Functional differential equations of neutral type with

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Functional differential equations of neutral type with integrable weak singularity: hydrogen thermal desorption model To cite this article: Yury V Zaika and Ekaterina K Kostikova 2017 J. Phys.: Conf. Ser. 929 012034

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International Conference PhysicA.SPb/2016 IOP Conf. Series: Journal of Physics: Conf. Series 1234567890 929 (2017) 012034

IOP Publishing doi:10.1088/1742-6596/929/1/012034

Functional differential equations of neutral type with integrable weak singularity: hydrogen thermal desorption model Yury V Zaika and Ekaterina K Kostikova Institute of Applied Mathematical Research, Karelian Research Centre, Russian Academy of Science, 11 Pushkinskaya street, Petrozavodsk, 185910, Russia E-mail: [email protected] Abstract. One of the technological challenges for hydrogen materials science (including the ITER project) is the currently active search for structural materials with various potential applications that will have predetermined limits of hydrogen permeability. One of the experimental methods is thermal desorption spectrometry (TDS). A hydrogen-saturated sample is degassed under vacuum and monotone heating. The desorption flux is measured by mass spectrometer to determine the character of interactions of hydrogen isotopes with the solid. We are interested in such transfer parameters as the coefficients of diffusion, dissolution, desorption. The paper presents a thermal desorption functional differential equations of neutral type with integrable weak singularity and a numerical method for TDS spectrum simulation, where only integration of a nonlinear system of low order ordinary differential equations (ODE) is required. This work is supported by the Russian Foundation for Basic Research (project 15-01-00744).

1. Introduction Interaction of hydrogen and its isotopes with solids has many applications [1–5]. It is sufficient to mention power engineering, protection of metals from hydrogen corrosion, design of chemical reactors, rocket and missile engineering. Hydrides help to retain substantial amounts of hydrogen: environmental-friendly energy source. Hence high expectations are attached to hydrogen batteries and motors that avoid using high pressures or low temperatures. Some special topics are considered in [6, 7]. Enthusiasts speak not only of hydrogen energy but also of hydrogen economy [4]. Transfer parameters depend on the process characteristics of producing the material batch, and one needs effective algorithms for processing measured curve instead of focusing on ‘tabular data’. We focus on the thermal desorption method and take into account only the limiting factors and the information capacities of the TDS experiment explicated below. 2. Mathematical model of TDS-experiment We consider hydrogen transfer through a test metal or alloy sample (plate of thickness ` ). Suppose heating is relatively slow and uniform so that the diffusion flow can be assumed to be proportional to the concentration gradient. The material is sufficiently homogeneous to neglect interaction of H with traps (micro-defects). Let us assume a standard model for diffusion: ct (t, x) = D(T )cxx (t, x),

(t, x) ∈ Qt∗ ,

(1)

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International Conference PhysicA.SPb/2016 IOP Conf. Series: Journal of Physics: Conf. Series 1234567890 929 (2017) 012034

IOP Publishing doi:10.1088/1742-6596/929/1/012034

where t is time, Qt∗ = (0, t∗ ) × (0, `); c(t, x) is concentration of dissolved atomic hydrogen. Diffusion coefficient D is an Arrhenius function of temperature T (t): D =  D0 exp −ED /[R T (t)] , D0 and ED are the preexponential factor and activation energy. Suppose the plate contacts with gas H2 and surface is potential barrier (see [2, pp. 177–206]). Taking into account (de)sorption processes, we obtain the following boundary conditions: c(0, x) = c¯(x), x ∈ [0, `], t ∈ [0, t∗ ],

(2)

c0 (t) = g(T )q0 (t), c` (t) = g(T )q` (t),

(3)

2 q˙0,` (t) = µs(T )p0,` (t) − b(T )q0,` (t) ± Dcx |x=0,` , (4)   b(T ) = b0 exp −Eb /[RT ] , s(T ) = s0 exp −Es /[RT ] . Here c0 (t) ≡ c(t, 0), c` (t) ≡ c(t, `) are boundary bulk concentrations of diffusing atomic hydrogen; q0 (t), q` (t) are concentrations on the surfaces (x = 0, `); g(T ) is the coefficient of local equilibrium between concentrations on the surface and near-surface bulk; µ is the gas kinetic coefficient; s(T ) is the parameter indicating that only a small part of incident hydrogen find its way on the surface (for brevity, we call s the adhesion coefficient and keep in mind that this coefficient describes the result of concurrent physisorption, dissociation, and chemisorption of molecular gas on the surface); p0 (t), p` (t) are pressures of molecular gas. Although hydrogen can be both in atomic and molecular forms,
we use atoms as the unit of amount: [c] = 1/cm3 , [q] = 1/cm2 , [Dcx ] = [J] = 1/cm2 s J = bq 2 . The quick dissolution model (3) is obtained from the more universal formulas     + −1 k − (T ) 1 − c0,` (t)c−1 max q0,` (t) − k (T ) 1 − q0,` (t)qmax c0,` (t) = ∓ D(T )cx 0,` .

