Formation of calcium sulfate through the aggregation of sub-3 nm anhydrous
primary species
Tomasz M. Stawski1,2*, Alexander E.S. van Driessche3,4**, Mercedes Ossorio4,
Juan Diego Rodriguez-Blanco5, Liane G. Benning1,2
1
Cohen Geochemistry, School of Earth and Environment, University of Leeds, LS2
9JT, Leeds, UK;
2
German Research Centre for Geosciences, GFZ, Interface
Geochemistry Section, 14473, Potsdam, Germany; 3Structural Biology Brussels, Vrije
Universiteit Brussel, Pleinlaan 2, 1050, Brussels, Belgium; 4Laboratorio de Estudios
Cristalográficos, Instituto Andaluz de Ciencias de la Tierra, CSIC-UGR, Avenida de
las Palmeras 4, E-18100, Armilla, Granada, Spain; 5NanoGeoScience, Nano-Science
Center; Department of Chemistry; University of Copenhagen, H.C. Øersted Institute,
C Bygn, Universitetsparken 5, DK 2100, Copenhagen, Denmark.
(Corresponding authors: *[email protected], **[email protected])
(Keywords: calcium sulfate, CaSO4; gypsum, bassanite, anhydrite; nucleation,
aggregation, self-assembly, growth; in situ small angle X-ray scattering, SAXS)
(Acknowledgments: This work was made possible by a Marie Curie grant from the
European Commission in the framework of the MINSC ITN (Initial Training
Research network), project number 290040. Access to the Diamond Light Source Ltd
(UK) for beamtime at beamline I22 was possible through a direct access grant to L.G.
Benning. Beamline staff of I22, A. J. Smith and N. J. Terrill, are thanked for their
competent support and advice.)
(Supporting Information Available:
S1. Saturation indices (SI) of gypsum, bassanite and anhydrite; S2. Mathematical
expressions for SAXS data modeling; S3. Monodisperse versus polydisperse hardsphere structure factor; S4. Structure factor during the transition between stages II and
III; S5. Evolution in the morphology of CaSO4 precipitates after 5400 seconds; S6.
Selected physicochemical properties of CaSO4·xH2O polymorphs; S7. Selected
structural aspects of the various CaSO4·xH2O polymorphs.)
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Abstract
The formation of crystalline calcium sulfate (CaSO4·xH2O) polymorphs from
aqueous solutions is assumed to occur via a single-step process following the classical
nucleation paradigm. However, although recent research contradicts this classical
picture and indicates that CaSO4·2H2O forms at room temperature through multiple
steps at different length and time-scales, these steps have so far not been quantified.
By using in situ and fast time-resolved small angle X-ray scattering (SAXS), we
demonstrate that the nucleation and growth of CaSO4·2H2O involves at the very
initial stages the formation of well-defined, primary species of < 3 nm in length (stage
I). Stage II of the reaction is characterized by the arrangement of these primary
species into domains, while in stage III these domains condense into larger
aggregates. Based on volume fractions and electron density considerations we
propose that the fast forming primary species from supersaturated aqueous CaSO4
solutions are composed of anhydrous Ca-SO4-cores. The first three stages of
nucleation and aggregation of the primary species are followed by a final stage (stage
IV), where the primary species grow within the aggregates, and eventually transform
into gypsum (CaSO4·2H2O). This final stage was also confirmed through
simultaneously collected wide-angle scattering (diffraction, WAXS) data, which
clearly show the growth of gypsum during stage IV only. Our results demonstrate that
CaSO4 formation is driven by the nucleation and aggregation of well-defined
anhydrous Ca-SO4-cores that transform through hydration into gypsum through a
complex nucleation and growth pathway.
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Introduction
Calcium sulfate has three crystalline phases with various hydration levels1:
CaSO4·2H2O (gypsum), CaSO4·0.5H2O (bassanite), and CaSO4 (anhydrite). Large
natural (often evaporitic) deposits of gypsum and anhydrite exist on Earth2, while
bassanite deposits3, 4 are rarely found due to its propensity to transform to gypsum.
Recently, the presence of significant amounts of bassanite and gypsum have also been
confirmed on the surface of Mars5, 6. The utility of CaSO4·xH2O phases in various
industrial processes is exemplified by the tremendous quantities (~1011 kg/annum
globally) used in the construction industry to produce the so-called plaster of Paris7-9.
On the other hand, the formation of CaSO4·xH2O mineral scales (e.g., precipitates on
pipe walls from supersaturated process waters) is a major problem in desalination
plants, wastewater treatment facilities, and oil and gas exploration.
Considering the relevance of this system, it does not come as a surprise that a
plethora of studies focused on elucidating the crystallization mechanism (i.e.,
nucleation and growth) of calcium sulfate polymorphse.g.
10-13
. Traditionally most
studies assume that the formation of crystalline CaSO4 from aqueous solutions occurs
via a single-step process (i.e., following the classical nucleation paradigm that the
stable phase is formed directly), with gypsum being the stable crystalline phase at
room temperature. However, recent research contradicts this classical picture and
extensive experimental data indicates that the precipitation of calcium sulfate at room
temperature involves multiple steps taking place at different length and time-scales
with gypsum as the final product14-17.
