The Fractal Ratio as a Metric of Nanostructure
Development in Hydrating Cement Paste
R.A. Livingston, W. Bumrongjaroen, and A.J. Allen1
Abstract. It is necessary to have appropriate metrics to quantify the development
of the nanostructure in Portland cement paste. The fractal ratio, calculated from
Small Angle Neutron Scattering (SANS) data, serves as such a metric. It expresses
the proportion of the volume-fractal surface area of calcium-silicate-hydrate gel
(C-S-H) to the surface-fractal surface area. The volume fractal develops in the
scale range from ≈ 5 nm to ≈ 100 nm, and it is associated with the formation of
outer product in the capillary pore space by the through-solution mechanism. The
surface fractal is attributed to the surface structure formed by colloidal particles on
solid substrates such as the Portland cement grains and fly ash particles. The evolution of this ratio over time provides insight into which types of hydration processes are dominant. Applied to study of the hydration of fly ash/Portland cement
mixes at later ages, the fractal ratio method showed that in every case, except two,
there was a reduced hydration rate due to the dilution effect. The two exceptions
involved fly ash fractions with sufficient CaO to generate significant C-S-H gel by
the alkali-activated reaction. In all cases the fractal ratio increased
with time, indicating the production of additional C-S-H through the topochemical
reaction.
1 Introduction
The macroscopic properties of concrete are determined by the development of the
nanometer scale structure in the cement paste. Therefore, in order to investigate
this development it is necessary to have appropriate metrics to quantify the nanostructure. There have been few such metrics available for characterizing cement
R.A. Livingston
Department of Materials Science & Engineering, University of Maryland, College Park,
MD, USA.
W. Bumrongjaroen
Vitreous State Laboratory, Catholic University of America, Washington, DC, USA.
A.J. Allen
Ceramics Division, National Institute of Standards and Technology,
Gaithersburg, MD, USA.
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paste. This has been due partly to the lack of analytical techniques that have
nanometer spatial resolution and also to the disordered structure of the material,
arising from its fractal structure. The metric that has been most used to date is the
specific surface area as measured by nitrogen isothermal adsorption (BET). However, this technique is controversial [1-3]. It has at least two major disadvantages:
the harsh preconditioning of the sample can induce changes to the nanostructure,
and it only provides the total surface area, while as discussed below, there are at
least two distinctly different surface area involved.
More recently Small Angle Neutron Scattering (SANS) has been applied to investigate cement paste microstructure. This technique uses cold neutrons to probe
the paste structure on length scales from 1 nm to >100 nm. This makes it possible
to observe the different types of nanostructure surface directly. In addition, SANS
does not require any pre-conditioning of samples, and it is nondestructive, which
enables repeated measurements on the same sample over time. Finally, it has the
unique capability of being able to hide and reveal individual phases by manipulating the contrast between the paste and the porewater solution. A recent review of
the literature on the application of SANS to investigate Portland cement paste microstructure is given in Allen et al. [4].
Figure 1 is an example of SANS data for a cement/ fly ash mix paste at 3
months of hydration. The horizontal axis is the magnitude of the scattering vector,
Q (where Q = (4π/λ)sinθ, λ is the wavelength, and θ is half of the scattering angle),
10000
BP6 data
-3.4
dΣ/dΩ(cm-1sr-1)
1000
~Q
surface-fractal
100
BP6 model fit
~Q-2.6
volume-fractal
10
1
SANS data with
fit background
scattering removed
0.1
0.01
0.001
Scattering
from gel
particle
-4
Q Porod
scattering
0.01
Q (A)-1
Fig. 1 Plot of SANS data for cement/fly ash paste
0.1
1
The Fractal Ratio as a Metric of Nanostructure Development
103
which is proportional to the reciprocal of the nanostructure length scale. The vertical axis is the differential macroscopic scattering cross-section, dΣ/dΩ, which is a
measure of the intensity of the scattered neutrons. It is essentially a spatial Fourier
transform of the microstructure. The curve shows several distinct regions related
to different features of the C-S-H gel. The plot can then be fitted with a mathematical model that provides several parameters which can be used to characterize
the nanostructure.
