thermally activated dislocation motion past point obstacles including the effects of dislocation...

R. J. DIMELFI and W. D. NIX: Thermally Activated Dislocation Motion 331

phys. stat. sol. (b) 58, 331 (1973)

Subject classification: 10

Department of Materials Science and Engineering, Stanford University, Stanford, California

Thermally Activated Dislocation Motion past Point Obstacles Including the Effects of Dislocation Shape

BY R. J. DIMELFI and W. D. NIX

Thermally activated motion of dislocations past point obstacles is studied by treating a special case of radially symmetric obstacles created by considering a position dependent dislocation line energy. Detailed consideration of the shape of the dislocation in the equilibrium and activated states is made by applying the techniques of the calculus of variations. The activation energy is calculated as a function of the applied stress and the shape and height of the obstacle. It is shown that the stress dependence of the activation energy is similar in form to that found for linear obstacles by Dorn and Rajnak. An estimate of the activation area, made by directly integrating the shape of the dislocation, compares favor- ably with the activation area found by considering the stress dependence of the activation energy. The shape of the dislocation in both the equilibrium and activated states is also calculated. Two athermal stress levels for the point obstacle problem are identified and studied. These involve dislocation bowing between obstacles and athermal cutting of the obstacles, respectively.

Die thermisch aktivierte Bewegung von Versetzungen an Punkthindernissen vorbei wird iintersucht mit der Behandlung eines Spezialfalles eines radial-symmetrischen Hindernisses, das durch die Beriicksichtigung einer ortsabhangigen Versetzungslinienenergie gebildet wird. Eine ausfuhrliche Betrachtung der Form der Versetzung im Gleichgewicht und der aktivierten Zustitnde wird durch Anwendung der Variationsrechnung durchgefiihrt. Die Aktivierungsenergie wird als Funktion der angelegten Spannung und der Form und Hohe der Hindernisse berechnet. Es wird gezeigt, dad die Spannungsabhangigkeit der Aktivie- rungsenergie in der Form der von Dorn und Rajnak fur lineare Hindernisse iihnlich ist. Eine Berechnung der Aktivierungsflache durch direkte Integration der Form der Versetzung stimmt befriedigend mit der Aktivierungsflache uberein, die bei Berucksichtigung der Spannungsabhiingigkeit der Aktivierungsenergie gefunden wird. Die Form der Versetzung wird sowohl im Gleichgewichts- als auch im aktivierten Zustand berechnet. Zwei athermische Spannungsniveaus fur das Problem der Punkthindernisse werden identifiziert und unter- sucht. Diese schlieden Versetzungskrummung zwischen Hindernissen bzw. athermisches Schneiden der Hindernisse ein.

1. Introduction The concept of thermally activated motion of dislocations over point obstacles

is often used to describe plastic deformation in crystalline materials. For example, Koppenaal and Arsenault [l] and Koppenaal [2] have used this idea to explain the strengthening of neutron-irradiated copper single crystals. Also, Tyson and Craig [3] employed the concept in their description of the plastic flow behavior of Zr-N alloys below 635 O K . Ilschner and Reppich [4] analysed the transient creep behavior of NaCl single crystals from the thermally activated glide point of view. A more recent example is the work of Borov and Gutmanas 151 who described the deformation of Pb single crystals a t 4.2 OK in this frame- work.

332 R. J. DIMELFI and W. D. NIX

The formalism of the thermodynamics of dislocation glide has its roots in the absolute reaction rate theory of Eyring and co-workers [6, 71. The actual basis for the treatment of dislocation motion was presented by Basinski [8]. The succeeding analyses such as those by Schoeck [9], Gibbs [lo, 113, Hirth and Nix [12] and recently Surek, et al. [13] generally followed his form in their descrip- tions of the thermodynamic parameters. None of the treatments of this problem have considered the dislocation shape in the vicinity of the obstacle during the course of this thermally activated process. The studies mentioned above envis- age the dislocation line being held in equilibrium under an applied stress by an array of obstacles. Each obstacle is imagined to exert a back force per unit length on some small straight segment of the dislocation line. I n some cases, as in Schoeck’s [9] work, the length of this straight segment is equal t o the obstacle spacing. I n others, its length is only a small fraction of the obstacle spacing and the remaining portion of the dislocation is allowed to bow between obstacles ; t,he entire dislocation is held in place by back forces on these small straight segments.

