaction between them becomes very important and the cooperative effects in this level come into play. As a result, the next level joins in and the defects of the next order appear. The rotations to observed in crystals lie within a wide range of angles, regardless of the crystal lattice type. Therefore, it is not crystallographic considerations that determine the nature of the rotations. From this, it follows that disclinations in crystals are partial, i.e. imperfect; they inevitably entail surface distortions, such as subgrain, grain or twin boundaries. The disclination power to can vary over a wide range. The disclination line characteristic scale also has a spectrum of length values, i.e. from /r = b (for loops enveloping twisted polymeric chains (Fig. 4(a)) to lr = 106-10Sb for macrorations during necking. However, the most typical scale for disclinations in crystals is the scale of the mesoscopic level with lr = 103-104 b.
Nb
¢M
(b)
(c)
-/ (d)
Fig. 4. Scale levels of plastic deformation in solids: (a) microscopic level (typical lengths being the size of a kink on a dislocation or the diameter of a disclination loop in twisted chains of a polymer); (b) mesoscopic level connected with dislocation substructures--a dislocation wall (wedge disclination dipole) or a pile-up (superdislocation); (c) structural level with translational or rotational motion of grains in polycrystals; (d) macroscopic level of the order of a sample size.
4. Different approaches to the calculation of energetic properties of disclinations To evaluate the elastic stresses and energies of screened disclinations, one can use linear or non-linear theory of elasticity, or apply the technique of computer simulation. The last two methods are more precise but more complex problems may be investigated exactly in the framework of linear elasticity. Therefore, it is necessary to determine the limits and the accuracy of the above three methods when analysing the properties of disclinations. If we take into account geometrical non-linearity (tensor of finite strains instead of tensor of small strains) and materials non-linearity (Murnaghane's law instead of Hooke's law), we can obtain the following expression for the elastic energy of a wedge disclination per unit length of a cylinder with a free surface [11]:
E~a
X/l(~. + ~)
2(2+2/~) /c
2R 2
[
l,]i
+x/~ [ 2(/].~~,u) 2
+ ~+/'t ( a 2 - Otl) + -3V2- - Vl Ct3] ~3R2 /~ 3~
(2)
A. E. Romanov
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Screened disclinations in solids
where ~c: ~o/2n characterizes the power of the disclination, R is the radius of the cylinder, and a, 51 and c~ are combinations of the linear elastic moduli ~ and /~ (Lame constants) and the second-order elastic constants v~, w and v3- In eqn. (2), the first term gives the well-known solution of linear elasticity. Usually, it is written in the form
(3) where D = G/2~(1 - v ) , with G the shear modulus and v being Poisson's ratio. The analysis shows that, for a not very large power of defects (for example, typically for partial disclinations of mesoscopic level ~o~<20°) and the secondorder constants typical for real solids, the stresses and energy of isolated disclinations are described by linear elasticity with an accuracy 10% or better. However, there is a set of qualitatively new non-linear effects for disclinations. As follows from eqn. (2), there is a difference in energies between positive and negative disclinations. Unlike the case in the linear theory, the mean dilatation of the body containing a wedge disclination is not equal to zero. This typical second-order
61
effect is evidently the main motive for the application of non-linear elasticity theory in elastostatic problems involving disclinations. It turns out that, in the secondorder theory, the dilatation 6 does not depend on the sign of o); a typical value for ~o= 10 ° is 6 = 10-3 and, for perfect disclinations with ~o = ~/3, the dilatation is d = 10-2-3 x 10 2. According to current ideas about the structure of amorphous materials, this result can be taken as being a rough estimate of the difference in densities between the amorphous and crystalline states of the material. There is another way to take into account nonlinearity when investigating the properties of disclinations, namely computer simulation [12, 13]. The idea of the molecular dynamics technique consists of the construction of an initial atomic configuration for the disclination, the introduction of the interaction potential between atoms and the performing of the relaxation procedure (Fig. 5). It has been shown [12, 13] that, for potentials characterizing metals, computer simulation of disclinations gives good agreement with the linear theory of elasticity (for defect energies and stresses). At the same time, the results of the non-linear approach are also confirmed. The energies of positive and negative disclinations are distinguished and non-zero total dilatation exists.