Coefficients k − , k + describe the intensity of the dissolution processes in the bulk and transfer to surface. If these processes are significantly faster than diffusion (for considered temperature range) then it is assumed that Dcx ≈ 0. Hence (if c, q  max) we obtain (3), where g = k − /k + . If the surface is isotropic (Ek− ≈ Ek+ ) then the coefficient g(T ) has weak temperature dependence. We can formally use the Arrhenius equation g(T ) = g0 exp{−Eg /[RT }], Eg = Ek− − Ek+ . Here activation energy Eg is not necessarily positive. The value θ(t) = q(t)/qmax means the degree of the surface covering. Flux density of atom H adsorption (dissociative hydrogen chemisorption on the surface) can be simulated by formula µs(T )p(t)[1 − θ(t)]2 in the balance equations (4). But we have θ  1 for low level of hydrogen. In the context of current TDS-experiment it is convenient to take [p] = Torr and so µ(T ) = (2πmkT )−1/2 ≈ √ 22 2.484 · 10 / T . Here we have [µ] = 1H2 /(Torr cm2 s), [T ] = K, k is the Boltzmann constant, m is mass of a hydrogen molecule. Saturation means that pressure p¯ of H2 on both sides of the plate is constant at T = const. Then (after a time) saturation concentration (¯ c) of dissolved atomic diffusivity hydrogen p √ c = g q¯). is established. Assuming zero derivatives in (4), we obtain c¯ = Γ p¯, Γ ≡ g√ µs/b (¯ Consequently the model is according to the Sieverts law adequacy range (¯ c ∝ p¯). Let us clarify the TDS-experiment. Plate of the material is placed into a chamber. Hydrogen gas is pumped into the chamber under relatively high pressure. The sample is heated to hasten sorption until the sample absorbs enough hydrogen (equilibrium gas saturation). Then heating current is shut off and the material is cooled. In such a case, the physico-chemical processes slow down sharply and significant amount of hydrogen remains inside the sample. The sample is monotonically and rather slowly reheated in vacuum. The heating rate is low and constant. Molecular hydrogen pressure in the chamber is measured by a mass-spectrometer. The desorption flux from the surface is defined by pressure. The desorption flux density is denoted by J(t) = b(t)q 2 (t) (usually J(t) ∝ p(t)). From now on the contracted notation will be used for simplification: b(t) ≡ b(T (t)), D(t) ≡ D(T (t)), s(t) ≡ s(T (t)). Parameter g of rapid dissolution (local surface-bulk equilibrium) usually is taken to be constant (Ek− = Ek+ ) near the TDS-peak. Symmetry conditions are fulfilled: p = p0,` , q = q0,` , c0 = c` , c¯(x) = c¯ = const. 2

International Conference PhysicA.SPb/2016 IOP Conf. Series: Journal of Physics: Conf. Series 1234567890 929 (2017) 012034

IOP Publishing doi:10.1088/1742-6596/929/1/012034

3. Functional differential TDS-equation Suppose the TDS-degassing model: r(t) ≡ µs(t)p(t), J(t) ≡ b(t)q 2 (t), ct = D(t)cxx , c(0, x) = c¯, c0,` (t) = gq(t), q(t) ˙ = r(t) − b(t)q 2 (t) + D(t)cx (t, 0). Vacuum system is assumed to be powerful enough. Hence resorption is neglected (r(t) = 0) for degassing. Values of J(t) are input data for the inverse problem of parametric identification. In the paper we only present numerical simulation of TDS-spectrum J = J(T ). Heating is usually linear (T (t) = T0 + vt). Heating rate v is less than 1 K/s. Heating is stopped (although degassing is continued) if upper temperature limit T (t) = Tmax is reached. Let us formulate the mathematical problem. Difference schemes for the boundary-value problem (with different types of defects) solution was presented in [8]. But a curve of TDSspectrum is determined by surface concentration (J = bq 2 ). Hence, we try to avoid the iterative solution of the boundary-value problem for the current approximations of the model parameters D0 , ED , b0 , Eb , s0 , Es , g. For this aim we perform the problem transformation. Finally, the modeling requires only integration of a system of low order ordinary differential equations (ODE). Rt We replace time by t0 = 0 Dds (keep previous notation t): ct (t, x) = cxx (t, x), c(0, x) = c¯, c0,` = gq(t),

(5)

cx |0 = −cx |` = q(t) ˙ + [J(t) − r(t)]D−1 (t).