These studies suggested the existence of various intermediate phase(s), e.g.,
amorphous and/or nano-crystalline precursors that are requisite precursors on the
pathway to gypsum formation. Such phases were detected using primarily (cryo-)
transmission electron microscopy (TEM) and infrared or Raman spectroscopies.
Jones18, based on infrared spectroscopy, suggested that gypsum forms from a longlived disordered precursor phase through a process in which water molecules moved
from disordered positions to those in which water could be associated with the
formation of crystalline gypsum, and possibly bassanite. Wang et al.14 showed by
TEM imaging of time-resolved quenched samples that the formation of gypsum was
preceded by bassanite, but also inferred an amorphous calcium sulfate as the first
phase to nucleate from solution. In our previous work16, using high-resolution (HR)
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TEM analysis of time resolved cryo-quenched samples, we documented the
homogeneous precipitation of bassanite nano-particles (5 to > 10 nm), which selfassembled into elongated aggregates and eventually transformed into gypsum. Our
fast quenching results implied that the first, and only, precursor was in fact not
amorphous, but nano-crystalline with a lattice spacing matching that of bassanite.
The primary reason for these ambiguities originates from the fact that it is
extremely challenging to target the elusive early stages of solids’ formation. Ideally, a
characterization technique (or combination of several) should provide us with (fast)
time-resolved in situ information at the nano-scale, without altering the crystallization
pathway and without the need to quench the solution to image particles. The recent
studies described above mostly relied on HR-TEM for imaging and analyses of the
pre-gypsum solid phase stages. This approach offers near-atomic resolution in the
best of cases but it presents the clear disadvantage of imaging any resulting particles
ex situ, after quenching, and the associated problems working under high vacuum
and/or of possible beam damage (and resulting artefacts). Current best available
strategies designed to reduce undesirable effects include cryo-quenching or direct
analysis of cryo-vitrified samples at different stages, yet even these suffer from
various artefacts14, 16, 17. Hence, true in situ and time-resolved characterization was not
achieved in any of the above approaches and thus caused the present day ambiguity
about the formation pathway(s) of calcium sulfate polymorphs.
To overcome this impasse we employed a genuine in situ and time-resolved
method to characterize all stages and species forming in solution in order to obtain a
complete picture of the calcium sulfate crystallization pathway. We performed
synchrotron-based
X-ray
small
and
wide-angle
scattering
experiments
(SAXS/WAXS) and followed the formation of solid phases from supersaturated
CaSO4 solutions from the very earliest stages of nucleation to the final crystalline
gypsum products. We show that instantly after solution mixing (i.e., reaching the
desired supersaturation levels) at all conditions relatively stable, well-defined sub-3
nm in length species (also referred to as scatterers and entities) formed. These entities
constitute the primary building blocks, which through self-assembly and aggregation
finally transform to gypsum. Based on these new results we propose a 4-stage
pathway for calcium sulfate formation from solution.
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Experimental Methods
CaSO4 was synthesized by reacting equimolar aqueous solutions of
CaCl2·2H2O (pure, Sigma) and Na2SO4 (> 99%, Sigma) at final concentrations of the
mixed solutions of 50, 75, 100, 150 mmol/L at T = 21 °C, and also for 50 mmol/L at
T = 12, 30, 40 °C. Prior to mixing, all solutions were equilibrated at the desired
reaction temperatures and filtered through 0.2 µm pore size polycarbonate filters to
remove possible impurities. The saturation indices with respect to the different bulk
calcium sulfate hydrates and at the salinities / ionic strengths (including contributions
for the Na+ and Cl- counter-ions) were calculated with PHREEQC19 using the LLNL
database (full details and info provided in Supplementary Information S1). All CaSO4
formation reactions were performed in a 200 mL temperature-stabilized glass reactor.
The reacting solutions were continuously stirred at 350 rpm, and circulated through a
custom-built PEEK flow-through cell with an embedded quartz capillary (ID 1.5 mm,
wall thickness ~10 µm) using a peristaltic pump (Gilson MiniPuls 3, flow ~10
mL/second). Typically an experiment started with 40 mL of a temperature-stabilized
CaCl2 aqueous solution inside the reactor. This solution was circulated through the
capillary cell while scattering patterns were collected continuously as described
below. CaSO4 formation reactions were initiated remotely through the injection of 40
mL of a temperature-stabilized Na2SO4 aqueous solution. Fast injection and mixing
(within 15 seconds) of the two solutions was achieved with the use of the fastinjection mode of a stopped-flow system (Bio-Logic SFM-400). Depending on
reaction conditions (supersaturation or temperature) reactions were followed up to 4
hours.