{
AO~Oa_
o]exoO-~o~o~.eo~,~o;..~ ° g ~ i O _ q P X i O ~ 0 i I N I g I ~IIP~
o°° °co ----
oo o
Fig. 5. Disclinations in two-dimensional crystallites. Results of computer simulation: (a) initial configuration for a negative perfect disclination of power ~o= -s~/3; (b) relaxed configuration with the additional defects generated at the surface of a crystallite; (c) relaxed configuration with partial disclinations of ~o~-20°; (d) amorphization of disclination core.
A. E. Romanov / Screeneddisclinationsin solids
62
With the help of computer simulations, some new effects of disclination screening have been discovered [13]. These are the amorphization of the disclination core when new disclinations are generated near the initial defect (Fig. 5(d)) and the appearance of a dislocation cloud formed by defects going from the surface of crystallite to the disclination (Fig. 5(b)). Both these processes lead to an increase in the number of defects and to a decrease in the energy of the crystallite under consideration.
5. Screened straight disclinations in an infinite continuum
other dipole (Fig. 6). For small values of a I and a 2 ( a l, a2"~.x2+y2), eqn. (5) transforms into the interaction law for edge dislocations which are macroscopically equivalent to these dipoles, i.e. 2 _y2 Fxb = 4Dtol tOzalaEx (xx +y2)Z
(6)
The physical consequences of the difference between eqns. (5) and (6) include the peculiarities of rotational structure development during plastic deformation.
6. Disclination loops in an infinite continuum
The elastic fields of straight disclinations in an infinite isotropic elastic continuum may be found in analytical form from the general relations of the theory of defects [14]. Typical for disclinations is the In divergence of stresses in the defect core and at large distances r from the defect line. Twist disclinations possess also the linear divergence along their lines. The In divergence of the long-range stresses can be disposed of by joining straight disclinations in dipole and other multipole configurations. One-axis dipoles and quadrupoles of wedge disclinations are selfscreened defect configurations. Their energies depend only on the distance between the disclinations inside the configuration. For example, for a dipole with axes of rotation shifted by the distances t 1 and t2 for each disclination, the energy is given by
Disclination loops are defect configurations with maximal screening. It is convenient to investigate the properties of circular disclination loops [8] and RDLs [9]. The elastic fields and energies of RDLs may be expressed through elementary functions. The analysis is easy in comparison with the case of circular loops, where special functions must be involved. The general expressions for the energy of RDLs with Frank vector components (.Oy and ~oz are .
+lul
+xxo In
-
ln(p+v)
V2
I-
E d = Do9313a 2 + 2a(t2 - / l ) + {2a(a +/2 - t t ) - t2tt}
+P('9218
) / X =x2-Rc Xo=X2 y = y 2 - R c yo=Y2 ~"2XXo____ x=x,+Rcxo=xlY=yl+Rcyo=Y,
(7)
xln(~a)+t22+t121n(R-~)12
(4)
13p3
E'°'=Dc°z24 2#(xx°+ yY°)+ 18 where 2a is the dipole arm, R c is the core radius of the dipole disclinations and R~ is an external screening parameter, e.g. the body size. From eqn. (4), for t2 = t I = 0 we obtain the energy of a two-axes dipole with unshifted axes of rotation and, for t 1 = - t: = a, the energy of the one-axis dipole. It is obvious that, in the last case, the energy has no dependence on R s. When analysing the interaction of disclination dipoles it is convenient to use their dislocation representation. The force of interaction between two dipoles with arms 2al and 2a 2 was determined using [8] Dtot to2 Fxa -
2
- v xxo
in(p + v) + v{(2yoVX +yo2U -v2x)
x ln(p + u) +(2XoUy+Xo2V-
u2y)
[{x 2 +(y+ax + a2)2}{x2 +(y-a1 -a2)2}]
xln[{x2+(y+al-a2)2}{x~e~~a2)2}] x In(/) + v)}]. =,:& .o-x=y=y=-g~yo=y~ (5)
where x and y are the coordinates of the center of
one of the dipoles with respect to the center of the
-I IX = X I + R¢lx 0 =x I
(8)
lY=Yl + R~lYo=Yl
where it is supposed that the loop is placed in the plane z = 0; x 2 and Xl, and Y2 and Yl are given by the projec-
A. E. Romanov
\
\,
/
63
Screened disclinations in solids
/
\,/
5
I--d-.. (a)
, / H / /
/
/
/
,,
,,(•
Y
|- -
(b)
............ .n ,o, :-~z_
~
~
m'
//
7~,(=)- mo~
~'~'n azc
/ ma~ ,-,o,
i
i
I
I
t
2
3
~
I
5 x/a
Fig. 6. The map and schematic diagram of interaction between two disclination dipoles. I and II are asymptotes (obtained from eqn. (6)) for q~0 and ~P2(the functions determining zero-value and minimal value of the force Fxd). The curves are given for the case a~ = 2a~. The region of attraction is shaded.
