Thermal Analysis of Solid State Amorphization: Predicted and Measured Reaction Enthalpies and Reaction Kinetics

Appendix A Instrumental and Thermal Lag

According to a simple RC scheme for the DSC apparatus [8, 12] and in the absence of a transformation involving enthalpy change, the heat flux to the sample ∂qs∂t, is given by the following equation:

where T_s is the temperature of the sample, T_a is the temperature imposed on the sample holder of the DSC apparatus (= T₀ + ϕt, where ϕ is the constant heating rate and t is annealing time) and R_s is the heat resistance from the sample holder to the sample via the sample container within the sample holder. A similar expression can be derived for the heat flux to the reference (here consisting of only an aluminium container), ∂qr∂t. In Ref. [12] it has been shown that the power difference readout, X, of the DSC apparatus can be assumed as:

where t_x is the power response time constant (instrumental lag) and ∂q∂t is defined as:

Using Eq. (A1) and its pendant for ∂qr∂t, a way to solve Eq. (A2) is given in Ref. [12]. Here, imposing the boundary condition that X(t = 0) = K, which incorporates the difference between sample and reference in the heat lost to the surroundings, the following expression for X is obtained:

where C_s and C_r are the heat capacities of sample (including the aluminium container) and reference and t_s = R_sC_s and t_r = R_rC_r.

The RC-times and K can be estimated from the beginning of the DSC measurement by fitting Eq. (A4) to a DSC run (either first run or second run ; cf. Section 2). This fit can best be made directly after the start of the DSC run, because then the difference in thermal lag between sample and reference can be appreciable, recognizing that in the present experiment the reference consists of an empty sample container. Results of such fits are shown in Fig. A1 for DSC runs for the three types of specimens studied. It follows that the RC times (data gathered in Table A1) range from 1 s to 4 s, and thus in view of the time scale pertaining to a DSC run, are negligible indeed.

Fig. A1

Fit of Eq. (A4) to the beginning of the first run of the DSC measurements with a heating rate of 10 K/min in order to obtain values for t_s, t_r and t_x (i.e., thermal lag of the specimen and the reference, and the instrumental lag, respectively) for the three types of Ni/Ti multilayers. (Fits to the second run give similar results.)

Appendix B Approximation of β

The state variable β (cf. Eq. (7b)) can be rewritten by partial integration as

where the origin of time corresponds with T = 0 K (see discussion above Eq. (9) in Section 4.2). The integral in the second term on the right-hand side of Eq. (B1) is written as (see below Eq. (B4):

By substituting Eq. (B2) into Eq. (B1) if follows that

where B(t) is given by

The essence of the above treatment is that for practical values of ϕ and Q (say ϕ = 1 to 20 K/min.; Q = 50 to 300 kJ/mol), B(t) can be taken as independent of time: B = 0.90 ± 0.02 as determined by numerical calculation after substitution of A(t) according to Eq. (B2). (Note that the origin of the time scale corresponds to T = 0 K).