(6)

Let q(t) be functional parameter, formula (6) is additional relation to the linear problem (5). Let us change variables so that boundary conditions in (5) become homogeneous: cˆ = c(t, x) − gq(t), cˆt (t, x) = cˆxx (t, x) + f (t), f (t) = −g q(t), ˙ cˆ(0, x) = ϕ(x) ˆ = 0, cˆ|0,` = 0. Write the solution using function of instantaneous source (Green’s function): Z tZ ` ∞ n n2 π 2 o nπx nπξ 2X exp − 2 t sin sin . cˆ(t, x) = G1 (x, ξ, t − τ )f (τ ) dξ dτ, G1 (x, ξ, t) = ` ` ` ` 0 0 n=1

Dynamical boundary conditions contain cˆx (t, 0): Z n n2 π 2 o X0 4g t cˆx |0 = − q(τ ˙ ) exp (τ − t) dτ , ` 0 `2

X0

≡

X n=1,3,5...

.

The series is divergent for τ = t so that term-by-term integration is meant. For original time t we obtain cx (t, 0) = cˆx (t, 0) = cx (t, `) = cˆx (t, `), Z n n2 π 2 Z t o 4gD X0 t 2 q(τ ˙ ) exp − 2 D(s) ds dτ . (7) q(t) ˙ = r(t) − b(t)q (t) − ` ` 0 τ The equation and the original boundary-value problem are equivalent. This means that the solution q(t) uniquely determines the solution c(t, x). An analogy to functional-differential equations of neutral type is due to impossibility to eliminate derivative q˙ in the right side of the equation due to divergent series. We are interested in the time interval [t1 , t2 ] ⊂ (0, t∗ ) near activity peak of thermal desorption (measurements for t near to 0, t∗ are uninformative). Resorption is neglected (r(t) = 0) for powerful vacuum system, thus the equation is simplified. Dimensionless parameters are more convenient for numerical modeling. We use replaces: Rt t0 = 0 D(s)ds/`2 , x0 = x/`, v = q/¯ q (¯ c = g q¯). But notation of t is not changed. We obtain: X0Z t  dv ˜ b(t)`2 q¯ 2 ˜ , b≡ . (8) v(τ ˙ ) exp −n2 π 2 [t − τ ] dτ , v(0) = 1, v˙ ≡ v(t) ˙ = −b(t)v (t) − 4g` dt D(t) 0 3

International Conference PhysicA.SPb/2016 IOP Conf. Series: Journal of Physics: Conf. Series 1234567890 929 (2017) 012034

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Initial saturation is conducted at high temperature T = T = const and pressure p¯ = const (to stimulate sorption). After establishing of equilibrium concentration we obtain  −1/2√ p = b(T )¯ q 2 , c¯ = g q¯ (g = const) ⇒ c¯ = gb0 µs¯ p T exp Eb [2RT ]−1 , µ(T )s(T )¯  ˜b(t) = q¯ b(t)`2 D−1 = c¯ b(t)`2 [gD(t)]−1 = ˜b0 exp −E˜[RT (t)]−1 , b p  E˜b ≡ Eb −ED , ˜b0 ≡ `2 D0−1 b0 µs¯ p T exp Eb [2RT ]−1 . Formally let ˜b(t) ≡ ˜b(T (t)) be Arrhenius parameter, here Eb −ED is activation energy. Usually Eb > ED and surface processes are faster speeded up on heating. (b˙ = . . . Eb T˙ , D˙ = . . . ED T˙ ), thus, E˜b > 0. The pre-exponential factor ˜b0 depends on all parameters D0 , b0 , Eb , s0 , Es (except ED ). Hence the coefficient ˜b changed monotonically and limited 0 < ˜b− ≤ ˜b(t) ≤ ˜b+ . Hereinafter let us suggest that the limit of surface concentration is at qmax ∼ 1015 − 1016 (monolayer in the context of geometrical statics). Where at the stage of initial saturation for selected model parameters we have q¯ > 1014 , for numerical simulations we must take into account the surface filling degree. In such a case, the formula µs¯ p = b¯ q 2 is substituted with the ratio −1 2 2 µs¯ p [1 − q¯qmax ] = b¯ q to estimate the initial value of q¯. For extracting singularity let us clarify why the quite accurate approximation of theP thermal desorption functional differential equation (8) still requires a large number of terms in 0 . The function X0  X0 X Θ(s) = 4 exp −n2 π 2 s , ≡ , n=1,3,5...