(c)
(d)
Fig. 8. External screening for wedge disclinations.
example, the stresses of an RDL at large distances r behave as r 4p or r-4p ~ for p~>l or p ~ 1, respectively, where p = c/d is the parameter of the loop shape (ratio of rectangular sides). In general, all the characteristics and properties of RDLs are strongly affected by their shape, in contrast to those of dislocation loops. If the loop shape is changed, the sign of the energy of interaction between a disclination loop and a source of internal stresses may even be reversed. 7. External screening for disclination defects
Fig. 7. RDL in the coordinate system used in calculation of the loop energy.
tion of the loop sides on the coordinate axes; u = x0 - x, v =Y0-Y, P = u2 +v2 (Fig. 7). Therefore, ~oz is responsible for, the twist RDL (~ozez is normal to the loop plane) and ~Oyfor the wedge R D L (Wyey is in the loop plane). As follows from the analysis, square loops have larger elastic energies than do circular loops of the same area, as is obvious from general physical considerations. Nonetheless, in most applications this difference (which does not exceed 25%) is insignificant. What is important is the similar dependence on the loop size, i.e. proportional to c 3 or a 3, where c is the side of the square and a is the circular loop radius. The asymptotes of stresses for circular disclination loops coincide with those for square loops within a factor of 3/~. RDLs permit one to study the effect of the shape on the elastic fields and energies of disclination loops. For
At present, there are solutions [8] for a set of problems concerning the calculation of stresses and energies of disclinations near free surfaces or interfaces in solids: for disclinations in uniform (Fig. 8(a)) and nonuniform [15] cylinders; for straight disclinations whose lines are parallel to a free surface (Fig. 8(b)); for straight wedge disclinations perpendicular to the surface of a half-space; for disclination loops and dipoles in a plate of finite thickness (Fig. 8(c)); for a straight wedge disclination in an elastic sphere [16]; for disclinations near grain boundaries (Fig. 8(d)). For example, for the wedge disclination, with o) displaced by a distance R~ with respect to the cylinder axis the Airy stress function, which permits one to find the stresses in a state of plane strain, is given by Do
'
, [ R2{(x_R~)2 + yZ}
x=-~ - {(x- Rl)Z+ fi} ,n [(x~, ~ q
(x 2 +y2)(R,2 - R 2 ) / R2 ]
+--~2 (9)
64
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Screened disclinations in solids
The introduction of any type of surface means the appearance of external screening for disclinations. It is obvious that different surfaces affect differently the elastic properties of disclinations, depending on their geometry and physical properties. Less trivial is the fact that external screening for disclinations is stronger than in the case of dislocations. If for dislocations, instead of the stress dependence o = 1/r at large distances r,> d (d is the distance from a defect line to the surface of a half-space), there appears o = 1 / r 2 - 1 / r 3, disclinations are characterized, instead of the stress divergence 0-~ In r, by the relationship 0 -~ 1/r 2 - 1 / r 3, depending on the type of disclination and the stress tensor component considered. The energy per unit length of a wedge disclination is proportional to the square of the typical screening parameter in the problem, e.g. to the distance d to a plane free surface, to the half-thickness of a plate h or to the radius R of a cylinder. In dependence on the body shape, the degree of screening diminishes in the following sequence: a cylinder (see also eqn. (2)) 1 Dof Ec= ~
(R2 ~ Rl2)2 _
(10)
a plate E h ~ 0.182Da)2h 2
(11)
