Theoretical analysis of particle size re-distribution due to Ostwald ripening in the fuel cell catalyst layer

2.1 Classical model of Ostwald ripening

Standard Ostwald ripening model, used as a basis of LSW theory [5, 6], assumes the changes in particle size due to dissolution and redeposition based on equation

with Δ = c − c_∞ being small difference between current (c) and saturated concentration (c_∞). D denotes the diffusion constant and α = 2σvc_∞/(RT) is determined by surface tension σ and atomic volume of the solute v, solubility c_∞, gas constant R and temperature T [6]. The bracketed term describes the effects, originating from Gibbs-Thompson equation (1), expanded for large r, while the size dependent factor D /r describes the diffusion of solute close to the particle surface under assumption that equilibrium concentration c_∞ is reached infinitely far from the particle.

According to (2) the particles with radius smaller than some limit value r_L = α/Δ will be dissolving while larger partilces will be growing. The particle size re-distribution resulting from this mechanism is calculated as [5]

with f (r, t) being PSD and particle growth rate (2) r˙(r)=drdt.

Under the assumption Δ ≪ c_∞, the asymptotic analytical solution for t −→ ∞ for time dependence of mean particle size r̄ (t) can be found, which obeys the equation

with β=8σc∞v2D9Rt. What is important is that PSD f (r, t) can also be expressed analytically as a function with specific scaling property

where

It is important to note that the same form of PSD g(x) is retained during system evolution, with only parameter x = r/r¯(t) changing with time as determined by equation (4). These two important results of LSW theory will be used for comparison with Ostwald ripening in electrochemical system, given in Section 3.

The improvements of LSW model are typically applied by taking into account the finite volume fraction of condensed particles, decreasing the diffusion distance and therefore altering the diffusion dependant term D /r while keeping the dissolution part of the model the same. This results in modification of PSD (6), but retaining the same scaling property (5). The mean size growth of form (4) retains the same functional dependency with only parameter β being modified [5, 8].

2.2 Platinum dissolution and Ostwald ripening in electrochemical environment

The models of Pt dissolution and redeposition in electrochemical environment, used to describe the Ostwald ripening in fuel cell catalyst layer (e. g. [16, 20, 21]), assume that Pt particles are being dispersed on the carbon surface, providing good electron conductivity, and embedded in the ionomer matrix through which the dissolved Pt ions can diffuse. In contrast to standard Ostwald ripening, the Pt dissolution and redeposition in fuel cell catalyst proceeds via equilibrium electrochemical reaction [16]

On the flat surface, the reaction approaches equilibrium at reference concentration c_ref and potential difference E_dis = 1.188 V against reversible hydrogen electrode (RHE) [16].Above a curved surface with radius of curvature r, the equilibrium potential is shifted according to Kelvin equation [16], derived from Gibbs-Thompson equation (1):

The shift in equilibrium potential is determined by surface tension σ, molar mass M, solid phase density ρ and Faraday constant F.

When out of equilibrium, the reaction (7) proceeds in direct or reverse directions, depending on voltage U, resulting in Pt dissolution or redeposition. For the case of spherical particle, this results in its growth or shrinkage according to equation [16]

The number of transferred electrons during dissolution reaction is n_e = 2, while the reaction asymmetry α = 0.5 is usually assumed in the literature [16, 17]. The expression (9) can therefore be further simplified to

with K0(U)=exp(αnF(U−Edis)RT) describing the voltage-dependent part of dissolution rate and K_r(r) = exp(αnσM2ρRTr)=exp(rKr) being size-dependent part.

One factor, affecting the dissolution rate but not taken into account in (9), is the coverage of Pt surface with stable oxide layer that passivizes the surface and therefore reduces the dissolution rate. This effects plays an important role in Pt dissolution during voltage cycling [23], but was in our case omitted to preserve the simplicity of the model and more clearly present only the most fundamental differences between classical and electrochemical Ostwald ripening.

