Supporting Information for the paper entitled:
“Evidence
of a fractal
percolating network during geopolymerisation”
by J. Rouyer and A. Poulesquen
I - Theoretical background
1. Oscillatory Rheology (OR)
Dynamic rheology is an approach which is widely used to measure viscoelastic
moduli. Taking a fixed pulsation 𝜔0 , a sinusoidal monochromatic strain is applied to
the sample under investigation and the resulting harmonic stress response is recorded.
The stress 𝜎 and the strain 𝛾 are expressed as follows in the linear regime:
𝛾(𝑡) = 𝛾0 sin(𝜔0 𝑡)
{
𝜎(𝑡) = 𝜎0 sin(𝜔0 𝑡 + 𝛿)
(1)
where 𝜎0 and 𝛾0 are the stress and the strain magnitude, respectively; the so-called loss
angle 𝛿 is the loss angle between the strain and stress signals. The viscoelastic moduli
𝐺′ and 𝐺′′, are classically defined by:
𝜎0
cos 𝛿
𝛾0
{
𝜎0
𝐺 ′′ (𝜔0 ) =
sin 𝛿
𝛾0
𝐺 ′ (𝜔0 ) =
(2)
Stress and strain magnitude and the loss angle can either be calculated directly from
the temporal signals or obtained by performing calculations in the angular frequency
domain. 𝜎0 and 𝛾0 are the spectral weight of the strain and stress Fourier transforms at
the given pulsation 𝜔0 . Likewise, the loss angle can be obtained by calculating the
difference between the phase angles in the strain and stress phases. Calculations in the
temporal and angular frequency domains yield strictly identical results.
When dynamic rheology methods are extended to a multi-decade angular frequency
range, it becomes a powerful mean of describing the rheological regime and the
mechanisms occurring between the components of the material under investigation. By
varying the angular frequency within the scope of the rheometer bandwidth, it is
possible to draw up the viscoelastic spectrum of the sample. Elastic versus viscous
moduli evolution can be used to describe the rheological behavior of the material using
well-known criteria such as the slope values, the parallelism or the crossover between
the curves of the moduli1; in this way, the evolution of the viscoelastic state, the state
of the gel and the relaxation times can be determined.
Successive iterations of several monochromatic angular frequencies within the useful
range can be performed during a relatively long period of time, depending on the
degree of refinement required. This duration will be at least equal to 182 minutes; for
example, in the case of four points per decade at angular frequencies ranging from 0.01
to 100 rad/s and a 10-cycle strain sine per angular frequency. In order to obtain an
accurate viscoelastic spectrum, the sample must show stable properties throughout this
time. Dedicated controlled environmental tools such as thermal devices, solvent traps
and convection ovens provide helpful means in order to maintain a consistent steady
state. However, this procedure is not at all suitable when spontaneous chemical
reactions or structural adjustment processes are liable to occur. More specifically, the
typical time taken by internal changes to occur has to be infinite in comparison with
the duration of the experiment. In the case of evolving materials, another means of
approaching this problem could consist of changing the mathematical formulation of
the harmonic strain load.
2. Time-Frequency-Resolved Rheology (TF2R)
This method, which was firstly described by Ghiringelli et al2, consists in using a
relatively short strain load, which contains an extended continuous angular frequency
range. It differs from the multi-wave technique3,4, where strain function is obtained by
summating a few monochromatic sine waves, resulting in a sparse spectrum (i.e. one
pair of parameters at each monochromatic frequency). The main advantage of the
TF2R method is that it can be used to measure a continuous spectrum in a multidecade frequency range experimentally in a relatively short time.
In order to illustrate how this approach is used, the frequency content of some strain
functions, which are commonly used in rheology, was studied from the signal
processing point of view. A peak holding experiment can be said to consist
theoretically of a Dirac function. By definition, the frequency content of this punctate
function is homogeneous in the Fourier domain5, as shown in Figs. 1 and 2, which
present the temporal and frequency curves, respectively. From the practical point of
view, these strains are not compatible with the experimental constraints, which involve
sampling limitations (maximum sampling frequency 𝑓𝑠 =1000 Hz) and quite-long spinup times due to the mechanical and geometrical inertia. To overcome these limitations,
a suitable square function or Gaussian function has to be chosen in order to smooth out
the beginning and the end of the rotation (as in the example presented in Fig. 1). The
theoretical advantage of applying an extremely short strain load corresponding to an
infinite frequency range amounts in practice to having a fairly narrow bandwidth. As
illustrated in Fig. 2, the square function and Gaussian function spectra give a
homogeneous bandwidth up to an angular frequency of 3 rad/s. Otherwise, higher
angular frequencies cannot be reached with these functions. In other words, the
experimental device acts like a low-pass filter when dealing with extremely short
strains.