8. Screened disclinations in rotational structures in crystalline solids The appearance and the development of rotational defect structures is characteristic for practically all crystalline materials, for different temperature ranges and for various regimes of mechanical loading. Block, cell, fragment and grain structures are related to rotational structures [8]. The main feature of all the above structures is the presence of misorientations between neighboring cells, fragments or grains. Very often, rotational structures realize themselves in the form of misorientation bands, which are regions of the crystal lattice misoriented with respect to the surrounding volume, with one dimension much smaller than the other two dimensions. One of the most important problems in the theoretical investigation of rotational structures is the description of their geometry and calculation of the internal stresses and latent energy in such structures. To find the solutions of these problems, it is natural to introduce a three-dimensional network of partial disclinations [10, 17, 18]. This network must be formed from screened disclinations at joints of grains (fragments, subgrains) (Fig. 9). In this way, fragment boundaries and subboundaries are identified with surfaces of partial disclinations and come into existence by the motion of the partial disclinations in the crystal interior.
a half-space 1 Ed=~ Dof d 2
1
(12)
?3
The energy of the disclination in a sphere is given by [16] E~ a = D~o2A ~
X
- , , =j
-.-4
--
(4m +3){ 8v2m 2 + 2m(5v 2 + 3 v - 1)+ (1 + v)( 1 + 2 v)}]
,j
(a)
I*-
ly'tO r,
(b)
(13)
where A is the sphere radius, v is Poisson's ratio and the bulky summation term is responsible for only small corrections (of the order of 10%) to the disclination energy. The external screening of defect elastic fields in small particles, thin films or in subsurface layers leads to decreases in the disclination energy down to values which are comparable with the energy of an individual lattice dislocation. Grain and phase boundaries have very low screening ability for individual disclinations but they can cause additional configurational forces acting on disclination dipoles (Fig. 8(d)) and loops.
WtO
(c)
(d)
Fig. 9. Disclinations in rotational structures: (a) disclination in the node of three grain boundaries of misorientations go1,go, and go3; (b) disclination separating the portions of a subboundary with different misorientations; (c) disclination dipole associated with a jog in the misorientation boundary; (d) disclination representing the termination of fragment or subgrain boundaries.
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Screeneddisclinations in solids"
The concept of screened disclinations permits one to describe the properties of misorientation boundaries themselves. The most promising use here is the application of the disclination model to large-angle grain boundaries, where the dislocation description does not work [19, 20]. Within the framework of this model and using a limited number of experimentally measured parameters (the energies 71 and 72 of the boundary at two favored misorientations c#1 and q~2), the dependence of the equilibrium grain boundary energy on the misorientation ep was evaluated as [20] [[
,,-,
,,
E'g- 16Jr: [{nf('~2)+ Z
Z
i=l / = i + 1
65
orientation may be found from computer models or experimentally. With the aid of eqn. (14), the dependence Eg(Cp)(Fig. 10) was calculated for symmetrical equilibrium tilt boundaries in AI [20]; these calculations are in good agreement with experimental results. Moreover, the disclination model permits one to investigate the socalled non-equilibrium state of grain boundaries. For this purpose, a random distribution of disclination dipoles should be considered. Owing to the nonuniformity of the dipole positions, the grain boundary contains an additional energy which may be comparable with the grain boundary energy in the lowest state.
{f(Hi-H,+2.