A slightly different approach was presented by Fleischer [14]. He developed an expression for the total force exerted by a tetragonal defect on an infinitely long straight dislocation as a function of the position of the defect. Treating an array of such obstacles spaced a distance L apart, he considered dislocation segments of length L to be activated over these barriers. Ono [15] also used this approach in his work.

The assumption of a straight activation length is either implicitly or explicitly assumed in all of the previous treatments of this problem. In the present work we suggest a model in which unique and distinct equilibrium and activated configurations are inherent. Thus, the local shape of the dislocation becomes an integral part of the calculation of the activation energy.

In their treatment of the Peierl’s mechanism, Dorn and Rajnak [16] presentcd a technique by which the optimum configuration of a kinked dislocation, during thermal activation over a Peierl’s barrier (linear obstacle), can be determined. This permitted them to obtain the activation energy to form the kink as a function of the applied stress. At the basis of their analysis was the concept that the interaction between the dislocation and the Peierl’s obstacle could be described as a position-dependent dislocation line energy. The calculus of variations was used to determine the optimum configurations of the dislocation line.

Similarly, the present model will treat dislocation-point obstacle interactions w-ith a line energy function concomitant with this type of barrier. Application of the calculus of variations in the appropriate manner for this problem will per- mit the determination of optimum configurations, from which the thermal activation parameters can be found.

It may be possible to numerically simulate this problem in its entirely. How- ever, the purpose of this work is to present a largely analytical approach to thr treatment of thermally activated dislocation motion ; thereby keeping in mind the physics of the problem, while including aspects that have been ignored previously. This analysis is presented in an effort to refine our understanding of the physical problem of thermally activated motion of dislocations past point obstacles.

I n the following presentation Sections 2 through 5 cover the analytical dc- velopment, and Section 6 is reserved for the results of the necessary numerical calculations.

Thermally Activated Dislocation Motion past Point Obstacles 333

2. Defect Geometry and the Total Energy Functional Essentially, the present model considers an initially straight dislocation line

which approaches a line of equally spaced point obstacles from far away under the action of an applied stress. Below a certain stress level, the dislocation will be pinned by the array of obstacles, and will thereby assume some equilibrium configuration. These point obstacles are assumed to be non-interacting and localized. By virtue of this assumption and symmetry, it is only necessary to treat the region around one of them. The local equilibrium configuration, shown schematically in Fig. 1, will be maintained until the dislocation moves, by thermal assistance, to an activated state. It can pass the obstacle from this state with no increase in the applied stress.

As mentioned previously, this analysis assumes that all of the interaction between the dislocation and the obstacle is included in a functional representa- tion of the line energy of the dislocation as it approaches the obstacle. For sim- plicity, and the maintainance of an analytical development, this line energy function will be assumed to be radially symmetric, centered a t the obstacle. The total energy change that is obtained when the dislocation moves from a position far from an obstacle to the equilibrium configuration under an applied stress is given by the increase in line energy less the work done by the applied stress. Referring to Fig. 1, this may be written generally as,

U = 2 [ I'(T) ds - To 6) - z b A ] ,

where r ( r ) is the functional dependence of the dislocation line energy, ds an element along the dislocation line, L the obstacle spacing, z the applied shear stress, I', the line energy of a straight dislocation given approximately by G b2/2, A the area swept out by the dislocation, b the value of the Burgers vector, and G the shear modulus. In polar coordinates, this expression becomes

where r and 0 are the coordinates as shown in Fig. 1, and r' = dr/d0. The final simplifying assumption is that the part of the dislocation line far from the obstacle assumes a uniform radius of curvature given by the simple Orowan bowing

