New Method for Extracting Diffusion-Controlled Kinetics from

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New Method for Extracting Diffusion-Controlled Kinetics from Differential Scanning Calorimetry: Application to Energetic Nanostructures Shijing Lu, Edward J. Mily, Doug Irving, Jon-Paul Maria, and Donald W. Brenner J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.5b03317 • Publication Date (Web): 18 May 2015 Downloaded from http://pubs.acs.org on May 27, 2015

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The Journal of Physical Chemistry

New Method for Extracting Diffusion-Controlled Kinetics from Differential Scanning Calorimetry: Application to Energetic Nanostructures Shijing Lu*‡ Edward J. Mily†, Douglas L. Irving∘, Jon-Paul Maria† and Donald W. Brenner*∘ Department of Materials Science and Engineering North Carolina State University, Raleigh, NC 27587 Keywords: Thermite, nanostructure, differential scanning calorimetry,

A new expression is derived for interpreting differential scanning calorimetry curves for solidstate reactions with diffusion controlled kinetics. The new form yields an analytic expression for temperature at the maximum peak height that is similar to a Kissinger analysis, but that explicitly accounts for laminar, cylindrical and spherical multi-layer system geometries. This expression was used to analyze two reactive multi-layer nano-laminate systems, a Zr/CuO thermite and an Al-Ni aluminide, that include systematically varied layer thicknesses. This new analysis scales DSC peak temperatures against sample geometry, which leads to geometry-independent inherent activation energies and prefactors. For the Zr-CuO system, the DSC data scales with the square of the bi-layer thickness, while for the Ni-Al system the DSC data scales with the thickness. This suggests distinct reaction mechanims between these systems.

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I. Introduction and Background Nano-structures from which energy is released by solid-state reactions are of interest from both fundamental science and application viewpoints. Thermites, for example, release energy via solid-state reduction-oxidation reactions involving mass transport between a metal oxide and a complementary metal.1–8 Similarly, multi-metal composites release energy by forming thermodynamically-stable alloys or intermetallic compounds.9–18 Nano-structuring these systems creates an extremely large interfacial area within a small volume, which in principle can be exploited to control power and sensitivity to temperature and external perturbations like shock loading. Designing such structures with predictive control of performance requires understanding how the kinetics of energy release from intrinsic chemical reactions are influenced by the nanostructure in a quantitative fashion. Techniques such as differential scanning calorimetry (DSC) and differential thermal analysis (DTA) are well established for probing reaction kinetics in solids. In a traditional powercompensated DSC experiment, the sample and reference materials are kept thermally insulated from each other but the temperatures of both are synchronized and elevated linearly with time. The difference in energy between the sample and the reference is recorded as a function of temperature. In DTA, identical heat flows to both the sample and reference material is maintained, and an uncompensated temperature difference between the two is recorded. Kinetic parameters including activation energies have been obtained from DSC using a Kissinger analysis19 and its derivatives.20–22 Existing methods for extracting reaction kinetics from DSC curves can be categorized into two groups, peak temperature methods that require the temperature at which the maximum transformation rate is reached (since it is often assumed that the maximum transformation rate is


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reached at the peak of the DSC exotherm), and isoconversion methods22 where the time to complete a certain degree of transformation is considered. In Kissinger’s original work,19 which is an example of the former type of kinetic analysis, an effective nth order transformation rate equation ⁄ = ⁄ 1 − was presumed where is the degree of transformation, n

is the order of the reaction leading to the transformation, and and are the activation energy

and pre-exponent associated with the transformation reaction, respectively. Assuming that the DSC curve reaches a maximum at the peak transformation rate, the dependence of peak temperatures ( ) and the heating rates ( = ⁄) can be found by setting the derivatives of the rate equation with respect to time to zero. The resulting expression gives the traditional Kissinger analysis

! ln = ln " − 1 ! which predicts that a plot of ln$ ⁄ % against 1⁄! will result a straight line with slope of

− .

In subsequent work it was shown that the Kissinger analysis can be generalized to processes whose transformation rate is a product of two functions, i.e.

= &. 2 Assuming that the temperature-dependent function has an Arrhenius form = / , integration of the general rate equation gives


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,+ + *

= * ≡ .$/ % 3 &

The kinetic function & on the right side of Eq. (3) depends on the reaction mechanism. While

its form is often unknown, up to a certain transformation fraction the integral .$/ % is assumed

to be a constant. This is known as the isoconversion assumption and correspondingly / is the

temperature at which 1/ is reached. The left hand side of Eq. (3) has no analytical expressions in closed form, although it is often approximated by different functions.23 For instance, the Kissinger–Akahira–Sunose (KAS) analysis19 assumes that

+ .$/ % = * ≈ ! !/

+

. 4

Considering that .$/ % is a constant under the isoconversion assumption, Eq. (4) suggests that

ln$ ⁄/ % ≈ − ⁄!/ + 5678., which is similar to Eq. (1) except that the temperature used here is at a fixed state of transformation rather than a peak temperature. Nevertheless, it has been shown that the maximum rate of transformation often happens at a fixed state of transformation.23,24 Despite the wide application and success of the Kissinger and related analyses in interpreting DSC and DTA data,7,12,25 they have two major limitations. First, the physical meaning of the apparent kinetic parameters retrieved from a Kissinger-like analysis is not always clear. To explore possible reaction mechanisms, thermal analysis data can be fit to an assumed & using