For the case of Pt in the LT-PEMFC, operating at about 80°C, the characteristic radius is rK=αnσM2ρRT≈4.3 nm [16]. Compared to typical particle size of r ∼ 2 nm in modern fuel cell catalysts [24], we conclude that the full exponential form of size dependence K_r(r) in (10) needs to be taken into account when describing Ostwald ripening in the fuel cell catalyst layer. By introducing a voltage dependent saturated equilibrium Pt concentration at the flat plane surface as

the dissolution rate can be further simplified to

where voltage dependent characteristic time was defined as

It is important to note that τ(U) decreases exponentially with increased voltage, and also depends on temperature T and dissolution rate k_dis = 3.4 × 10⁻⁹ mol m⁻² s⁻¹, with its value taken from Ref. [16]. At equilibrium voltage U = E_dis,0 with K₀ = 1, the characteristic time τ₀ ≈ 1.4×10⁵ s ≈ 37 h, defining the time scale of system evolution.

The reaction rate (9) can be used to model the Ostwald ripening by assuming an ensemble of N particles of various sizes, distributed according to size distribution ∫ f (r) = dN/dr with N = f (r)dr. In a mean-field model, proposed by Darling [16], all the particles in the ensemble are coupled to the same value of Pt concentration c_Pt²⁺ in the interparticle space, with no non-uniformity or diffusion effects taken into account. This approach can to some extent be justified by the specific conditions in the fuel cell catalyst, with relatively small distances between particles ( d=V0/N3∼10 nm where V₀ is the catalyst volume [24]) and large Pt diffusivity in the ionomer ( Dpt2+= 3.4 × 10⁻¹¹ m²/s [17]), resulting in characteristic diffusion time τdif=d2Dpt2+∼10−6s. Since this value is orders of magnitude smaller than τ₀, the omission of diffusion effects is a reasonable assumption.

Since particles of all sizes exchange the Pt ions with the shared reservoir, the time derivative of concentration cpt2+ is determined by the integration over all particles in the distribution

with r˙ (r) = dr/dt being size dependent dissolution rate (9). V₀ represents the volume of interparticle space, available for dissolved Pt, which is considered to be constant in existing electrochemical Ostwald ripening models [20] as well as in LSW theory [5].

A pseudo steady-state Pt ion concentration cpt2+, denoted c_S, can be determined by solving the equation (14) for dcPt2+dt=0, resulting in

where functional F [f (r)] was introduced, with its value defined solely by the shape of PSD f (r). Note that c_S is not a real steady-state value since even at this conditions, the PSD is changing and therefore also the Pt ion concentration. Similarly to classic Ostwald ripening model, the particles smaller than some limiting size r_L, determined as r˙(rL)=0:

will be dissolving and the dissolved Pt will be redeposited on particles larger than r_L. The time evolution of particle size redistribution due to Ostwald ripening in electrochemical system is governed by the same continuity equation (3) as in classical case, but with particle growth rate (9) being used.

2.3 Comparison between classical and electrochemical Ostwald ripening model

The expansion of electrochemical reaction rate (12) under assumption r>>rK,Kr(r)≈1+rKr. yields a result similar to (2):

where

Although the bracketed term in equation (17) is very similar in both cases, note that the expansion for small r_K/r is not justified in fuel cell catalyst with r_K ∼ 5 nm is larger than mean particle size. The assumption of small Δ˜(U) can also hardly be justified, with F [f (r)] ∼ 1 again being true only for large mean particle sizes r̄. Both arguments clearly highlight the importance of the use of original electrochemical particle growth rate (9) in Ostwald ripening modeling in the fuel cell to obtain physically relevant results.

Also note that size dependent prefactor D/ r in equation (2), is in equation (17) substituted by voltage dependent prefactor, indicating the differences in the origin of mass transfer mechanism between classical and electrochemical Ostwald ripening.

2.4 Numerical methods

To highlight the differences between cases of classical and electrochemically driven Ostwald ripening, the equations of electrochemical particle growth rate (9) was incorporated into equation for PSD (3) and Pt ion concentration evolution (14) and solved numerically by discretization of PSD: f(ri,t)→fi(t) and using symmetric finite difference method [25] to estimate the derivative of the product in the right hand side of the equation (3):