Fig. 1. The configuration of a dislocation line near a local obstacle (schematic)

I x2l2 I


relation

when the dislocation is in equilibrium. The center of curvature of this bowed section is determined inherently by the optimization procedure, assuming that i t lies on the line of symmetry bet,ween two obstacles. Referring once more to Fig. 1, the total energy functional is given by

(2 L* - ro cos 8,) r, sin 8, + (L* - r, cos 0,) +

1 R2 R2 2 2 +-a - -sinor cosa , (3)

where ro and 0, correspond t o the point where the optimized dislocation shape coincides with and is tangent to the uniformly bowed section, n is the angle shown, which is geometrically related t o r,, and L* is the distance between the arbitrary starting point of the straight dislocation and the obstacle. It is to this total energy functional that the previously mentioned optimization procedure, introduced by Dorn and Rajnak [16], will be applied.

3. Optimization Procedure and the Activation Energy According to the calculus of variations, the optimum dislocation shape (or

shapes), i.e., that which gives extremal values for the energy functional U, must satisfy the Euler condition, which is for this problem:

This differential equation obviously has a first integral given by

Letting u = (1 + Ff/ri2, and combining terms gives,

r r ( r ) t br2 U 2 1 ' -+ -. - = c

The integration constant, C,, must be determined from knowledge of the shape a t some point along the dislocation. This can be done at 8 = 0 by applying the following auxiliary physical argument.

If we balance the forces, evaluated a t 8 = 0, due to the applied stress and the line tension of the dislocation against those arising from the obstacle interaction, we can write


where r (0 ) is r evaluated a t 8 = 0, and R(0) is the radius of curvature a t that point. This same expression can be obtained by carrying out the differentiation of the Euler condition (a), if it is assumed that the dislocation line is normal to the line 0 = 0 a t that point, and hence that rb=o = 0. The radial symmetry of the line energy function indicates that R(0) = r (O) , so we then have

for our force balance. For a particular function r ( r ) , we can solve this equation for r (0 ) . Using this value with (6), and noting that u = 1 by definition, since r& = 0, we get for C,:

(9)

At the other end point, (r,, O,), the condition of tangency dictates that r'+ 03

as 8 -+ 8,; and thus, by definition u -+ 00 at this point. Solving (6) for u gives

z b r2(0) c, = r (0 ) T ( r ( 0 ) ) 4- ~ 2 .

The condition u + 00 can therefore only be satisfied by

We now have the radial coordinates of the two endpoints of the dislocation line in an optimum shape. The y-coordinate (Fig. 1) of the center of curvature of the bowed section can be obtained from geometry, and is given by

The angles 8, and 01 are also determined and are

A closer inspection of the expression for u (equation (10)) obtained above re- veals that, for the same value of the constant C,, there are two values of r which give u = 1. Thus, there are two values of r (0) . Since they both satisfy the Euler condition, they both must correspond to a point on an optimum shape (0 = 0 ) , and hence to an energy extremum. Therefore, one of these, the larger one, must correspond to the equilibrium shape (r(O)), and the other t o the activated shape ( r * ( O ) ) . Note, this is not true a t the other end point. There is only one value of r, which gives u = 00. Thus, both curves have this point in common.


We are now in a position to rewrite the total energy expression in a more use- ful form. From (3) we have

1- from both sides and recalling that u = vl + (r' /r)z, we get

Changing the variable of integration gives

/- Noting that r' = & r pu2 - 1, and choosing the positive sign for x > 0 (Fig. 1) and recalling that u = r T ( r ) / [ C , - (t b r 2 ) / 2 ] (equation (lo)), we get,

10

z b r2 + - c1 a r2 P ( r ) C, - (t b r 2 / 2 ) 2 dr

r - -~ - + mo, e,) + 0,. (18) - -~ 2 / = = : (C1 - (z l'(r))z br2/2))2 - 1 u = r (0 ) .I

This expression reduces to

We can now obtain, by assuming a form for 17(r), the total energy of the equilibrium and activated states without explicitly knowing the shapes, r(f3). The difference between these quantities gives the activation energy, or the energy which must be supplied thermally for the dislocation to pass a point obstacle under a given applied stress. The integral of (19) can be evaluated numerically.