In particular, there are two types of fractal structure that develop, defined by different fractal dimensions. A mass- (or volume) fractal develops in the scale range
from ≈ 5 nm to ≈ 100 nm, and has been attributed to a disordered diffusion-limited
aggregate structure composed of 5-nm colloidal C-S-H gel particles. The scattering
arises because there is a neutron scattering contrast between the solid C-S-H gel particles and the pore fluid in the ≈ 5 nm gel pores between them. The volume-fractal
component is associated with the formation of the outer product in the capillary pore
space by the through-solution mechanism during the nucleation and growth period
of the hydration reaction. It is quantified in terms of the surface area of the volumefractal component per unit volume of paste, and it is symbolized by Svf.
The other type of fractal is attributed to the fractally-rough surface structure
produced by colloidal particles on solid substrates such as the Portland cement
grains and fly ash particles. It is referred to as a surface fractal, and it is also
measured in terms of its surface area per unit paste volume, Ssf. While this
structure co-exists with the volume-fractal, the scattering associated with it is only
observable at lower values of Q than the volume-fractal, because it has a steeper
Q-dependence and extends to larger scale lengths, as shown in Fig 1.
Whereas the individual surface areas, Svf and Ssf, can vary stochastically between samples, taking the ratio of Svf to Ssf, can significantly compensate for this
sample to sample random variation. The fractal ratio was first introduced by Allen
and Livingston in 1995 to study the effects of silica fume on Portland cement
paste microstructure during early hydration [5, 6]. More recently, it has been used
to investigate the effects of fly ash morphology and chemical composition on the
developing microstructure of the hydrating cement system at longer hydration
times [7]. Some results of this application of the fractal ratio are described below.
2 Experimental Approach
The addition of fly ash to Portland cement paste produces competing effects: it
contributes C-S-H gel through the pozzolanic and alkali-activated reactions, but
dilutes the C-S-H contribution of the main Portland cement reaction. To investigate these effects, SANS was applied to several density-fractionated samples of a
lignite-type (Ottumwa, IA) and a bituminous-type (Brayton Point, MA) fly ash in
a Portland cement paste with a mass fraction of 20%. The experimental details are
provided in Bumrongjaroen et al. [7]
The SANS measurements were performed at the National Institute of Standards
and Technology (NIST). The SANS instrument, as well as the data acquisition
and analysis methods have been described elsewhere [6]. To study the evolution of
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the cement paste microstructure over time, the SANS measurements were made
after hydration for 7 d, 28 d, and 90 d. By using 3 different configurations of the instrument, an overall Q-range of 0.003 Å-1 to 1 Å-1, or 0.03 nm-1 to 10 nm-1, was covered. This range is sufficient to probe features in the size range from 1 nm to 100
nm. Finally, the two-dimensional data set was reduced to one dimension by circular
averaging to give dΣ/dΩ.
The SANS intensity data were then modeled over various Q ranges using two
different models. In the first model, the Q-4 Porod scattering law terminal slope,
observed at high Q after flat-background subtraction (see Fig. 1), was used to determine the total internal surface area per unit sample volume, ST. The value of ST
was then used with the second model which represents the full fractal system. This
model combines a mass- or volume-fractal scattering term, and a surface-fractal
scattering term along with several other factors related to the colloidal particles in
the gel. It incorporates a significant number (9) of adjustable parameters. However, different terms dominate at different parts of the scattering curve, and hence
only about 3 parameters are needed to determine the fit in any one region. A full
description of the models and the fitting procedures is also given in Bumrongjaroen et al. [7].