All in situ and time-resolved SAXS/WAXS measurements were carried out at
beamline I22 of the Diamond Light Source Ltd (UK). Experiments were performed
using a monochromatic X-ray beam at 12.4 keV and two-dimensional scattered
intensities were collected at small-angles with a Dectris Pilatus 2M (2D large area
pixel-array detector20). Transmission was measured by means of a photodiode
installed in the beam-stop of the SAXS detector. A sample-to-detector distance of
4.22 m allowed for a usable q-range of 0.1 < q < 3.8 nm-1. The scattering-range at
small-angles was calibrated against silver behenate21 and dry collagen standards22. For
the wide-angle measurements we used a HOTWAXS detector (a photo-counting 1D
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microstrip gas chamber detector23). The WAXS detector was calibrated with synthetic
and highly crystalline silicon (NIST SRM 640C), and with commercial gypsum and
bassanite powders (Sigma Aldrich).
The reactions were followed in situ from the very early stages and up to the
point when a final crystalline phase was fully developed and no more changes in the
WAXS peak intensities were observed. Furthermore, for each experiment we also
measured a series of backgrounds and reference samples including: the empty
capillary cell, cell filled with water, and cell filled with the initial, unmixed CaCl2 and
Na2SO4 solutions at the various used concentrations and temperatures. In all
simultaneous SAXS/WAXS measurements, the acquisition time per frame varied
between experiments (from 1 to 30 seconds/frame) and this time frame was based on
previously off-line tested reaction times for the various conditions. The triggering of
SAXS and WAXS frame acquisition was synchronized between the two detectors, so
that a given frame in SAXS corresponded to the one in WAXS. Most of the recorded
2D SAXS patterns were found to be independent of the in-plane azimuthal angle with
respect to the detector (i.e., scattering patterns where circular in shape), showing that
the investigated systems could be considered isotropic. In those patterns, pixels
corresponding to similar q regardless of their azimuthal angle where averaged
together, and hence the 2D patterns were reduced to 1D curves. In several cases
dependence between the scattering intensity and the in-plane azimuthal angle was
observed (i.e., the scattering patterns were elliptical in shape). This indicated
preferred orientation of the scatterers in the investigated samples. Therefore, selected
angle-dependent 1D scattering curves were obtained by averaging of pixels with
similar q and limited to ca. +/- 3° angle off the direction indicated by the chosen
azimuthal angle: the equatorial and meridional directions of the elliptical 2D patterns.
SAXS data processing and reduction included primarily masking of undesired pixels,
normalizations and correction for transmission, background subtraction and data
integration to 1D. These steps were performed using the Data Analysis WorkbeNch
(DAWN) software package (v. 1.3 & 1.4) according to I22 guidelines24.
For WAXS data, in order to increase the signal-to-noise ratio, the collected diffraction
patterns were averaged together maintaining the same proportion of added frames for
the total course of the experiment for each dataset, thus allowing for more accurate
characterization of the potential calcium sulfate phases present in solution. The in situ
diffraction indicated after a long induction time only the presence of gypsum. The
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time-resolved WAXS patterns were fitted using XFit-Koalariet25, which allowed us to
extract the degree of crystallization, α, over the course of the formation of gypsum .
Results
General analysis of the time-resolved scattering patterns
We investigated the formation of solid CaSO4 phases from various solutions
with concentrations of up to 150 mmol/L and at temperatures between 12 and 40 °C.
In all cases changes in SAXS and WAXS intensities and shapes of the patterns were
observed over time and these changes were correlated with supersaturation levels and
temperature (Table 1).
Table 1. Overview of physicochemical conditions and CaSO4 development stages observed in our in situ and timeresolved SAXS/WAXS experiments. The time-periods listed for WAXS correspond to the induction times at
which the intensity of the (020) diffraction peak of gypsum were above background. Note the change in the time
length/frame between experiments; these time frames and total experimental lengths were chosen based on our
previous work16 and match the increase in reactions rates at the higher temperatures and higher supersaturations.
50
75
100
150
[CaSO4] [mmol/L]
[CaSO4] [mmol/L]
[CaSO4] [mmol/L]
[CaSO4] [mmol/L]
1 s/frame,
600 frames
SAXS: stages III-IV
WAXS: ~140 s
1 s/frame,
600 frames
SAXS: stages III-IV
WAXS: ~60 s
1 s/frame,
600 frames
SAXS: stages III-IV
WAXS: ~40 s
12 °C
30 s/frame,
480 frames
SAXS: stages I-IV
WAXS: ~1500 s
21 °C
30 s/frame,
400 frames
SAXS: stages III-IV
WAXS: ~750 s
30 °C
15 s/frame,
500 frames
SAXS: stages II-IV
WAXS: ~650 s
40 °C
15 s/frame,
240 frames
SAXS: stages II-IV
WAXS: ~400 s
The initial analysis of the time-resolved SAXS patterns indicated that the scattering
features for all experiments evolved similarly, regardless of experimental conditions.
Thus, for simplicity, below we discuss the changes in the structural information
contained within the scattering patterns from the 50 mmol/L CaSO4 experiments
measured at 12 °C. At this temperature, and supersaturation, the reaction kinetics
were relatively slow and both the SAXS and WAXS patterns were collected at a time
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resolution of 30 seconds/frame and for a total time length of 4 hours. This allowed us
to capture all details of the reaction from the earliest stages to the final products.
Consequently, this experiment was used to develop and validate the model for the
entire reaction pathway.