The range of particle sizes with discretization Δr = 0.125nm was selected to be between r_min = 0.125nm (to avoid divergence of dissolution rate at r = 0) and r_max = 15 nm (the largest particle size in aged fuel cell catalyst [26]). According to [6], the initial shape of PSD should play no role in asymptotic behaviour of the system and therefore a normal distribution of particles was assumed, with initial mean particle size r̄(0) = 2 nm and standard deviation σ(0) = 0.5nm [24]. In the absence of diffusion processes, the density of particles in the matrix should play no role in asymptotic system behaviour since equilibrium Pt concentration c_Pt²⁺ is expected to retain its pseudo steady-state value (15). This assumption was tested by choosing three different initial particle densities: n_A = 2.4 × 1024/m³ being typical density in real fuel cell catalyst, and smaller particle densities n _B = 2.4×1019/m³ and n_C = 2.4 × 1014/m³. Since the electric potential U plays an important role in effective Pt solubility in the ionomer (12), the whole range of potentials between 0 and 1.4 V, relevant in the fuel cell, was tested at each particle density. The integration times were chosen such that the mean particle size increased beyond the characteristic radius r_K, which is long enough for the PSD to evolve to its asymptotic shape.

3.1 Particle growth rate - Ostwald ripening regime

The first system property of interest was the time evolution of mean particle size, calculated from numerically calculated distribution f_i(t) as

The results for all three particle densities n _A, n _B and n_C are plotted in Figure 1 for values of electric potential U between 0 and 1.4 V in an intervals of 0.05 V as a function of renormalized time t′ = t/τ(U), scaled by voltage dependent characteristic time τ(U), defined in (13).

Figure 1

The increase of mean particle size r as a function of dimensionless time t′ = t/τ(U) for voltage in range of 0 to 1.4 V in intervals of 0.05 V, and for particle densities n_A = 2.4 × 1024/m³, n_B = 2.4 × 1019/m³ and n_C = 2.4 × 1014/m³. At low voltages, the growth follows power law (21), typical for Ostwald ripening. The best fit function is plotted as a black dotted line.

It can be seen from Figure 1 that for low voltages, the growth rates for all particle densities exhibit identical behaviour in renormalized time t′, indicating that the growth mechanism is identical in all cases. Furthermore, the results can be fitted perfectly with a model function, similar to (4)

where p, r₀ and β′ are fitting parameters. The black dotted line in Figure 1 shows the best fitted function, obtained at fitting parameters

The fitted exponent p = 2.92 indicates a very similar growth rate as for standard Ostwald ripening (4), where p = 3. The fitted initial radius r₀ = 2.19nm takes a value reasonably close to the mean radius of initial distribution r̄(0) = 2 nm. Note that constant coefficient β′ in renormalized time t′ implies that growth coefficient β = β′/τ(U) in physical time t (see Eq. (4)) increases exponentially with the voltage U, which can be explained by increased solubility of Pt at elevated voltage (11), as suggested by LSW result, where β is proportional to solubility c_∞. The exact value of β′ depends on other system parameters, but since it does not affect the general growth law, it was not further analyzed.

The growth profile of mean particle size in Figure 1 shows a close resemblance between classical and electrochemical Ostwald ripening. However, it can be further be discerned from Figure 1 that at increased voltages, the growth rate is drastically increased, indicated by almost instantaneous growth in mean particle size for large voltages. This indicates the particle growth mechanism notably different from Ostwald ripening, which will be further explored and analyzed in the next section.

3.2 Particle growth rate - high voltage regime

The process of fast, non-Ostwald particle growth at high voltage values, was further analyzed by plotting the mean particle size as a function of renormalized time t′, with focus on short time scales, shown in Figure 2. The mean particle size in this regime shows exponential growth, with the growth rate increasing with voltage even in renormalized time scale. This behaviour is significantly different from classical Ostwald ripening and calls for more detailed explanation, which can on qualitative level be provided by the the relationship between saturated concentration of Pt c_S (15), electric potential U and mean particle size in the system (15).

Figure 2

The increase in mean particle size as a function of renormalized time t′ for small values of t′ = t/τ(U) for voltage in range of 0 to 1.4 V in intervals of 0.05 V, and for particle densities n_A = 2.4 × 1024/m³, n_B = 2.4 × 1019/m³ and n_C = 2.4 × 1014/m³. The systems at sufficiently high voltage U exhibit an exponential growth of particles for all particle densities.