9. Disclination mechanisms of plastic deformation and hardening +
mdl~'l + nd2y2
Db2~o2n
H
JrH
(14)
where H = m d l +nd 2 is the period of the boundary under consideration, dl and d2 (X2 = ~d2/H) are the periods of favored boundaries (the dimensions of the structural elements in favored boundaries) and Hi=yi:~/H are coordinates of disclination dipoles which appear in junctions of structure elements of two different types. These coordinates and the number of structure elements (m and n) are determined in the framework of the model and are connected with the misorientation q). In eqn. (14), the last term is caused by the contribution of the disclination cores and the special function
When moving, disclinations contribute to plastic deformation. To calculate the amount of strain e, the geometry of disclination motion must be known in detail. To obtain an elementary estimate of strain rate, one can apply the following relationship: ,~= O¢olrV
(15)
where 0 is the density of screened disclination systems (dipoles) with the power ~o, V is their velocity and lr is the characteristic size (the dipole arm 2a). The motion of two-axes dipoles of wedge partial disclinations is the main mechanism of rotational plastic deformation (Fig. 11). The terminated band of misorientation is modelled as a disclination dipole [21]. The dipole movement occurs when edge dislocations
/
f(t) =16 f ( u - t ) ln(]2 sin ul) du [I
was introduced and investigated in the work [19]. The set of faw)red boundaries for the given axis of misRg
E36.87° /i,
>7.~ %
[~o-~,,j ~
(a)
@
•. '. v
0--
)0
Fig. 10. Energy of symmetrical tilt boundaries in A1 as function of misorientation angle q~ around the [001] direction: disclination model [20]; e, experimental data [20]; o, computer simulation.
(b) Fig. 1 1. Disclination mechanism of plastic deformation: (a) contribution of disclination dipole motion to the change in shape of a sample; (b) micromechanism of wedge partial disclination dipole motion.
66
A. E. Romanov
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Screened disclinations in solids
travel from the volume to the planes where partial disclinations are located (Fig. 11(b)). The disclination approach permits one to describe all the details of the development of misorientation bands in crystals and other solids. It permits one to connect the parameters of a band (the thickness 2a and misorientation q~= to) with the density p of mobile dislocations and the applied stress a e. For example, for the dislocation density, the following limitations exist:
p~/z < ~ < a p b
(16)
where b again is the value of the lattice dislocation Burgers vector. The following estimates for the parameters of a dipole have been found [21]: 2 a = 0 . 1 - 1 . 0 / ~ m and to=0.01-0.1 for p = 1 0 9 - 1 0 m cm -2. These values correspond to experimental observations [17] of misorientation bands in crystals. The evolution of the dislocation-disclination ensemble may be investigated in the framework of the kinetic approach [22, 23]. The equations of twoelement, reaction-diffusion kinetics look as follows: a2___P ~ t = F(p) - L( O) Bp 2 - mpO + D Ox2
O0 - - - - Q ( O)+ ttMpO at
(17)
Q(0), L(0) and F(p). Two cases are of particular interest. The first case, with L ( 0 ) = I , F(p)=Ap and Q(0) = QO, gives time-periodic uniform solutions for densities of dislocations and disclinations. This explains periodic variations of material properties (for example, microhardness) with plastic deformation under active loading [22]. The second case, i.e. L(O)=O, F(p)=P and Q(O)=QO 1/2, leads to the formation of spatially periodic, stationary dislocation and disclination patterns [23]. For a small deviation in the defect densities from the stationary uniform solutions Ps and 0s of eqns. (17) and (18), the characteristic pattern period is
Iu(D/M) 1/2 x , = 2 : r / 2 p / Q _ QB/M 2
Numerical estimation of XT again leads to the values typical for the mesoscopic level; i.e. XT= 0.1-1.0/~m. Hardening when rotational structures begin to form is connected with the resistance to dislocation motion in the elastic fields of screened disclinations. At the stage of active rotational deformation, the hardening appears to be due to the interdisclination interaction (Fig. 12). Typical for disclination hardening is the dependence of deforming stresses
a = flGto (18)
Here p and 0 are the respective densities of dislocations and disclinations. The function F(p) is the source term for dislocations; in the case of Frank-Read sources, F(p)=P=const., and in the case of the double-cross slip mechanism of dislocation multiplication, F(p)=Ap. The second term on the right-hand side of eqn. (17) describes the annihilation of mobile dislocations. The function L(O) reflects the possible catalytic role of disclinations in the process of dislocation annihilation: L(O)= 1 in the absence of catalytic properties and L(0) = 0 in their presence. The interaction of disclinations with mobile dislocations leads to a decrease in the dislocation density p, with the rate MpO, and to the formation of disclination defects. The coefficient/a must be much less than unity, since, for the generation of a disclination defect, it is necessary to have a large number of dislocations [22]. The function Q(0) depends on microscopical mechanisms of disclination disappearance in the absence of other defects. It is also supposed that the mobility of dislocations can be described with the aid of a diffusion-like term. The parameters of the model A or P, B, M, D and/~ must be determined by microscopical considerations. The eqns. (17) and (18) model various physical situations when one chooses different dependences
(19)
(20)
where fl is the geometry factor determined by the actual type of disclinations interacting. Considering eqn. (20) helps to explain deviations from the well-
(a)
(b)
-
(c)
Io1
(d)
Fig. 12. Hardening mechanisms owing to disclination interactions: (a)-(d) possible arrangements of disclination interactions.