4. Dislocation Shape and tho Activation Area The equilibrium and activated shapes ought to be obtainable from the above

formalism. With

and with (10) we have


Hence the equilibrium and activated shapes are

and

respectively. However, due to the singularities a t both end points, i.e., g( r ) = 0 a t 8 = 0, and g ( r ) -+ co as 8 + O,, it is difficult t o obtain an unambiguous numerical relation for the dislocation shape.

Let us consider the differential equation (20) once again. Both the equilibrium and activated shapes must obey this equation. Also, they both have a common initial point (ro, €lo). Clearly, the states must coincide as 0 decreases from B0. When the value of 6 decreases to the point (0 = y) where r = r(O), (dr/dO) = 0, and the equilibrium dislocation shape follows a circular path as 0 -+ 0. Along the activated shape, denoted by r * ( 8 ) , the value of r * continues to decrease from r(0) to r*(O) as 8 + 0. Equation (20) shows that dr*/dO increases to a maximum and then decreases to zero again a t r * (O) . Fig. 2 shows, schematically, the equilibrium and activated states obtained from the above development. It should be noted that if the dislocation were to assume, a t 0 = 0, a position between r (0 ) and r*(O), equation (20) would give dr/dOle,,, > 0. This means that an auxiliary force would be required to maintain the dislocation in this configuration (Fig. 2). Thus, only those values obtained by the optimization procedure (i.e., r(0) and r*(O)) require no extra force to maintain equilibrium. Further, a value of P at O = 0 less than r* (O) gives imaginary values for dr/d8le=o, implying that the dislocation has moved passed the obstacle. Features similar to those described above were observed by Dorn and Rajnak [16] in their treatment of linear obstacles.

The shape concepts presented above aid in the calculation of the area swept out by the dislocation when it moves from the equilibrium to the activated state. This is the activation area, often appearing in the literature as the stress de-

Fig. 2. Schematic illustration of the shapes of the dislocation in the equilibrium and activated states. The dashed

line shows the dislocation in an unstable configuration

2 2 physics (b) 66/1

338 R. J. DIMELBI and W. D. NIX

pendence of the activation energy [4]

Referring to Fig. 2, this quantity can be determined by evaluating the following integral

dA* r2(0) - r*2 dr* 2 (dr*/d6)

Changing the variables of integration, and letting ~ = ~ -- we get

Since dr*/d8 is a known function of r* (equation (20)) , (24) can be integrated numerically. If we choose small increments, Ar*, we can carry the integration to a point very close to the singular point (r* = r*(O), dr*ld8 = 0) and thus obtain a good estimate of the activation area. In addition, the shape of the activated state from 8 = ly to 8 = 0 can be determined incrementally by noting that

Thus for an area increment AA* we can obtain the change in angle for a given change in r*.

5. Athermal Stress

The determination of equilibrium and activated states presumes that t,he applied stress is below a critical level a t which the dislocation could pass the obstacle without thermal assistance. In this model there are two such athermal stress levels. One of these is described by the Orowan bowing relation. When the radius of curvature, R, of the uniformly bowed section reaches its minimum value, L/2 , the dislocation can bow through the obstacles without further increase in stress. Thus, this athermal stress level is given by

The other athermal stress level is the one a t which the dislocation can cut, through the obstacle. In order to determine this stress, a re-examination of the treatment presented earlier is necessary. Recalling (9), we see that C, is a unique function of r(O) for a particular value of stress. The relation between C, and r(O) is shown graphically in Fig. 3a. Since the relative minimum in the curve satis- fies the equation


b

Fig. 3. a) Schematic illustration of the dependence of the integration constant C, on r(O), (see equation (S)), b) effect of the stress on the relation between the integration constant C, and r(O), (see equation (9)), showing the conditions for reaching the athermal stress, TAC