For a system with narrow PSD where σ ≪ r with mean particle size r̄ at voltage U, the saturated particle concentration can, following equation (15), be approximated as

that is increasing with electric potential and decreasing with particle size. In standard Ostwald ripening regime, the increase in particle size and decrease in saturated Pt concentration are balanced by the deposition of dissolved material on particles. In electrochemical environment at high voltages, however, the saturated concentration (23) can be so high that, according to the model, the amount of Pt dissolved in the interparticle space highly exceeds the amount of Pt present in form of particles. In this conditions, the increase in mean particle size, resulting in the lowering of saturated concentration, cannot be balanced by the decrease of Pt ion concentration since the rate of Pt redeposition on particles is insignificant compared to the total amount of total dissolved Pt. The increasing particle size leads to further increase in oversaturation, faster Pt redeposition and consequent exponential particle growth.

The exponential growth can be qualitatively explained if constant Pt concentration (23), not affected by the particle growth, is assumed for the system of uniform particles of size r¯=r0. The equation for particle growth rate in renormalized time t′ (12) can for small changes in particle size δr(t′) = r(t′) − r₀ be approximated as

This results in exponential growth of particles δr(t′) ∝ exp λt′ with exponent λ=rK2r02Kr(r0) being independent of voltage U. It is important to note that this result is applicable only if constant concentration c_Pt²⁺ is assumed, which is the case only when the particle density in the system is to low to be able absorb the exceeding amount of dissolved Pt ions.

Although this simple analysis explains the system behaviour on a qualitative level, the fitting of the results of numerical analysis show a linear increase of exponent with voltage λ(U) ≈ 3300 + 28000 V⁻¹ × U and much larger values as predicted by equation (24), for initial state r₀ = 2 nm predicting the exponent λ ≈ 40 s⁻¹. This discrepancy is probably caused by the effects of finite spread of PSD, with larger particles growing much faster due to larger difference between their equilibrium concentration and actual Pt ion concentration in the system.

Both Figure 2 and the explanation of exponential particle growth by Pt oversaturation suggest that the transition voltage from Ostwald to exponential growth regime depends on the particle density of the system. This assumption is further explored by plotting the renormalized time t′ = t/τ(U), needed for the mean particle size r to grow above the characteristic size r_K, as a function of voltage U, shown in Figure 3. We see a sharp drop in growth time when the system transits to exponential growth regime at sufficiently high voltage, which indeed depends on particle density n (solid lines in Figure 3).

Figure 3

Renormalized time t′ needed for the system to reach mean particle size r¯=rK, as a function of electric potential U for particle densities n_A = 2.4 × 1024/m³, n_B = 2.4 × 1019/m³ and n_C = 2.4 × 1014/m³. The transition voltage U_tr depends on particle density n, supporting the explanation that the exponential growth occurs when the mass of dissolved Pt in saturated solution m_dis significantly exceeds the mass of solid Pt in form of particles m_sol. Transition voltages U_tr, calculated under assumption m_sol = m_dis, are marked by dashed lines.

To further confirm the proposed explanation based on dissolved Pt oversaturation, the predicted transition voltages were calculated based on few simple assumptions. As a limiting case it was assumed that the initial mass of dissolved platinum m_dis = c_SV0M is comparable to the mass of solid Pt in form of particles, msol≈4πr03ρN, with N being the total number of particles in the system. By equating m_dis = m_sol, expressing particle density as n = N /V and calculating the saturated Pt concentration numerically from (15), the transition voltage U_tr between both regimes can be determined:

We see that the transition voltage U_tr indeed increases with particle density n. The values of U_tr, calculated by numerical integration of F [f (r)] (15) for particle densities n _A, n _B and n_C, are plotted in Figure 3 as solid vertical line for each. The results show a good agreement between predicted and observed transition voltage, further indicating the validity of proposed explanation of exponential particle growth.

Despite sound and plausible interpretation of exponential growth, the actual relevance of this effect in real fuel cell systems is highly questionable. As explained, the process is observed only when the amount of dissolved Pt in initial state exceeds the amount of solid Pt, which is a highly unrealistic situation. Nevertheless, a very similar situation might be present in the fuel cell catalyst during a fast decrease of voltage, when the amount of dissolved Pt strongly exceeds the equilibrium concentration and fast Pt redeposition takes place. However, as seen in Figure 3, the dynamics during exponential growth take place on a time scale of 10⁻⁵τ(U), where the effects of Pt diffusion might play a much more prominent role and should therefore be included in the model. Furthermore, the mean field approach (24), assuming a uniform distribution c_Pt²⁺ of dissolved Pt, might be unjustified, calling for additional improvements of the existing models. The results therefore indicate that the system behaviour at high voltages, often found in fuel cells, calls for more refined approach to Pt dissolution to describe the effects not covered in mean field approach typically used in fuel cell degradation models [16, 18].