In order to span all the frequencies covered by the rheometer bandwidth in a relatively
short time, the use of the chirp function was found to provide the best solution. Here
we focus on a four-decade bandwidth ranging from angular frequencies of 0.01 to 100
rad/s. A chirp is a pseudo-harmonic function obtained by applying a continuous
frequency sweep with time. In order to maximize the signal-to-noise ratio at the lower
frequencies, an exponential frequency sweep was used, which was expressed as
follows:
𝜔
𝛾(𝑡) = 𝛾0 sin ( 𝐾1 (𝑒 𝐾𝑡 − 1)),
with 𝐾 =
𝜔 ∈ [𝜔1 , 𝜔2 ],
𝜔
ln 2
(3)
𝜔1
𝑇1
The angular frequency sweep, which starts from 𝜔1 to 𝜔2 . 𝑇1 , corresponding to the
duration of the sweep, is classically equal to the period of the lowest angular
frequency, and hence 𝑇1 = 2𝜋 ⁄𝜔1 , where 𝐾 is the sweeping rate of the strain and 𝛾0 is
the magnitude of the strain. An example of a 𝑇1 = 628 seconds chirp is shown in Fig. 3.
The frequency distribution is continuous over the bandwidth of interest, as shown in
Fig. 2. The spectral weight of the chirp is always greater than the other loads involved.
It is worth noting that the magnitude of the strain is the same in all the temporal
functions. The energy contributed by each frequency is greater than with the other
short loads tested such as Dirac, square and Gaussian functions.
However, a typical noisy oscillation appears in the chirp spectrum at the upper limit of
the high frequencies. In order to obtain an unperturbed bandwidth, the upper boundary
is moved to 𝜔𝑢 = 400 rad/s. In the same way, the lowest angular frequency is
decreased to 𝜔𝑑 = 10-3 rad/s. The spectrum of the extended bandwidth is plotted in
Fig. 2; at the lowest frequencies involved, the spectral weights have been maximized
and the bandwidth of interest is devoid of undesirable artifacts. It can be noted that
expanding the bandwidth (keeping the same duration) firstly results in a higher
sweeping rate and secondly, in a slight overall decrease in the spectral weight.
In the general case of linear viscoelastic materials, the stress response is defined by the
convolution product (⊗) of the complex shear modulus and the strain velocity:
𝜎(𝑡) = 𝐺(𝑡) ⊗ 𝛾̇ (𝑡)
(4)
𝐺(𝑡) is the so-called relaxation function of the sample. The use of the convolution
product makes it possible to shift from the temporal domain to the Fourier domain (~),
and the previous relation becomes:
𝜎̃(𝜔) = 𝑖𝜔𝐺̃ (𝜔)𝛾̃(𝜔)
(5)
The method used to model the viscoelastic moduli, which has been presented in detail
in [23], gives the following two expressions:
𝜎̃(𝜔)
]
𝛾̃(𝜔)
𝜎̃(𝜔)
𝐺 ′′ (𝜔) = ℑ [
]
𝛾̃(𝜔)
{
𝐺 ′ (𝜔) = ℜ [
(6)
A snapshot of the viscoelastic behavior of the material under investigation can
therefore be obtained by calculating the stress-to-strain ratio in the previously defined
bounded Fourier domain.
3. Mutation number in the case of TF2R
As mentioned in previous section, the possibility to obtain meaningful rheological
measurements depends on the properties consistency of the sample during the time
required by the measuring apparatus. This experimental prerequisite makes necessary
to adapt the duration 𝑇 of the chirp to the changes in the properties of the material. The
mutation number of
6,7
is therefore introduced in order to predict the changes liable to
occur during the pseudo-harmonic loading. In the present case, both moduli will result
in the following two dimensionless numbers (𝑁𝑚𝑢 ):
𝑁 ′ 𝑚𝑢 (𝑡) =
𝑇 𝜕𝐺 ′
𝑇 𝜕𝐺′′
and 𝑁 ′′ 𝑚𝑢 (𝑡) =
′
𝐺 𝜕𝑡
𝐺′′ 𝜕𝑡
(7)
The smaller the mutation number is, the more negligible the mutations will be in the
analysis of the experimental results. Negligible non-linear effects were observed with
small mutation numbers2, and the moduli were accurately determined within the
duration 𝑇 as long as 𝑁𝑚𝑢 was smaller than 15% 6; if this finding is confirmed, the
sample can be assumed to be quasi-stable during the experiment. One rad/s
monochromatic paste setting kinetics (Fig. 1 in the main text) were therefore used to
calculate the mutation number during a period of 𝑇1 = 628 seconds. It can be seen from
′
′′
Fig. 5 that the patterns of evolution of 𝑁𝑚𝑢
and 𝑁𝑚𝑢
were very similar: only the
mutation number of the elastic modulus was therefore taken into account. The highest
value recommended – 0.15 – was greatly exceeded during the process, and the chirp
initially included in equation (3) therefore turned out to be unsuitable. In order to make
the length of the chirp comply with the recommended value, the duration 𝑇1 was
decreased whenever the mutation number was greater than 0.15. Two other chirp
durations were therefore chosen: that of the shortest chirp observed, and an
intermediate duration, which were defined as follows: 𝑇2 = 368 s and 𝑇3 = 108 s. An
experimental procedure integrating the three appropriate durations was defined at
specific times during the setting process, as shown in Fig. 5, where they form a
stepwise pattern.