A. E. Romanov
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Screened disclinations in sofids
known dependence a = a G b p 1/2. For the disclination mechanisms of hardening, one can obtain a = p " with ~
67
pentagonal axis (motion of the disclination to the particle surface) and splitting of the pentagonal axis (formation of new partial disclinations) (Fig. 14). This conclusion is in good agreement with the available experimental results.
10. Disclinations in non-crystalline materials 11. Conclusions
If a misorientation band moves in composite or in polymer materials, its front can be modelled as a dipole of wedge partial disclinations, similar to that in homogeneous crystals. The peculiarities of these materials are connected with their plastic anisotropy. Dipole motion in the matrix of the composite occurs by the redistribution of dislocations. However, screened disclination defects are left behind by the dipole when it passes plastically undeformed inclusions, fibers or plates. The stresses and energies of these defects, the origin of which is similar to the origin of Orowan dislocation loops near obstacles to dislocation motion, may be easily found on the basis of the properties of screened disclinations. Transformations of the defects may lead to the destruction of strengthening inclusions or fibers [8]. In amorphous materials, a three-dimensional disclination network is an essential part of their structure. This network appears as a result of the formation of non-crystallographic groups of atoms, which are stable at small sizes, i.e. decahedrons and icosahedrons (Fig. 13) [24]. Structural disclinations in an amorphous body are defects of the first order at the microscopic scale level. The following model of the amorphous state may be proposed. There is a "sea" of positive disclination of small power w+ = 2 : r - 10 arcsin(3-1/2) ~ 7020 ' (Fig. 13(b)) with negative disclinations of power w_ = 2 arcsin(3 1/2),~2:r/5 (Fig. 13(c)) sprinkled in it. The total disclination charge is equal to zero. In such a disclination system, effective screening takes place, which results in the finite density of the latent energy of amorphous solids 5 e = ~ . Dw 2 ~5
G 120(l-v)
(21)
When compared with experimental results, this latent energy proves to be equal to the energy released during the crystallization of metallic glasses. Topological disclinations of power w+ also may be observed in pentagonal nanoparticles. Owing to the presence of a disclination, non-uniform elastic stresses are generated in the particle interior. These stresses may then relax in different ways. It was shown in ref. 25 that the principal relaxation channels in pentagonal particles are dislocation creation, displacement of the
Disclinations in solids exist only in the form of screened configurations with lower self-energies than for isolated disclinations. Partial disclinations, which may be observed in defect structures of deformed solids, manifest themselves on the mesoscopic level, i.e. 0.1-1.0/zm. Their generation and motion are the results of collective effects in dislocation ensembles. The type of boundary conditions strongly affects the behavior of isolated disclinations in solids. The energy of wedge disclinations is determined by the screening parameter, e.g. the thickness of a plate, the size of a small particle, the diameter of a cylinder, etc. In self-screened disclination systems, the energy decreases, owing to the interaction with defects of opposite sign (disclination loops, dipoles or multipoles) or owing to the interaction with defects of different type (dislocation ensembles). In contrast to dislocation loops, the elastic fields and energies of disclination loops depend essentially on the loop shape. Models based on the properties of screened disclinations are important when considering deformation, work hardening and fracture of polycrystals under high degrees of plastic strain. Screened disclinations strongly contribute to the properties of small particles, amorphous solids and grain boundaries.