3.3 Particle size distribution

In this section we will present the properties of PSD f (r) during time evolution in electrochemical Ostwald ripening regime, calculated numerically as explained in Section 2.4, and compare them with the results of LSW theory, presented in Section 2.1. This will be done by comparing the moments of distribution function, namely standard deviation σ and skewness λ, defined as [22]

where average value is defined as 〈 y(r) 〉=∫y(r)f(r)dr and r¯=〈 r 〉. Note that with this definition, the standard deviation is expressed in units of length while skewness is dimensionless.

It follows from the scaling properties of PSD (5) that the standard deviation of LSW distribution increases linearly with mean particle size and that skewness is constant, with values calculated from (6):

Relatively small standard deviation and large negative skewness are typical characteristics and a common point of criticism of LSW solution, since more symmetric and broader distributions are usually observed in experiments [5].

The properties, listed in (27), are compared with numerical results for electrochemical Ostwald ripening by plotting σ and λ as a functions of mean particle size r for all three particle densities at voltage levels between 0 and 1.4 V in intervals of 0.05 V. In Figure 4, the standard deviation is plotted as solid lines for voltages below the values U_tr (25), dotted for U slightly above U_tr, and as dashed line for U > U_tr. It is clear from the plot that for U < U_tr, the system obeys linear scaling relation between σ and r, marked by orange dotted line. Compared to LSW relation (black dotted line), the factor of proportionality is larger, namely σ≈0.30r¯.

Figure 4

Standard deviation of PSD σ, plotted as a function of mean particle size r during system evolution at particle densities n_A = 2.4 × 1024/m³, n_B = 2.4 × 1019/m³ and n_C = 2.4 × 1014/m³ for voltages between 0 and 1.4 V. Two distinct relations between named quantities can be observed, relating to Ostwald ripening growth (linear, solid lines) for voltages U < U_tr and to exponential growth regime (dashed lines) at U ^≫ U_tr. Systems at voltages only slightly above U_tr (dotted lines) transit during evolution from exponential growth regime into Ostwald ripening regime. The results for electrochemical Ostwald ripening show linear dependence between σ and r (orange dots), similar to LSW theory (black dots).

Above the transition voltage U_tr, the relation between σ and r is no longer linear, but interestingly, the relation is still independent of either voltage or particle density, indicating fundamentally the same growth mechanism for all voltages U > U_tr.

It is interesting to observe that at certain conditions with U≳Utr (dotted line), the system starts in exponential growth regime, but once the sufficient amount of Pt is redeposited, transits to the Ostwald ripening regime. We see that the relation between σ and r after transition to Ostwald regime is practically identical for all systems, regardless of the growth regime at the start of system evolution. This is very important observation, indicating the stability and robustness of the PSD during Ostwald ripening, independent of initial conditions of the system. This property of the observed system will be further highlighted in Section 3.4.

Similarly as in case of standard deviation, the skewness λ exhibits different behaviour in different growth regimes, as seen in Figure 5. What is interesting is that λ is changing with the growth in particle size, which is in contrast with predictions of standard Ostwald ripening theory (27), where constant negative λ is predicted. What is even more surprising is that the sign of skewness in Ostwald ripening regime at U < U_tr is positive, which is opposite to the LSW result, where the size distribution is characterized by strong negative skewness. The results is important since negative skewness is often used as an indication of Ostwald ripening in the analysis of experimentally measured PSDs, which might lead to incorrect interpretation of growth mechanism in fuel cell catalyst [27]. The experimental measurements on aged fuel cell catalyst samples [26], however, typically show positive skewness, confirming the validity of the modelled results.

Figure 5

Skewness of PSD λ, plotted as a function of mean particle size r during system evolution at particle densities n_A = 2.4 × 1024/m³, n_B = 2.4 × 1019/m³ and n_C = 2.4 × 1014/m³ for voltages between 0 and 1.4 V. Similarly as for standard deviation, the relation between quantities is different for Ostwald ripening regime (solid line, U < U_tr) and exponential growth regime (dashed line, U ^≫ U_tr), with systems transferring from exponential towards Ostwald ripening regime (dotted line, U≳Utr ) at sufficiently low Pt oversaturation.