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a “trial-and-error” procedure. This method, however, is prone to inaccuracies because experimental data can often be fit to more than one kinetic function with apparently equal success.23,26 In addition, for practical reasons simple and empirical algebric expressions, such as the nth order equation, are used. Those equations are almost all based on approximations27-29 and are considered proper largely because the actual kinetic functions are too complicated to be derived. A recent exception is the work of Liu and Barmak, who have incorporated diffusion dynamics into a different analysis of DSC curves.30 Instead of using non-specific kinetic model functions, our model addresses the importance of diffusion in solid reactions for nano reactive materials. A second limitation is that in general the effects of geometry on reaction kinetics is not considered. This limitation is especially pertinant when DSC analysis is used to understand solidstate reaction kinetics in nanostructures. Our model is able to account for geometry in multilayered laminate, cylindrical and spherical structures In this paper, an explicit form of a non-isotherm function is proposed that is based on Fick’s diffusion equation. This form is used to derive analytical expressions for DSC curves and the dependence of peak temperature on heat rate that leads to a Kissinger-like analysis that can account for the effects of multi-layered laminate, cylindrical and spherical geometries. The utility of this new relation is demonstrated by analyzing experimental DSC data for Zr – CuO nanolaminate thermites and a multi-layer Ni/Al aluminide stack. Using this method, the data for many different bi-layer geometries are captured in a single fit, which allows intrinsic activation energies to be extracted, and to understand and predict the influence of physical structure. II. Formulation The starting point is the one-dimensional mass diffusion equation


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9:;, 9 = 9:;, = ; = >; ? @ 5 9 9; 9; where the dimensionless Schvab-Zeldovich variable :;, describes the degree of atom mixing

at position ; and time .11,14,31,32 The initial conditions and :;, are defined so that for the region initially containing the species responsible for exothermic reaction :;, 0 = 1, while

throughout the region to which this species diffuses initially :;, 0 = −1. The quantity C in Eq. (5) denotes the geometry and the corresponding coordinate system in which Eq. (5) is defined. Values of m=0, 1 and 2 correspond to laminates (Cartesian coordinates), wires (cylindrical coordinates) and spheres (spherical coordinates), respectively (see Figure 1). Diffusion is assumed normal to the interfaces between the initial regions for each structure (i.e. quasi-one-dimensional diffusion), and the kinetic parameters for diffusion are assumed to be the same for both regions. The diffusion coefficients in the experimental system will depend not only on the materials and initial microstructure (e.g. grain boundary diffusion), but they will evolve as the concentrations and structure evolves. Describing all of this in detail would require a complicated numerical calculation, and even then it is not clear if all of the unknown details would be adequately captured. Instead our approach gives what is essentially an “average diffusion coefficient”, which as shown below is similar in magnitude to known self-diffusion coefficients. For the laminates this leads to a constant value of :;, > 0 = 0 at the interface between the different regions. This boundary condition is imposed on the solution for the cylindrical and spherical geometries, as well as the natural boundary condition 9: ⁄9 E|GH = 0. Although prior studies suggest that thermite reactions, for example, are often regulated by oxygen diffusion,33–35 Eq. (5) does not discriminate with respect to reacting species or diffusion


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mechanisms. Based on the assumption that mass diffusion is the rate limiting process, DSC curves can be defined as IJ = ΔL

9 * :;, Ω , 9 N

?O: = IJ + 6

where IJ is the heat releasing rate, ΔL is the enthalpy change for the chemical reactions, Ω

denotes the reaction region, is the initial temperature and is the heating rate.

Figure 1. Illustration of sample geometries considered. Left: Laminate, Center: Cylinder, Right: Sphere.

The slab geometry consists of a bilayer with thickness Q and periodic boundaries (i.e. an

infinite number of layers). The reaction zone Ω is every other layer of the laminate, and Ω = R. With this geometry an analytical solution to Eq. (6) can be obtained using a Green’s function36 as IJ =

2D

TUΘ

\

W1 + 2 X Y Z + Y Z^_ ` 7 ]H

b

Θ ≡ * ? ,

[

[

Y ≡ −

c[ deb


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where f is the order to which the analytical solution is truncated. The peak temperatures ( ) can

be obtained by finding the temperature at which the derivative of Eq. (7) with respect to is zero, ?O: g = 0. 8 h

Assuming a diffusion coefficient with form ? = ? / , and that the activation energy for mass diffusion is much larger than the Boltzmann factor at the peak temperature ( ≫ ! ),

an approximate solution to Eqs. (7) and (8) can be found as

2 ln

Q ? + = − ln + ln . 9 ! ! Qk Qk !Θ

In this expression Θ is a constant corresponding to the laminate, Qk is the bilayer total thickness,

and = 1 ℃/Cf7 is a nondimensionalization factor for heating rates. Details of the

mathematical derivation from Eq. (5) and (6) to Eq. (9) are included in the Supplemental Material. In principle Eqs. (5) and (6) can also be solved for the cylindrical and spherical geometries using a Green’s function. However, it was found that the resulting expressions are complicated and it is impractical to find a simple analytical solution to Eq. (6) or Eq. (8). Recognizing this, Eq. (6) and Eq. (8) were solved numerically using a finite difference method with various system sizes and heating rates. The resulting DSC peak temperatures fit Eq. (9) very well (an error

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