Temperature Dependence of the Dislocation Contribution to the Modulus Defect

The dislocation contribution to the modulus defect AG/G has been found to be temperature dependent. It is shown that a temperature dependent modulus defect in reasonable agreement with experiment can be obtained by calculating the thermal ly averaged displacement 5 for a string model with Cottrell pinning point interactions using cl assical statistical mechanics. The experimental results of Bauer and Gordon (1) show that the elastic modulus in a deformed NaCl crystal has a different temperature dependence from that of a subsequently x-irradiated crystal. It is known that x-irradiation produces pinning points which immobilize dislocations in this material, so that the difference in the elastic modulus before and after irradiation (the modulus defect) is due to vibrating dislocations. The modulus defect obtained by Bauer and Gordon at 90KHz is displayed in Fig. 1. The dislocation contribution to the modulus defect (AG/G) in NaCl increases rapidly with temperature, increasing by over a factor of two as the temperature increases from 15 to 70K. Comparison (Fig. 1) to ultrasonic test data taken at 10 MHz by the authors indicates that the effect is frequency independent in NaCl. Thompson and Holmes (2) found AG/G for copper increases by a factor of four to six from 4K to room temperature. The similarity of the effect in these two different materials suggests that it is universal. These large changes in the modulus defect with temperature could not be readily explained by the Granato-Lucke theory ( 3 ) , which has been so successful in rationalizing other dislocation effects. According to their theory, the modulus defect at low frequency and strain amplitudes is given by (1) AG/G = 8Gb2 3!nnL2 n'+C where G is the shear modulus, A the dislocation density, C is the line tension, b is the Burgers vector, n is an orientation factor, and L is the dislocation mean loop length. If the dislocation density and loop length are assumed to be constant with temperature, then net increases of order only 20% are expected due to decreases in the line tension and elastic constants with increasing temperature. The inability of the Granato-Lucke theory to explain the temperature dependence of the modulus defect poses a serious problem, as the model has been verified by a vast number of experiments encompassing a wide range of experimental conditions. It was therefore surmised that the model is valid and that the dislocation loop length could Vary with temperature. The simple dislocation double loop model shown in Fig. 2 was employed to calculate the temperature dependence of the modulus defect. The dislocation double-loop consists of an edge dislocation of length a with unbreakable pinning points at x=0,2a and a pinning point at x=a binding the dislocation according to the Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19851038 CIO-168 JOURNAL DE PHYSIQUE Cottrell interaction. The applied stress causes the dislocation to bow out, producing the double loop configuration. With increasing stress amplitude or thermal fluctuations, the dislocation can move out of the Cottrell field, sweeping out an increased area and hence increasing the dislocation contribution to the crystalline strain. Qualitatively, the configuration can produce an increase in the effective loop length of about a factor of two with increasing temperature or stress, leading to an increase of about four (from Eqn. 1) in the modulus defect. Quantitative calrulations of the temperature dependence of the modulus defect for a dislocation double loop were made. The modulus defect for a solid containing N double loops is where y is the average displacement of the dislocation under the applied stress. Using the vibrating string model, the equation of motion for the dislocation displacement in the presence of the Cottrell potential Uc(y) is where Uc(y) is taken after Cottrell (4) to be In static equilibrium at absolute zero in temperature, the overall potential energy of the double-loop configuration can be derived by integration of three force terms due to line tension, the Cottrell potential, and the applied stress over the dislocation length to obtain U/U = + + where a = ~ba2/2Cr~, 6 = Uo~/Cro2 O 36 6 6 1+S2 The quantity is a normalized stress, 6 is the normaliied loop length, and S=y/ro is the normalized separation between the pinning point and the dislocation at.the origin. The modulus defect is then (6) AG/G = (GNb6Cro3/UOo) (43 + S). The individual contributions to the potential energy are shown in Fig. 3, and for low applied stress it is clear that the dislocation will be essentially bound by the equilibrium between the short range Cottrell potential and the applied stress at absolute zero temperature. From Eqn. 5, S =a/(l+6) at equilibrium at absolute zero, to first order. The thermally averaged dislocation displacement at finite temperatures is then derived from classical statistical mechanics. The calculation is analagous to that for thermal expansion of a solid, with applied stress in this case'providing the asymmetric contribution to the potential. The Cottrell interaction is highly anharmonic, necessitating numerical calculations in-general. In the low stress limit, the average can be approximated to yield where is the thermally averaged displacement of the dislocation in the Cottrell potential . The average displacement increases from CS> = :/(l+~) at low temperatures to CS> = a at high temperatures. Using the following values of the parameters appropriate to NaC1; U =O. lev, G=10 1 2 dynes/cm2, b=2A, a=lum, and C=+Gb2, a value of 6=400 was lerived. Hence, the model predicts increases of a factor of four in AG/G, in qualitative agreement with experiment. Quantitative numerical calculations were performed for al1 temperatures over a range of dislocation loop lengths and Cottrell interaction strengths. The best fit to the data with the Cottrell pinning pints constrained to lie at y=O in Fig. 2, however, led to an activation energy of about 0.025ev which is somewhat lower than value O.lev normally assumed for NaCl. Lucke and Schlipf(5) have suggested that in fact pinning point displacements should have a Gaussian distribution at thermal equilibrium. According to their results, the width of the Gaussial for the symmetric double loop considered here is given by w=(k~al~)% or in terms of Our normalized parameters, (~TB/UO)%. The pinning points are assumed here to be immobilized below a temperature of To, leading to the calculated temperature dependence of modulus defect shown in Fig. 4 for several values of To. Results are in good quantitative agreement with experiment for NaCl for kTo/Uo=0.2 and other parameters as defined above. The significance of the results is that temperature-dependent dislocation motion can be described by the Granato-Lucke vibrating string model using straight forward classical statistical mechanics, without resort to ad hoc assumptions or more complex models. Without this demonstration, the use of the vibrating string model for the interpretation of damping results would be in serious question.