3 Results
The fractal ratios for the pastes are plotted as a function of time in Fig. 2. For
comparison the fractal ratio of the pure Portland cement paste (CCRL) is also plotted as a solid line. At a given hydration time, the fractal ratio is determined by the
cumulative Portland cement hydration and pozzolanic action processes that have
occurred up to that time. It can be seen that the values for the Brayton Pt mixes
are all lower than the pure cement control (CCRL). For the Ottumwa samples, the
values for two of the mixes are also lower than the control, but the other two, OT5
and OT6, exceed it. These two fractions have the highest potential for the alkaliactivated (AA) or self-cementing reaction, based on their calcium contents. Conversely, none of the Brayton Pt fractions have significant AA potential. Therefore
the AA reaction of the fly ashes appears to be an important factor in the development of the microstructure in the mixes. The reduced values of the fractal ratios
of the other samples relative to the pure cement is an indication of the dilution effect caused by replacing some of the cement with fly ash.
In every case, the fractal ratio increases with time. Examination of the individual curves reveals that this trend is the result of both an increase in Svf and a decrease in Ssf. Since the total fractal surface area, ST, is essentially constant, this
implies that the surface fractal is being transformed into the volume fractal.
By the (hydration) time these measurements were made, the formation of the
outer product was essentially complete, and for both the cement and the fly ash any
continuing hydration would be due primarily to the topochemical reaction. As the
reaction progresses, the reacted zone becomes thicker, more closely resembling a
volume fractal. However, the rate of reaction would be lower for the fly ash particles
than for the Portland cement because of the requirement for dissolution of Ca from
The Fractal Ratio as a Metric of Nanostructure Development
105
Fig. 2 Time dependence of the fractal ratio (Svf/Ssf) for Ottumwa (left) and Brayton Pt
(right) fly ash cement pastes. (Vertical bars are estimated standard deviations.)
calcium hydroxide (CH) through the surrounding gel to the reactive surface of the
fly ash. Thus, the overall rate of reaction of a mix would be reduced relative to the
pure cement CCRL. The only exceptions are OT5 and OT6, which have enhanced
rates of reaction. However, these fractions also can generate significant amounts
of C-S-H by AA, which does not involve Ca diffusion.
Another consideration is the effect of the particle size distribution (PSD). Portland cement grains are typically tens of micrometers in diameter. The PSDs of all
of the fly ash fractions contain many finer particles, in some cases smaller by an
order of magnitude. Examination of the details of the individual SANS curves
reveals that most of the fly ash/cement states start off at 7 d with Ssf larger than the
control. This can be explained by the fact they would have larger total surface
areas because of the presence of fine fly ash particle sizes. In turn, this provides
more substrate for the C-S-H gel comprising the surface fractal. Conversely, the
Svf values for the mixes are all initially lower than the control, due to the reduction
in the production of outer product because of the dilution effect. The combination
of these two effects produces an initial ratio that would be lower than that of the
pure cement paste.
4 Conclusions
It is necessary to have appropriate metrics to quantify the development of
nanostructure in cement paste. SANS provides several advantages over nitrogen
BET surface area measurements for this purpose, including the ability to discriminate between different types of surfaces, avoidance of harsh pre-conditioning of
samples, and ability to make repeated measurements on the same sample over time.
SANS yields several related to the nanostructure. In particular, the fractal ratio
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expresses the proportion of the volume fractal surface area to the surface fractal
surface area. The evolution of this ratio over time provides insight into which
types of hydration processes are dominant.
Applied to study of the hydration of fly ash/Portland cement mixes at later ages
(7 d, onwards) the fractal ratio method showed that in every case, except two,
there was a reduction in the rate of reaction due to the dilution effect. The two
exceptions involved fly ash fractions with sufficient CaO to generate significant
C-S-H gel by the alkali-activated reaction. In all cases the fractal ratio increased
with time, indicating the production of additional C-S-H through the topochemical
reaction.
Concerning industrial applications of these results, neutron scattering methods
are not practical for use on a routine basis because they have to be done at
specialized facilities. However, the SANS fractal ratio approach provides insights
that can be used to develop reliable routine methods for characterizing fly ash
reactivity.
Acknowledgments. The authors gratefully acknowledge the participation of Dan Neumann
in numerous discussions about the fractal ratio. We acknowledge the support of the
National Institute of Standards and Technology, U.S. Department of Commerce, in providing the neutron research facilities used in this work.
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