Similarly as in the plot of standard deviation σ, we see the systems with U≳Utr starting in exponentially growth regime evolving gradually to stable Ostwald growth regime if the level of initial oversaturation is sufficiently low (dotted line in Figure 5).

As seen in Figure 5, the skewness of the size distribution in exponential growth regime is changing a sign from positive to negative value during growth. This processes is highlighted in Figure 6 by plotting the scaled PSD at several mean particle sizes between 2 and 7nm as a function of particle size, scaled by mean size: r/r¯. To preserve the transparency, only the results for particles density n_A will be plotted at two different voltages, below transition voltage at U₀ = 0.4 V (solid lines), and above U_tr at U₁ = 1.4 V (dashed line). To simplify the comparison of the results, all distributions are scaled to have the same maximal value, f˜(r)=f(r)/max[ f(r) ].

Figure 6

Normalized PSDf˜(r), plotted as a function of reduced size r/r¯ for various values of mean particle size r during Ostwald ripening (solid line) and exponential growth (dashed line) system evolution. The distribution during exponential growth shows much more pronounced negative skewness due to fast deposition of Pt on large particles, promoting their growth and skewing the distribution towards the right. The distribution during Ostwald ripening is more symmetric and does not change significantly during system evolution. The distribution is compared with LSW result [6], showing smaller standard deviation σ and more pronounced skenwess λ.

As expected, the shape of PSD is significantly different in different growth regimes. In Ostwald growth regime at U₀ (solid line in Figure 6), the shape of the function does not change significantly during growth, which is expected behaviour during Ostwald ripening. For high voltage U₁, on the other hand, the skewness of the distribution is much more significant, evolving from positive towards negative value (dashed line in Figure 6). Especially interesting is a local peak in distribution, emerging at small particle sizes. This is explained by fast deposition of Pt on large particles and slow dissolution of small particles in highly oversaturated environment, resulting in broader distribution with small left-over particles, being dissolved only after the Pt concentration decreases sufficiently. Numerically calculated distributions are further compared to the LSW result (Eq. (6)), plotted as a cyan dotted line in Figure 6. As already explained, the analytic results is much more skewed and narrower than numerically calculated distributions in either electrochemical Ostwald ripening or exponential growth regime.

The experimentally measured PSD, e.g. [26, 28], show more resemblance to results, obtained numerically in this study, than to highly skewed LSW result. Note, however, that in real fuel cell systems the particle size redistribution is affected by competing processes of Ostwald ripening and particle agglomeration, both of which happen simultaneously and at similar conditions [29]. It is therefore difficult to directly compare the results of this study to the experimental measurements, and similar analysis, focused on particle agglomeration, will be needed to further highlight the combined effect of both processes.

3.4 Effects of initial conditions

As seen in Figures 4 and 5 for the systems just above the transition voltage U_tr (dotted lines, U≳Utr ), even though initially in exponential growth regime, the transition to Ostwald ripening regime might occur after sufficient time, with system retaining standard relations between standard deviation, skewness and mean particle size. This behaviour supports the argument, given in LSW theory [6], that the asymptotic behaviour of the PSD undergoing Ostwald ripening is universal and independent of initial conditions [5].

To further test this assumption in electrochemical system, the Ostwald ripenign of the system was modelled with with several different initial PSDs with initial mean size between 2 and 5nm and standard deviations between 0.5 and 1.25 nm, and with particle density n_A. The evolution of each initial state in Ostwald ripening regime at U = 0.4 V is presented as a σ − r̄ plot in Figure 7. It is clear from the plot that the system evolves towards the same linear relation between mean particle size and standard deviation, regardless of initial PSD.

Figure 7

Standard deviation of PSD σ, plotted as a function of mean particle size r¯ during system evolution from various initial distributions at voltage U = 0.4 V and particle density n_A = 2.4 × 1024/m³ (dashed black lines). The convergence of lines towards the same linear relation (red solid line) indicates the robustness and stability of PSD during Ostwald ripening and the generality of the results.

These results confirm that the LSW assumption of universal asymptotic behavior also holds for Ostwald ripening in electrochemical systems. This establishes the relations between mean particle size, standard deviation and skewness, presented in Section 3.3, as a general and universal result in asimptotic limit t −→ ∞, independent of detailed properties of initial PSD.