Fig. 1A shows the overview of the scattering patterns for the first 5400
seconds (1.5 hours) of the 50 mmol/L experiment. In the initial part of the reaction the
scattering intensity increased by several orders of magnitude. Although the
experiment lasted for nearly 4 hours (i.e., 14370 seconds), after ~5400 seconds the
overall SAXS signal remained relatively constant increasing during the final 2.5 hours
of reaction by less than 5%. From this data four characteristic stages, which are
described in detail below, could be distinguished:
(I)
30 to 120 seconds: formation of small primary entities/scatterers as evidenced
through the change in I(q) ∝ q-1 for q > 1 nm-1 and I(q) ∝ q0 for q < 1 nm-1;
(II)
150 to 390 seconds: development of a structure factor, manifested by the
decrease of intensity at q < 0.3 nm-1, indicating interactions between the
previously formed primary scatterers;
(III) 420 to 1500 seconds (and up to 5400 seconds): formation and development of
large scattering features evidenced through the increase in intensity at q < 1.0
nm-1; this change followed a dependence of I(q) ∝ q-3
> -a > -4
; the overall
intensity and shape of the curves at q > 1.0 nm-1 corresponding to the scattering
from the primary species remained relatively unchanged at this stage;
(IV) Growth of the primary species after ~1500 seconds manifesting itself at q > 1
nm-1 by the shift of the scattering curves towards higher q values and the
gradual decrease in intensity towards I(q) ∝ q-4.
For the experiments at higher supersaturations (75 – 150 mmol/L) and higher
temperatures (21, 30 and 40 °C), due to the increased rate of the reactions16, it was
not always possible to distinguish all of the above described 4 stages (Table 1).
Nevertheless, the start of the reaction in the experiment at the lowest supersaturation
(12 °C and 50 mmol/L; Fig. 1A and bottom pattern in Fig. 1B) could be compared
with selected patterns at similar time frames from experiments at a higher temperature
(21 °C) and higher supersaturations (50 – 100 mmol/L; Fig. 1B). This comparison
(Fig. 1B) reveals the differences in the early stages of the formation of the initial solid
entities through the fact that each curve shows different characteristic intensities at q
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< 1 nm-1 (increase with concentration and temperature). This indicates that at
comparable reaction times the as-formed structures at 21 °C and higher
supersaturation experiments had already reached a more advanced stage of
development. The low-q profiles of these patterns resemble those of stage III for the
50 mmol/L, 12 ºC experiment (Fig. 1A). Important to note, however, is the fact that at
q > 1 nm-1 all compared scattering curves have nearly identical profiles (Fig. 1B).
This clearly demonstrates that despite the differences in solution supersaturations, the
primary species that formed in solution were very similar in terms of shape and size.
The differences in intensity scaling in this q-range are attributable to different volume
fractions of the primary scatterers forming at different supersaturations (i.e., at higher
supersaturations more scatterers exist in the solution).
Fig. 1. A) Time-resolved in situ SAXS patterns from the formation of solids in an experiment starting with an
initial concentration of 50 mmol/L CaSO4 and equilibrated at 12 °C. The significance of the marked stages I – IV
is explained in the text; the inset in A shows a selected 2D scattering curve and indicates the significance of the I(q) dependence of the scattering exponents q (dashed lines) pointing out the characteristic features in scattering as
described in the text; B) Comparison of in situ SAXS patterns collected at various physicochemical conditions (see
inset legend); Shown also is the change in the I(q) dependence of the scattering exponent to emphasize the
invariant in the high-q part of the data (dashed lines) indicating that at q > 1 nm-1 the pattern originates from
practically identical scatterers.
The formation and initial self-assembly of primary species: stages I and II in detail
In order to establish a model that explains all stages (Fig. 1A) we first discuss
in detail the 390 seconds corresponding to stages I an II of the reaction (Fig. 2A). The
first four scattering curves (30 – 120 seconds) contain a clear Guinier region26, 27 i.e.,
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a plateau in I(q) ∝ q0 for q < 1 nm-1, and a region for which I(q) ∝ q-1 for ca. 1.5 nm 1
> q > 3.0 nm-1 (see also Fig. 1B). The fact that a I(q) ∝ q0 dependence is evident at
low-q for q → 0 indicates that the scattering originated from non-aggregated and noninteracting individual species. Furthermore, the occurrence of the I(q) ∝ q-1
dependence (at high-q) is typically attributable to scattering from elongated,
anisotropic shapes26, 28. Therefore, for stage I, the scattering curves could be best
fitted with Eq. 1, which includes the analytical expression for a cylindrical form
factor26 Pcyl(q,R,L) (detailed forms of this expression can be found in Supp. In.: S2,
Eq. S2.1):
I (q, R, L,φ) = φVpart ( Δρ) Pcyl (q, R, L) →
2
(1)
2
1
⎡
⎤
2J1 (qRsinα)
sin( qL cosα) Δρ is
where
theR,radius,
ϕ the
of primary
→ Pcyl L
(q,isR,the
L) length,
= ∫ F R2 (q,
L,α)sinα
dαvolume
= ∫ ⎢ fraction
⋅ 1 2 scatterers,
⎥ sinα dα
0
0
⎣ qRsinα
⎦
2 qL cosα
the scattering length density difference between the scatterers and the matrix
π/2
π/2
(solvent), and Vpart is the volume of a single particle (scatterer).