which is essentially the same as the local force balance equation (8), the value of r(0) a t the minimum corresponds to the equilibrium value of r(0) used through- out the anilysis. The relative minimum value of C, (C, = C1,min) is thus the particular value of the constant that was used. The ot'her value of r(0) giving the aame value of C, = C1,,,in corresponds to the radius r*(O). Fig. 3b shows the general relation between C, and r(0) for different values of stress. The two values of r (0) which correspond to converge as the stress is increased. The value of stress where they coincide is the athermal stress, T A G . This stress can be determined by requiring that

be solved simultaneously with (27). This gives a second athermal stress inde- pendent of that due to bowing.

6. Results of Numerical Calculations

In this analysis the point obstacle-dislocation interaction is described by a radially symmetric position-dependent line energy function. A reasonable form for this function was chosen to be

where F, is the maximum value of the dislocation line energy at r = 0, and a is an appropriate scaling factor.

Before presenting the results of the numerical calculations using the above function, it may be interesting to compare the nature of the obstacle, as described by (29), with other point obstacle treatments. As mentioned earlier, some previous analyses [14, 151 have handled this problem by calculating the linear elastic interaction between the point defect and a perfectly straight dislocation, thereby computing the total force on the dislocation as it passes the obstacle. This total force calculation can also be made for the obstacle described in this work. Referring to the straight dislocation depicted in Fig. 1, the force per unit length at a part,icular point on the dislocation is

dl" F - --case, dY

u -


Thus; the total force is 00

- - o o

Integrating, and letting 6 = y/a and Fo = 4 (T,,, - To), we get F - = f exp (- 5 2 ) PO

for our total force. Barnett and Nix [17] have recently made improved calculations of the elastic

interactions between screw dislocations and tetragonal defects, including ani- sotropic effects. Fig. 4 shows a comparison of some of their results with (30). It can be seen that the force-distance relation derived from the present model is reasonably similar to those based on long-range elastic interactions between screw dislocations and tetragonal defects.

Using the expression for r ( r ) (equation (29)) in (19), and integrating numerically, the activation energy, A U , can be obtained for various values of applied stress. Fig. 5 shows the stress dependence of the activation energy. The point on the graph corresponding to AU = 0 was determined separately by a calculation of ZAG from (27) and (28). The form of this graph is very similar to the stress dependence of the activation energy obtained by Dorn and Rajnak [IS] for the linear obstacle. The actual values of AU calculated ranged from 0.22 to 0.47 eV for the values of (Tal - To) and a used in this analysis.

The activation area, A*, was determined numerically using (24). It was necessary to cut off the summation just before the singularity (at r* = r*(O)) was reached, but since very small increments in r* were used, a good estimate of t h e area was obtained. Fig. 6 shows the activation area as a function of the

I b I t 085r I

0 017 012 03 04 05 116 07 018 119 [LlZkb/($-$ l-

Fig. 4. Force-distance curve obtained in this work (equation (30)) compared to those vation energy. Obstacle parameters: obtained by Barnett and Nix [17] for tetragonal defects in the rock salt structure and

Fig. 5. The stress dependence of the acti-

r, = 7 x dyn, r,, - r,, = 1.68 x 1 0 - 4 dyn,

in tungsten L/2 = 3 x cm, a = 9 x lo-* cm


Fig. 6. The stress dependence of the activation area as determined both from (24) and from the slopes of Fig. 5. Obstacle parameters: To = 7 x los6 dyn, T, - To = 1.68 x dyn,

L/2 = 3 x 10-6 cm, a = 9 x cm

applied stress. Included in the figure are values of A* determined by the relation

1 BAU A* = -___ b t?r ’

as obtained from slopes of the AU vs. z curve (Fig. 5). Both the actual values and the stress dependence of the activation area, determined by these two methods, are in reasonable agreement. I n this model, the equilibrium and activated configurations approach coincidence as the applied stress approaches zAC. It is therefore expected that the activation area decreases with increasing stress,

As mentioned earlier, the shape of the dislocation in the equilibrium and activated states in the vicinity of the obstacle can be determined by (25). Fig. 7 is a plot of this local shape. The points shown for 8 > y are from a numerical integration of (21a). The remaining shape, corresponding to larger angles, will be as shown schematically in Fig. 1.