has finite values for s > 0. The series is rapidly converging for large s. Formally, substituting s = 0 (the integration variable reaches the upper limit of t) we obtain a divergent series. This can be fixed by term-by-term integration. We have max |zi (t)| = O(n−2 ) (n = 2i − 1, i ≥ 1), wherefore convergence is slow. Let us run some transformations using the theory of Jacobi theta functions. More specifically, we will focus on X∞ θ3 (t, x) = 1 + 2 exp{−n2 π 2 t} cos(2nπx), t > 0. n=1

 √ P 2 We have an alternative presentation [10] for x = 0: θ3 (t, 0) = +∞ −∞ exp − n /t / πt. The series on the left is rapidly converging for large t. But the series on the right P is rapidly converging for small t. Here we are only concerned with small t. Defining θ(t) = exp{−πn2 t} √ √ P P (n ∈ Z, t > 0) we obtain the formula θ(1/t) = tθ(t) or t exp{−πn2 t} = exp{−πn2 /t} (n ∈ Z). This relation is known as functional equation for theta function [11]. Let us run auxiliary transformations. The series is split into odd and even terms: [πt]−1/2

X+∞

=1+2

−∞

X0

X∞  exp − n2 /t = 1 + 2 exp{−n2 π 2 t}

+2

n=1

X∞ k=1

X0 X+∞  exp{−k 2 π 2 4t} = 2 + [4πt]−1/2 exp − n2 /4t . −∞

With this in mind, the last series is subtracted from the first series and the result is doubled. The following expansion is obtained for s > 0: X0  1−Q Θ(s) ≡ 4 exp −n2 π 2 s = √ , πs

Q(s) ≡ −2

X∞ n=1

2

q n , q ≡ − exp{−(4s)−1 }.

The series Q is rapidly √ converging for small s. As s → +0 we have Q → 0 and the integrable singularity Θ ≈ 1/ πs. Since the series is alternating, we have the inequalities Q > 0 (more exactly Q ∈ (0, 1)) and Q ≤ 2|q|. The graph for function Q(s) is an S-shaped saturation curve 4

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and Q(1) ≈ 0.999. Using the above-described presentation of Θ(s) we obtain the TDS functional differential equation of neytral type (terminology from [9]): v(t) ˙ = −˜b(t)v 2 (t) − κ

Z 0

t

1 − Q(t − τ ) p v(τ ˙ ) dτ π[t − τ ]


t ≥ 0, v(0) = 1 .

(9)

Numerical method and computer simulation To be specific in the paper, we use experimental data for nickel and steel 12Cr18Ni10Ti [3]. The assumed values of model parameters are the following: (steel) ` = 0.1 cm, T0 = 300 K, b0 = 1.28 × 10−9 cm2 /s, Eb = 90 × 103 J/mol, D0 = 3.09 × 10−4 cm2 /s, ED = 31 × 103 , g = 50 cm−1 , s0 = 0.6, Es = 110 × 103 , c¯ = 6 × 1017 at.H/cm3 , T = 1170 K, T˙ = 0.5 K; (nickel) ` = 0.1, T0 = 300, b0 = 1.53 × 10−14 , Eb = 43.2 × 103 , D0 = 7.5 · ×10−3 , ED = 40 × 103 , g = 100, s0 = 1.8 × 10−2 , Es = 61.4 × 103 , p¯ = 37.4 Torr, T = 770, T˙ = 0.5. The main role in degassing dynamics belongs to quadratic desorption. So let us approximate the integral term in equation (9). The aftereffect horizon here is h < 1 (τ ∈ [t, t − h] for dimensionless time, Q(1) ≈ 0.999). Let us fix h ∼ 0.3−0.4 due to continuously differentiable √ function Q(s)/ s graph and trapezoid rule for numerical integration. Then, TDS-equation (9) can be approximated by ODE system of low order one-by-one on time segments of length h. The greatest contribution to the integral is made by the value v(τ ˙ ) (τ ≈ t) due to non-limited (but integrable) singularity. Thus, the quadratic approximation v(τ ˙ ) ≈ v(t) ˙ + A[t − τ ] + B[t − τ ]2 is used due to function v˙ concavity. Let us consider the current period of time t ∈ [kh, (k + 1)h], Rt ˙ ) dτ = v(t) − v(kh) (here suppose k ≥ 0. The conditions v(τ ˙ )|kh = v(kh) ˙ and kh v(τ v(t) ≈ v(kh) + v(t)[t ˙ − kh]) determine the sought values of A(t), B(t) (constant on τ ): v(τ ˙ ) ≈ v(t) ˙ +2