For all scattering profiles shown in Fig. 2A, at q > 1 nm-1, neither the shape of the
profiles nor the intensity values of the scattering features changed, but at q < 0.3 nm-1,
the intensity systematically decreased between 150 and 390 seconds (stage II). We
attributed this change to an evolution of the inter-particle structure factor S(q), which
resulted from the increase in particle-particle interferences between the primary
species in solution. For these elongated primary species, an expression for the
scattering intensity was derived by combining a form factor Pcyl(q) with a structure
factor S(q) using a decoupling approximation29 (Supp. In.: S2, Eq. S2.2). In our
analysis we considered a simplified structure factor, SHS(q), for scatterers in solution
that only accounts for the interactions through the hard-core repulsive potential30 (Eq.
2):
SHS (q,v, ReHS ) =
1
1+ 24v {G(q,v, ReHS ) / (2qReHS )}
(2)
The exact form of the G(q,v,ReHS) function can be found in Supp. In.: S2, Eq. S2.3.
SHS(q) depends on the local volume fraction of interacting neighboring scatterers, v,
and the effective hard-sphere radius, ReHS, which represents the typical distance
between neighboring primary scatterers31, 32.
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Fig. 2. SAXS patterns obtained from the experiment with a 50 mmol/L CaSO4 solution reacted at 12 °C for up to
14370 seconds showing different parts of the reaction. A) SAXS patterns of the first 390 seconds (stages I & II);
B) Stage III is represented by the SAXS patterns between 420 and 840 seconds (solid lines represent best fits as
described in the main body text; please note that the directions of the time-axes in A) and B) are different); C)
intensity change in the SAXS pattern collected at 720 seconds and, D) at 1500 and 3000 seconds; in C) and D)
various fits are also shown (dashed and solid lines according to the inset legends); E) Progressive change in the
intensities in the SAXS patterns between 750 and 5400 seconds, showing the I(q) dependencies and the change in
scattering exponents (dashed lines); F) Degree of crystallization, α, vs. time from the corresponding WAXS
measurements derived from the change in the area of the (020) WAXS reflection of gypsum measured
simultaneously with the corresponding SAXS patterns.
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Our initial fitting, which included the hard-sphere model described above (Eq.
2) also indicated that lengths, radii, and φVpart(Δρ)2 pre-factors (Eq. 1) of the
cylindrical form factor were constant throughout stages I and II of the reaction. Since
their geometry remained constant we can assume that the electron density of these
primary entities did not change either. Furthermore, due to the fact that the overall
φVpart(Δρ)2 remained constant, it follows that the volume fraction φ of these entities
invariably must have remained constant. In addition, our data indicate that ReHS >> Rg,
where the latter is the radius of gyration of a cylinder (Supp. In.: S2, Eq. S2.4). This
implies that, on average, the distance between primary particles was sufficiently large
to prevent the aggregation of the species. By taking into account the constancy of
φVpart(Δρ)2, and the fact that ReHS values are large (> 8 nm), we proposed that these
primary species could form domains of locally increased scatterer number densities
separated by regions depleted of scatterers, i.e., local species number density
fluctuations in the solution.
Hence, ReHS could be interpreted in terms of the Debye-Hückel screening length
separating individual primary scatterers, surrounded by an electric double layer,
which are randomly oriented with respect to each other. Within these domains, the
arrangement of the primary species is most likely not homogeneous. This is a
consequence of the dynamic character of the electric double layer and its dependence
on the anisotropic geometry of the elongated primary species. We propose that these
heterogeneities can be modeled by assuming a length distribution of the effective
hard-sphere radius, and due to the fact that ReHS >> Rg, ReHS is independent of the size
and geometry of the primary scatterers. Therefore, the length distribution does not
affect the form factor of primary species but only their structure factor33. This concept
is mathematically expressed in Eq. 3 for <SHS(q)>, where the size distribution D(r) is
combined with the hard-sphere structure factor using a local monodisperse
approximation34:
< SHS ( q,v,< ReHS
∫
>,σ ) >=
∞
0
D ( < ReHS >,σ,r ) ⋅ SHS ( q,v,r ) dr
∫
∞
0
D ( < ReHS >,σ,r ) dr
(3)
where <ReHS> denotes the intensity averaged effective hard-sphere radius, and σ the
corresponding standard deviation. The structure factor expressed by Eq. 3 is further
referred to as a polydisperse hard-sphere structure factor, <SHS(q)>, and for D(r) we
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used a Zimm-Schulz-Flory distribution35 (see details in Supp. In.: S2, Eq. S2.5). In
Supp. In.: S3, we also discuss the difference in fitting of monodisperse and
polydisperse variants of the hard-sphere structure factor for a selected data set.