Values of zAC were calculated for two values of obstacle “strength” as measur- ed by Fo = 4 (r, - To). As discussed previously, there is another athermal stress level, zAn, resulting from the maximum bowing stress relation. zA obviously depends on the obstacle spacing, whereas ZAC does not. Fig. 8 is a plot of T A B vs. 2/L. Includedin this graph are the two values t A C mentioned above. The smaller value corresponds to the lower obstacle “strength” (Po = 6 7 . 5 ~ x dyn). This value of Fo was used in the calculations of AU and A* discussed earlier. It can be seen in Fig. 8 that, for this value of F,, the athermal stress is given by t~ c when the spacing L < 2.8 x for L > 2.8 x 10V om. Thus, for closely spaced obstacles, this model allows the possibility of an athermal stress below that obtained from the bowing relation.

ca, and is given by ZA

Fig. 7. Equilibrium and activated shapes as determined by (21a) and (25). Obstacle parameters: To = 7 x 10-6 dyn, T, - To = 1.68 x dyn,

L/2 = 3 x 10-6 cm, a = 9 x cm

342 R. J. DIMELEI and W. D. NIX: Thermally Activated Dislocation Motion

I /-----

T ~ ~ , 6=60*7~ajn 2 .s ‘ I----- bD

Fig. 8. The dependence of the athermal stress levels on the obstacle spacing

7. Conclusions The analysis presented here has considered the thermodynamics of disloca-

tion glide past point obstacles. The nature of the treatment has allowed the dislocation shape to be taken into account ; a feature which has been neglected in all earlier studies of this problem.

The model is capable of yielding reasonable, self-consistant values of physical and experimental parameters such as activation energy and activation area. It also includes the predictions of two separate athermal stress levels, above which glide is possible without thermal assistance.

Aehmowledgements

The authors wish to express their appreciation to Prof. D. M. Barnett for his important contributions to this work. The support of The Metallurgy Branch of the Atomic Energy Commission through Contract No. AT(04-3)326PA-17 is greatly appreciated.

References [l] T. J. KOPPENAAL and R. J. ARSENAULT, phys. stat. sol. 17,27 (1966). [2] T. J. KOPPENAAL, Acta metall. 15,681 (1967). [3] W. R. TYSON and G. B. CRAIG, Canad. MetalsQuart. 7, 119 (1968). [4] B. ILSCHNER and B. REPPICH, phys. stat. sol. 3,2093 (1963). [5] 17. S. BOROV and E. Yu. GUTMANAS, phys. stat. sol. (b) 54,413 (1972). [6] H. EYRING, J. chem. Phys. 4, 283 (1936). [7] S. GLASSTONE, K. J. LAIDLER, and H. EYRING, Theory of Rate Processes, McGraw-Hill

[8] Z. S. BASINSKI, Acta metall. 5,684 (1957); Phil. Mag. 4,393 (1959). [9] G. SCHOECK, phys. stat. sol. 8, 499 (1965). 101 G. B. GIBBS, phys. stat. sol. 5, 693 (1964). 111 6. B. GIBBS, phys. stat. sol. 10, 507 (1965). 121 J. P. HIHTH and W. D. NIX, phys. stat. sol. 35,177 (1969). 131 T. SUREK, M. J. LUTON, and J. J. JONAS, to be published. -141 R. L. FLEISCHER, J. appl. Phys. 33, 3504 (1962). 151 K. ONO, J. appl. Phys. 39, 1803 (1967). 161 J. F. DORN and S. RAJNAK, Trans. MS AIME 230,1052 (1964). 171 D. M. BARNETT and W. D. NIX, t o be published.

Publ. Co., New York 1941.

(Received April 19, 1973)

thermally activated dislocation motion past point obstacles including the effects of dislocation...

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