XXXI International Mineral Processing Congress 2024 Proceedings/Washington, DC/Sep 29–Oct 3 3347
from 250K to 300K, a further increase in the dynamic
signals is observed, as well as subtle changes between the
data collected at 350K and 400K. At 500K and 550K,
the dynamic signal increases but then decreases at 600K.
Contrarily, as shown in Figures 3(b) and 3(c), for Al2O3 –
LDH and Fe-LDH the dynamic signals, as expected,
continuously increase with temperature. The existence of
complex behavior with Al(OH)3 – LDH indicates the pres-
ence of loosely bound water molecules and hence it behaves
more dynamic at much higher temperatures.
For the quantitative description of the QENS signals,
these were described as:
𝐼(𝑄, 𝐸) =[A0(𝑄)𝛿(𝐸) +(1 − A0 (𝑄))𝑆(𝑄, 𝐸)]
⨂𝑅(𝑄, 𝐸) +𝐵(𝑄, 𝐸) (1)
where A0(𝑄) is the elastic fraction of the scattering signal,
d(E) is a Dirac delta function accounting for the elastic
scattering (zero energy transfer), S(Q, E) is the dynamic
structure factor (or dynamic scattering function), R(Q, E) is
the instrument resolution function, and B(Q, E) is a linear
background. Here, S(Q, E) has been described by a Cole-
Cole distribution function:
cos`
sin`
S E E
E E
E E0
1
1 2d 2
2
0
1
0
2
rC^Qh ra
ra
C^Qh
C^Qh
C^Qh =
+
-
+
-
-
-a
-a
^Q,Eh
d
d
^1-ah
n
n
n
j
j
(2)
In eq. 2, Γ(Q) is the half-width at the half maximum of
the curve, and “α” is a stretching factor (0 ≤ α ≤ 1 that,
in the lower limit case, α =0, leads eq. 2 to a Lorentzian
function. The stretching factor accounts for the presence
of a distribution of the dynamic motions around the mean
value. Finally, the Q-dependences of the Γ(Q) were fitted
with a jump-diffusion model:
Qh DQ 1 2
0
2 'DQ
x C =+
^(3)
where t0 is the residence time between jumps amid two
sites, and D is the diffusion coefficient of the populations
of water under analysis. If t0 =0, the equation leads to a
Fickian diffusion.
The evolutions of Γ(Q) as functions of Q2 at the dif-
ferent experimental temperatures are presented in Figure 4
for Al(OH)3 – LDH and Figure 5 for Al2O3 – LDH and
Figure 6 for Fe-LDH. The values obtained for D and t0 are
displayed in Table 1 and the Arrhenius plot in Figure 7.
In the latter, we obtained activation energies of the
dynamic processes whose diffusion coefficients followed an
Arrhenius-like behavior in at least three temperatures.
In general, the temperature-dependent behavior of the
diffusion coefficients agrees with the discussions previously
made based on the elastic intensity scans. Al(OH)3 – LDH
presents a complex scenario at temperatures above 350K
due to sequential scaping of water populations from the
sample. Still, between 250K and 350K, the diffusivities
present Arrhenius-like behavior with an activation energy
of 17.1 ± 2.1 kJ.mol–1. Here, even though it is not reason-
able to compute the activation energy for Al2O3 – LDH in
the same temperature range, it is very likely that this sample
has similar activation energy to Al(OH)3 – LDH. That is,
considering the data points at 300K and 350K, one would
obtain a value of 15 kJ.mol–1.
Qualitatively, the diffusion coefficients determined
for Al(OH)3 – LDH around room temperature are quite
high and very much comparable with bulk water (~30 x
10–10 m2/s). Nevertheless, this comparison ought to be
made with caution since, we have not been able to describe
Figure 3. Temperature dependence of the QENS spectra collected for (a) Al(OH)
3 – LDH, (b) Al
2 O
3 – LDH samples, and (c)
Fe-LDH samples
from 250K to 300K, a further increase in the dynamic
signals is observed, as well as subtle changes between the
data collected at 350K and 400K. At 500K and 550K,
the dynamic signal increases but then decreases at 600K.
Contrarily, as shown in Figures 3(b) and 3(c), for Al2O3 –
LDH and Fe-LDH the dynamic signals, as expected,
continuously increase with temperature. The existence of
complex behavior with Al(OH)3 – LDH indicates the pres-
ence of loosely bound water molecules and hence it behaves
more dynamic at much higher temperatures.
For the quantitative description of the QENS signals,
these were described as:
𝐼(𝑄, 𝐸) =[A0(𝑄)𝛿(𝐸) +(1 − A0 (𝑄))𝑆(𝑄, 𝐸)]
⨂𝑅(𝑄, 𝐸) +𝐵(𝑄, 𝐸) (1)
where A0(𝑄) is the elastic fraction of the scattering signal,
d(E) is a Dirac delta function accounting for the elastic
scattering (zero energy transfer), S(Q, E) is the dynamic
structure factor (or dynamic scattering function), R(Q, E) is
the instrument resolution function, and B(Q, E) is a linear
background. Here, S(Q, E) has been described by a Cole-
Cole distribution function:
cos`
sin`
S E E
E E
E E0
1
1 2d 2
2
0
1
0
2
rC^Qh ra
ra
C^Qh
C^Qh
C^Qh =
+
-
+
-
-
-a
-a
^Q,Eh
d
d
^1-ah
n
n
n
j
j
(2)
In eq. 2, Γ(Q) is the half-width at the half maximum of
the curve, and “α” is a stretching factor (0 ≤ α ≤ 1 that,
in the lower limit case, α =0, leads eq. 2 to a Lorentzian
function. The stretching factor accounts for the presence
of a distribution of the dynamic motions around the mean
value. Finally, the Q-dependences of the Γ(Q) were fitted
with a jump-diffusion model:
Qh DQ 1 2
0
2 'DQ
x C =+
^(3)
where t0 is the residence time between jumps amid two
sites, and D is the diffusion coefficient of the populations
of water under analysis. If t0 =0, the equation leads to a
Fickian diffusion.
The evolutions of Γ(Q) as functions of Q2 at the dif-
ferent experimental temperatures are presented in Figure 4
for Al(OH)3 – LDH and Figure 5 for Al2O3 – LDH and
Figure 6 for Fe-LDH. The values obtained for D and t0 are
displayed in Table 1 and the Arrhenius plot in Figure 7.
In the latter, we obtained activation energies of the
dynamic processes whose diffusion coefficients followed an
Arrhenius-like behavior in at least three temperatures.
In general, the temperature-dependent behavior of the
diffusion coefficients agrees with the discussions previously
made based on the elastic intensity scans. Al(OH)3 – LDH
presents a complex scenario at temperatures above 350K
due to sequential scaping of water populations from the
sample. Still, between 250K and 350K, the diffusivities
present Arrhenius-like behavior with an activation energy
of 17.1 ± 2.1 kJ.mol–1. Here, even though it is not reason-
able to compute the activation energy for Al2O3 – LDH in
the same temperature range, it is very likely that this sample
has similar activation energy to Al(OH)3 – LDH. That is,
considering the data points at 300K and 350K, one would
obtain a value of 15 kJ.mol–1.
Qualitatively, the diffusion coefficients determined
for Al(OH)3 – LDH around room temperature are quite
high and very much comparable with bulk water (~30 x
10–10 m2/s). Nevertheless, this comparison ought to be
made with caution since, we have not been able to describe
Figure 3. Temperature dependence of the QENS spectra collected for (a) Al(OH)
3 – LDH, (b) Al
2 O
3 – LDH samples, and (c)
Fe-LDH samples