The aggregation of the primary species: stage III
In stage III (Fig. 1A), in the q > 1 nm-1 region of the scattering patterns
between 420 and 840 seconds only negligible variations in shape and intensity were
observed (Fig. 2B). On the other hand at q < 1 nm-1 a characteristic increase in
intensity occurred, indicating a gradual growth of larger scattering features. Beyond
420 seconds, the overall increase in scattering at q < 1 nm-1 followed a I(q) ∝ q-3 > -a > 4
dependence indicating that the signal could be attributed to scattering from rough
fractal surfaces. Bale & Schmidt and Wong & Bray36,
37
derived expressions
approximating scattering from such features (Eq. 4):
Γ(5 − Ds )sin[π(3 − Ds ) / 2] −6+Ds
I SF (q, Ds , A) = A ⋅ SF(q, Ds ) ⇒ A ⋅ SF(q, Ds ) = A ⋅
q
(4a)
3 − Ds
Γ(5Γ(5
− D−s )sin[π(3
− D−s )D/ s2]
Ds )sin[π(3
) / 2]−6+D−6+D
Ds )Ds )⇒ ⇒
A ⋅ SF(q,
Ds )D=s )A=⋅ A ⋅
q qs s
AF(q,
⋅ SF(q,
A ⋅ SF(q,
3 −3D−s Ds
(4b)
where A is a constant proportional to the surface area of the scattering features (such
as crystal surfaces or pore surfaces etc.) and Γ denotes a gamma function.
Furthermore, the parameter Ds is a surface fractal dimension, where Ds = 2 represents
smooth surfaces and Ds
3 represents very rough fractal surfaces (with Ds = 3 being
a singularity).
As it is further evidenced by the detailed analysis of the φVpart(Δρ)2 and the
normalized φ(Δρ)2 pre-factors for the primary species (Eq. 1), the growth of the
surface fractal features proceeded throughout the entire stage III, while φ (Δρ)2
remained constant. This indicates, that the growth of the surface fractals can be
interpreted in terms of the aggregation of primary species, rather than the growth of
larger structures at the expense of the smaller ones. Thus Eq. 4, is as such applicable
only to the low-q part of our data, since it implies that ISF(q
∞) = 0, and does not
allow for the surface fractal structure to be internally composed of the smaller primary
scatterers. Therefore, Eq. 4 was re-written to include the contribution from the smaller
structural units.
In order to do this, let us consider a collection of primary species in which the
individual units start to form larger, much denser packed aggregates. These new
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aggregates would, at short length-scales, be composed of primary species, and be
arranged in such a way that at larger length-scales the structure would appear to be a
solid coarse surface. Hence the intensity at low-q would be dominated by scattering
from the external envelopes of these large aggregates. Scattering intensity from such
morphologies can be described by an expression that combines the form factor of the
primary scatterers with the structure factor of the aggregate (Fig. 2C). This way, the
following necessary conditions can be fulfilled: in the low-q regime Pcyl(q → 0) = 1
and for high-q S(q → ∞) = 1. To best fit our data the following semi-empirical
expression for the resulting structure surface fractal factor, SSF(q), was derived (Eq.
5):
SSF (q, Ds , A') = 1+ A'⋅ SF(q, Ds )
(5)
In this equation SF(q,Ds) is defined as in Eq. 4, where A was defined as being
proportional to the total surface of all scattering features. In contrast, in Eq. 5, Aʹ is
proportional to the relative surface of all scattering features formed from the
aggregating primary species and is normalized against the form factor in order to
fulfill the condition that SSF(q → ∞) = 1. We interpret the occurrence of a structure
factor expressed by Eq. 5 in terms of a dynamic transition from a state dominated in
the bulk of the sample by <SHS(q)>–type interactions to a state where denser
aggregates prevail, and for which the low-q scattering intensity is dominated by their
surfaces. Thus, the scattering patterns collected after 420 seconds, when the intensity
starts to increase at low-q, mark the onset of very rapid aggregation of the primary
scatterers, which in stage II only loosely interacted forming domains of widely-spaced
individual scatterers. This process is dominated at low-q by the scattering from the
growing surfaces of these aggregates. It is worth noting that the inter-particle
interactions between 420 and 510 seconds were also affected by the minor
contribution from <SHS(q)>. This minor effect is discussed in Supp. In.: S4 where we
also present how this contribution affected our fitting and model.
Growth of primary species
During stages III and IV (Fig. 1A) the increase in intensity is dominated by a
change at q < 1 nm-1. We modeled these changes in terms of surface fractal
contributions using the above described scattering model which fitted well all patterns
up to ~1500 seconds. However, as the system further developed, the scattering curves
!