With a view to investigate the effects of using shorter chirp durations in the TF2R
method, preliminary tests were conducted on PDMS samples (Annex A in the main
text). These tests led us to reduce the frequency range of the viscoelastic spectrum,
depending on the chirp duration. The spectral range adopted was therefore [0.01, 100]
rad/s in the case of 𝑇1 , [0.05, 100] rad/s in that of 𝑇2 and [0.1, 100] rad/s in that of 𝑇3 .
In addition, the possibility of obtaining reliable moduli at frequencies of less than 0.1
rad/s depends not only on having sufficiently long chirp durations, but even more on
the order of magnitude of the torque measurements2. The low sensitivity of the torque
transducer can make difficult to obtain the theoretical full-bandwidth characterization
expected.
II - Testing the TF2R method on a PDMS sample
PDMS is a silicon-based organic polymer, which was used as a viscoelastic test
material since it is non-evolving and shows excellent long term stability. The
viscoelastic spectrum was measured at angular frequencies ranging from 0.01 to 100
rad/s using oscillatory rheological methods at 25°C and a strain magnitude equal to 101%. Four points per decade were determined. Measurements were performed using
plate-plate geometry with a diameter of 2.25 cm. TF2R was then carried out with the
exponential chirp strain loading at angular frequencies ranging from 𝜔𝑙 = 0.001 rad/s
to 𝜔𝑢 = 400 rad/s and 𝛾0 = 10-1 %. The superimposed viscoelastic spectra obtained
using the TF2R method and standard dynamic rheological methods are presented in
Fig. 4 in the case of the three chirp durations (𝑇1 = 628 s, 𝑇2 = 368 s and 𝑇3 = 108 s).
The relative error 𝛿𝐺 between the standard method and the TF2R method is entered in
Table 1. The discrete points in the OR were accurately matched at angular frequencies
greater than 0.3 rad/s with a relative error of less than 4 % with all the chirp durations
tested; the crossover moduli could therefore be accurately determined.
As shown in Table 1 and Fig.6, the longer the chirp duration is, the more accurately
this technique can be expected to give fine moduli values at low frequencies. The
viscous modulus was accurately assessed in general, but a limitation emerged as
regards the description of the elastic modulus at the lower frequencies tested, when the
measurements were disturbed by strong oscillations (Fig.6 for 𝑇3 ). This undesirable
effect can be attributed to the small number of sine periods present in the lower
pulsation range: the lack of redundancy reduces the accuracy of the TF2R method at
the lower frequencies.
It was therefore decided to reduce the bandwidth, depending on the chirp durations
used; [10-2; 102] rad/s in the case of 𝑇1 , [5.10-2; 102] rad/s in that of 𝑇2 and [10-1; 102]
rad/s in that of 𝑇3 . Particular attention will therefore have to be paid to how the
spectrum is interpreted in the lower angular frequency range. These preliminary
experiments provide a useful framework for applying the TF2R method to
geopolymerization processes.
Table 1 Relative error 𝜹𝑮 between the standard method and the TF2R method with the three
chirp durations (𝑻𝟏 = 628 s, 𝑻𝟐 = 368 s and 𝑻𝟑 = 108 s) to determine the viscoelastic moduli.