(a)
(b)
(c) Fig. 1 3. Disclinations in packing of tetrahedra: (a) deficit of solid angle in packing five tetrahedra with a common edge; (b) compact decahedron with disclination of power 09+; (c) disclination of power ~o_ introduced into the decahedron.
A. E. Romanov
68
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Screened discfinations in solids
( (a)
(b)
(c) Fig. 14. Mechanisms of screening of disclination elastic fields in pentagonal nanoparticles: (a) generation of lattice dislocations; (b) motion of the axis of pentagonal symmetry to the surface; (c) splitting of the pentagonal axis and formation of two disclinations.
Acknowledgments The present article was prepared in Max-PlanckInstitut fiir Metallforschung (Stuttgart) during my tenure of a fellowship from the Alexander von Humboldt Foundation (Bonn). The support of both these organizations is gratefully acknowledged.
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3 K. Anthony, U. Essmann, A. Seeger and H. Trfiuble, in E. Kr6ner (ed.), Mechanics of Generalized Continua, Springer, Berlin, 1968, p. 355. 4 M. N. Salomaa and G. E. Volovik, Rev. Mod. Phys., 59 (1987)533. 5 V. Volterra, Ann. Ecole Norm. Sup. Paris, 23 (3)(1907) 401. 6 J. Friedel, Dislocations, Pergamon, London, 1964. 7 E R. N. Nabarro, Theory of Crystal Dislocations, Clarendon Press, Oxford, 1967. 8 A. E. Romanov and V. I. Vladimirov, in E R. N. Nabarro (ed.), Dislocations in Solids 9, North-Holland, Amsterdam, 1992. 9 N. A. Pertsev, A. E. Romanov, V. I. Vladimirov, Philos. Mag. A, 49(1984) 591. 10 V. E. Panin, V. A. Likhachev and Yu. V. Grinyaev, Structure Levels of Deformation in Solids, Nauka, Novosibirsk, 1985 (in Russian). i 1 V. I. Vladimirov, I. A. Polonsky and A. E. Romanov, Soy. Phys. Tech. Phys., 58 (1988) 882. 12 M. Doyama and R. M. J. Cotterill, Philos. Mag. A, 50 (1984) L7. 13 L. V. Zigilei, A. 1. Mikhailin and A. E. Romanov, Phys. Met. Metall., 66 (1988) 56. 14 R. de Wit, J. Res. Nat. Bur. Stand. A, 77 (1973) 607. 15 V. G. Gryaznov, A. M. Kaprelov, I. A. Polonsky and A. E. Romanov, Phys. Status Solidi B, 167 ( 1991 ) 29. 16 I. A. Polonsky, A. E. Romanov, V. G. Gryaznov and A. M. Kaprelov, Philos. Mag. A, 64 ( 1991 ) 281. 17 V. V. Rybin, Large Plastic Deformation and Fracture of Metals, Metallurgia, Moscow, 1986 (in Russian). 18 W. Bollmann, Philos. Mag. A, 57(1988) 181. 19 J. C. M. Li, Surf. Sci., 31 (1972) 12. 20 V. Yu. Gertsman, A. A. Nazarov, A. E. Romanov, R. Z. Valiev and V. I. Vladimirov, Philos. Mag. A, 59 (1989) 1113. 21 V. I. Vladimirov and A. E. Romanov, Sov. Phys. Solid State, 20 (1978) 1795. 22 B. K. Barakhtin, S. A. lvanov, I. A. Ovid'ko, A. E. Romanov and V. I. Vladimirov, J. Phys. D, 22 (1989) 519. 23 A. E. Romanov and E. C. Aifantis, Scr. Metall. Mater., to be published. 24 D. R. Nelson, Phys. Rev. B, 28 (1983) 5515. 25 V. G. Gryaznov, A. M. Kaprelov, A. E. Romanov and I. A. Polonsky, Phys. Status Solidi B, 167 (1991) 441.