14!
showed considerable changes in their shape at q > 1 nm-1 indicating the growth of the
primary species. As illustrated by Fig. 2D, the scattering curve at 1500 seconds is
fitted with a cylindrical form factor and surface fractal structure factor SSF(q). Even
though at 3000 seconds the low-q part of the scattering curve could still be fitted by
SSF(q), at high-q the form factor changed. Thus, the expression for the cylindrical
form factor was no longer valid and the scattering curves between 1500 and 5400
seconds (see Fig. 1A) were fitted only partially with an expression for the scattering
intensity including SSF(q) (Eq. 5). The auxiliary cylindrical form factor expression
was only used to extract the φVpart(Δρ)2 (Eq. 1) pre-factor values, and this enabled the
correct scaling of the structure factor. For scattering expressed by a form factor on its
own, the φVpart(Δρ)2 pre-factor corresponds to I(q=0) because Pcyl(q=0) = 1, which is
no longer true when scattering at low-q is modified by a contribution from a structure
factor. However, because by definition S(q → ∞) = 1, we could use an approximate
form factor to derive the values for φVpart(Δρ)2 (Eq. 1) by simultaneously
extrapolating the form factor to low-q and the structure factor to high-q. This way the
pre-factor is the minimized parameter that is matching the mutual scaling of both
factors (see also Figs. 2C&D). As a result of this fitting the characteristic parameters
that represent growth of surfaces throughout stage IV could be obtained.
In Fig. 2E a comparison of selected data sets between 750 and 5400 seconds is
shown. Up to 1500 seconds, the high-q part of all scattering curves (at q > 1 nm-1)
was very similar and followed a I(q) ∝ q-1 dependence. This indicates that a
cylindrical form factor could be used to fit these parts of the curves. On the other
hand, the contribution from the surface fractal with I(q) ∝ q-3 > -a > -4 increased by one
order of magnitude between 750 and 1500 seconds. It is worth noting that for the
patterns collected 1500 seconds after the onset of the reaction, it became increasingly
difficult to develop a model to describe scattering at high-q, yet, the continuous
changes could still be analyzed qualitatively. In the high-q range, between 1500 and
3000 seconds (Figs. 2D&E), the scattering curves showed characteristic “curling
down”, and with increasing reaction time the scattering intensity followed a I(q) ∝ q–a
< -1
dependence with the exponent -a gradually approaching a value of -4 at 5400
seconds. The increase in intensity of the scattering features from the surface fractals
continued, reaching values that were two orders of magnitude larger than the one at
1500 seconds (for q > 1 nm-1 in Fig. 2E). Between 3000 and 5400 seconds the
!
15!
difference in the surface fractal contributions was very small (< 10 %), and between
4500 and 5400 seconds practically no further growth of those features was detected at
low-q. Importantly, throughout this particular time period a significant change shape
of the scattering curves was observed at high-q (for q < 1 nm-1 in Fig. 2E), which
continued throughout the experiment. This evolution indicates that up to ~1500
seconds the formation of the scattering features was related to various rearrangements
of the primary species inside the aggregates and the formation of surface fractals as
evidenced primarily by the changes in the low-q region of the scattering patterns. On
the other hand, 1500 seconds marks the onset of a major internal rearrangement of the
primary species, referred to as stage IV. Interestingly, when analyzing the WAXS
data this point also marks the time frame when the first diffraction peaks for gypsum
emerged in the WAXS signal (Fig. 2F). This period preceding the appearance of
diffraction signals in WAXS patterns is referred to as the induction time and indicates
the onset of the crystallization of a new phase, as shown for other systems38, 39. In our
case this coincides with the onset of the formation of gypsum (Fig. 2F). This
crystallization process is evidenced through the evolution of the parameter α as a
function of time, which is evaluated from the change in area of the (020) reflection of
gypsum. These processes are also reflected in the evolution of the SAXS patterns, in
particular an increase in the exponent –a in the I(q) ∝ q–a < -1 dependence and the fact
that the exponent is approaching a value of -4. This suggests a complex growth
process of the primary species and a likely change in their sizes and morphologies in
stage IV. This possibly also involved the oriented rearrangement of scatterers, which
was evidenced by the development of the anisotropic scattering patterns after 5400
seconds (for details see Supp. In.: S5).
The structural characteristics of the primary species, aggregates and final particles
In Fig. 3A, the evolution of the radius and length (R and L) of the formed
primary species, that are characterized by a cylindrical form factor, is shown up to
~1500 seconds. Between 30 and ~800 seconds, the length of the formed entities, L,
remained constant with a value of ~2.8 nm. Subsequently, these elongated primary
species gradually grew in length reaching ~4 nm at ~1500 seconds. The average
radius of these species, R, was < 0.3 nm and it remained constant up to ~1500
seconds. One needs to point out that within the measured q-range the part of the form
!
16!
factor contributing to the scattering from the cylindrical cross-section is represented
only by the data points at q > 3 nm-1. Hence, a circular shape for the cylindrical cross
section is only assumed for the sake of fitting (since the actual information is not
contained in the data). The employed models were also used to fit the scattering
curves from the earliest stages in all other experiments (Table 1). For all cases the
resulting fits yielded R = 0.2 ± 10% nm and L = 2.7 ± 12% nm, confirming that
independent of reaction conditions these individual, well-defined elongated primary
species constituted the primary building blocks for the larger aggregates.