Pulsation
0.01
0.03
0.13
0.32
0.80
2.00
5.02
12.62
31.70
79.62
𝛅𝑮′ for 𝑻𝟑 [%]
(-)
(-)
-13.56
2.08
1.44
0.63
3.08
3.00
2.87
2.40
𝛅𝑮′′ for 𝑻𝟑 [%]
(-)
-30.42
0.22
1.58
3.59
2.17
2.41
2.58
3.46
2.16
𝛅𝑮′ for 𝑻𝟐 [%]
(-)
30.77
2.92
0.37
3.20
1.15
2.60
2.42
2.44
1.57
𝛅𝑮′′ for 𝑻𝟐 [%]
-8.75
3.45
1.17
2.03
2.41
2.43
2.95
3.49
3.95
5.04
𝛅𝑮′ for 𝑻𝟏 [%]
-86.89
-1.33
0.16
2.44
3.23
1.36
1.93
2.18
2.47
3.09
𝛅𝑮′′ for 𝑻𝟏 [%]
-0.93
-0.69
1.18
2.05
2.57
1.89
2.90
3.16
3.82
1.42
[rad/s]
Relative errors were calculated using: 𝛿𝐺 = 1 − 𝐺𝑇𝐹2𝑅 /𝐺𝑂𝑅 – The symbol (-) denotes aberrant values in the TF2R results. Modulus
values tended to diverge at low frequencies, as shown in Fig.A.
Supporting information references
1.
Macosko CW. Linear Viscoelasticity. In: Rheology: Principles, Measurements, and
Applications. Wiley-VCH; 1994:109-126.
2.
Ghiringhelli E, Roux D, Bleses D, Galliard H, Caton F. Optimal fourier rheometry.
Rheol. Acta 2012;51(5):413-420.
3.
Ma J, Lin Y, Chen X, Zhao B, Zhang J. Flow behavior, thixotropy and dynamical
viscoelasticity of sodium alginate aqueous solutions. Food Hydrocoll. 2014;38(0):119128.
4.
Nachbaur L, Mutin JC, Nonat A, Choplin L. Dynamic mode rheology of cement and
tricalcium silicate pastes from mixing to setting. Cem. Concr. Res. 2001;31(2):183192.
5.
Smith DC. An introduction to distribution theory for signals analysis.: Part II. The
convolution. Digit. Signal Process. 2006;16(4):419-444.
6.
Winter HH, Morganelli P, Chambon F. Stoichiometry effects on rheology of model
polyurethanes at the gel point. Macromolecules 1988;21(2):532-535.
7.
Mours M, Winter HH. Time-resolved rheometry. Rheol. Acta 1994;33(5):385-397.
doi:10.1007/bf00366581.
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Winter HH, Morganelli P, Chambon F. Stoichiometry effects on rheology of model
polyurethanes at the gel point. Macromolecules. 1988;21(2):532-535.
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Mours M, Winter HH. Time-resolved rheometry. Rheol Acta. 1994;33(5):385-397.
Supporting information figures
Fig. 1 Normalized graphs of the Dirac delta function (black dotted circle), the 1-second wide square
function and the 0.425-standard deviation Gaussian function.
Fig. 2 Spectral curve of the Dirac delta, the square function, the Gaussian function and the
exponential chirp functions used to illustrate the interest in the TF2R method. These curves
correspond to the module of the Fourier transform of the normalized function. This qualitative
comparison shows that the chirp gives a large spectral weight within a wide pulsation range.
Fig. 3 Representation of the exponential chirp defined with the following parameters: 𝜔1 =0.01 rad/s
and 𝜔2 =100 rad/s and 𝑇1 = 628 seconds. The corresponding spectral curve is shown in Fig. 2 (thin
continuous line).
Fig. 4 Evolution of the mutation numbers calculated with the 1-rad/s viscoelastic moduli measured
during the setting process (Fig.4) and calculated using equation (7). The values plotted correspond to
an experimental time of T = 628 second, which is the duration of the exponential chirp defined by
equation (3) and shown in Fig.3. Mutation numbers have to be less than 0.15 in order to obtain
reliable measurements within this time; this figure illustrates the situation where the experimental
duration of the chirp had to be decreased.
Fig. 5 Analysis of the mutation number as a function of the moduli led to adapting the chirp
duration to obtain a mutation number smaller than 0.15. First the shorter duration was found
to be equal to 108 seconds (a median-long duration). The chirp duration was then adapted to
the threshold meaningful mutation number, which was set at 15%. This fig. gives the chirp
durations chosen during the geopolymerization process, depending on the increase in the
moduli.
Fig. 6 Comparison between the standard OR and the TF2R methods. A 108 (a), 368 (b) and 628
seconds (c) chirp duration was employed to measure the continuous viscoelastic spectra by TF2R
(plain and dashed lines). The OR determines moduli iteratively at discrete pulsation (squares and
circles) and takes about 182 minutes to obtain 4 points per decade within a four decade pulsation
range.