v(t) ˙ − v(kh) ˙ v(t) ˙ − v(kh) ˙ [t − τ ] − 3 [t − τ ]2 , t − kh [t − kh]2

τ ∈ [kh, t].

(10)

Represent √ the integral as a sum τ ∈ [0, kh], τ ∈ [kh, t]. For the second additive, the singularity v(τ ˙ )/ t − τ is explicitly integrated √ by substituting (10). The trapezoid rule is used for the integral without singularity (Q(+0)/ +0 = 0). For approximation of the first integral Z 0

kh

X0 Z kh  1 − Q(t − τ ) p v(τ ˙ ) dτ = 4 v(τ ˙ ) exp −n2 π 2 [t − τ ] dτ π[t − τ ] 0

we use only several series terms (for definiteness n = 1, 3) on the right-hand side due to the exponent n2 . Thereby, the negative additives (v˙ < 0) are dropped. Let us compensate for that by replacing kh with t, and introduce in addition variables w1,2 (see below). As the result we obtain the ODE system:   κ ˆ 2 Q(t − kh) √ 2 ˆ + t − kh v(kh) ˙ (11) v(t) ˙ = −b(t)v (t) + √ 2 π 15 X Z t ˜b κ √ √ − 4κ ˆ exp{−n2 π 2 [t − τ ]}v(τ ˙ ) dτ , ˆb ≡ , κ ˆ≡ , 32κ √ t−kh 32κ √ 1+ 15 π 1+ 15 t−kh n=1,3 | 0 {z } π w1,2 (t),

wi (0)=0

w˙ 1 (t) = . . . − [4κ ˆ + π 2 ]w1 (t) − 4κ ˆ w2 (t),

w˙ 2 (t) = . . . − 4κ ˆ w1 (t) − [4κ ˆ + 9π 2 ]w2 (t).

The ellipsis stands for the right-hand side of formula (11). The sought function v(t) = q(t)/¯ q 0 0 (t = t , t ∈ [kh, (k + 1)h], k ≥ 0) is computed by the standard Runge-Kutta 4th order method for integration of the ODE system (the authors used Scilab software). The numerically obtained 5

International Conference PhysicA.SPb/2016 IOP Conf. Series: Journal of Physics: Conf. Series 1234567890 929 (2017) 012034

IOP Publishing doi:10.1088/1742-6596/929/1/012034

TDS-spectra nearly agree with experimental curves. If a TDS-spectrum has two peaks, then w3,4 should be additionally used to improve modeling accuracy. For a comparison, lines with marks in Fig. 1,2 illustrate TDS-spectra simulated by numerical solution (based on difference scheme) of the original distributed boundary-value problem with dynamical boundary conditions (the algorithm, including the model with reversible trapping in the bulk, was presented in Ref. [8]).

J(T) .10 -14

v

[J] = 1/cm2s

difference scheme

Steel

TDS-equation

Ni Steel

Ni

w2

w1 Figure 1. Phase trajectories

difference scheme TDS-equation

T Figure 2. TDS-spectrum, numerical simulation

Conclusions The paper presents a thermal desorption functional differential equation of neutral type with integrable weak singularity for modeling the degassing TDS-spectrum of hydrogen-presaturated structural materials. The proposed numerical method allows using a standard procedure for low order ODE system solving as an alternative to solving the original non-linear boundary-value problem with dynamical boundary conditions. One can thus quickly model TDS-spectra for a wide range of parameters, reveal the limiting factors and explore the material also under extreme conditions (where a natural experiment would be problematic). The main physical sense of the results is the following. One cannot a priori rule out fluctuating thermal desorption, not even for defect-free materials with minor trapping of the diffusing hydrogen (the diffusion model in this paper is free of additives associated with traps). The problem for future research is to differentiate the peaks arising from “surface-volume” interaction dynamics from those induced by hydrogen release from traps with different binding energies. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]

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