The above mentioned ~30% increase in size at ~1500 seconds of the formed
elongated primary species was also confirmed when the change in the pre-factor of
the cylindrical form factor (φVpart(Δρ)2, Eq. 1) was evaluated. At the very beginning
of the reaction (see arrows in Fig. 3B at 30 seconds) a very sharp increase between 30
and 60 seconds is observed. Following this initial sharp growth, the φVpart(Δρ)2 value
remained constant until ~800 seconds, after which up to ~1500 seconds its value
increased gradually by ~20%. The initial sharp increase originates from the formation
of the primary species during the solution injection/mixing period (~15 seconds). The
normalization of the pre-factor against the scatterer volume yields the product ϕ(Δρ)2
(Fig. 3B). Between 60 and ~1500 seconds, ϕ(Δρ)2 maintained an average constant
value of ~3.4·1019 cm-4 (shown by the solid horizontal line in Fig. 3B). This indicates
that the primary scatterers formed in the first 30 seconds, and that after 60 seconds the
ϕ(Δρ)2 product already reached a plateau and remained constant until ~1500 seconds
(Fig. 3B). This constant behavior of the ϕ(Δρ)2 product and assuming that the electron
density of the primary scatterers remained constant, despite the apparent increase in
ϕVpart(Δρ)2 after ~800 seconds, indicates that no new primary scatterers were formed.
In addition, during the latter stages of the reaction (stages III and IV; Fig. 3D),
between 1500 and 5400 seconds, the pre-factor value ϕVpart(Δρ)2 gradually increased
by another ~70%, which we attribute primarily to the increase of the average volume
of individual particles in stage IV. The changes in the pre-factor can be considered as
a measure of the progress of reaction. Interestingly the pre-factor values after 1500
seconds did not yet reach a plateau, which suggests that internal re-arrangement and
growth of particles in stage IV was not completed at 5400 seconds. As mentioned
before, SAXS patterns were collected up to 14370 seconds but the change in the last
!
17!
~2.5 hour accounted for a maximal 5% change in overall intensity (see also Supp. In.:
S5).
Fig. 3. Evolution of the fitting parameters as a function of time in seconds for: A) elongated scatterers with
lengths, L, and cross-sectional radii, R, up to ~1500 seconds; B) pre-factors φVpart(Δρ)2 and φ(Δρ)2 up to ~1500
seconds; the average of the φ(Δρ)2
product is marked with a horizontal solid line; C) pre-factor φVpart(Δρ)2
evolution up to 5400 seconds; D) mean effective hard-sphere radii <ReHS> and their standard deviations, σ
characterizing <SHS(q)>; E) local volume fractions, v, characterizing <SHS(q)>; F) surface area contribution, Aʹ, and
surface fractal dimension, Ds, characterizing <SSF(q)>. Note that up to 600 seconds the Aʹ and Ds values exhibited
very large uncertainty, due to the limited contribution of large scatterers in the low-q range of the scattering
patterns. For Ds any value between 2 and < 3 would produce reasonable fits, but the yielded Aʹ values for all Ds > 2
were significantly out of trend and are thus not shown.
In the q < 1 nm-1 region, the intensity of the initial scattering patterns (up to
120 seconds) followed a I(q) ∝ q0 dependence, and thus we fitted those curves with
an expression for the cylindrical form factor and a S(q) = 1. However, already the next
scattering pattern (at 150 seconds) revealed a small characteristic decrease in intensity
!
18!
at low-q (Fig. 2A), and thus all subsequent scattering patterns were fitted after
introducing the polydisperse hard-sphere structure factor <SHS(q)> (Eq. 3). This
allowed us to evaluate the evolution of the mean effective hard-sphere radii values,
<ReHS>, together with the corresponding standard deviation values, σ (Fig. 3D). The
results show that, <ReHS> varied between ~8 and ~11 nm, with a gradual increase
after 240 seconds and a maximum at 360 seconds, with a final value around ~10 nm.
On the other hand, the standard deviation, σ, shows a positive trend from an initial
σ/<ReHS> value of ~7% increasing up to ~40%. This indicates an increase in variance
of the inter-particle correlation. The evolution of the local volume fraction parameter,
v, is plotted in Fig. 3E, and it is seen that the overall values of v increased from ~1%
at 150 seconds to ~4.5% at 360 seconds, after which a decrease to 2.5% at 510
seconds. These trends suggest that changes in the structure factor could be associated
with the formation of locally interacting domains of primary species, where the
individual entities were separated from each other by on average 2<ReHS>. Therefore,
the local volume fraction parameter v, expresses the degree of correlation within the
particle domains, with larger values indicating denser and more extensive domains
and reaches a maximum at 360 seconds. After 420 seconds a gradual increase in
scattering intensity started (see Fig. 2B), which we attributed to the formation of
fractal surfaces. This inherently involved a gradual decrease of the contribution from
<SHS(q)>, and marked a decrease of v until 510 seconds (see also Supp. In.: S4).
Finally, from the surface fractal contribution, SSF(q), to the fitting (Eq. 5), we
evaluated two additional independent parameters characterizing the structure factor:
Aʹ and Ds. The evolution of both these parameters as a function of time (Fig. 3F)
shows that Aʹ is proportional to the surface area of growing scattering units and
follows a sigmoidal growth profile with a rapid increase (one order of magnitude)
between ~900 and ~2300 seconds, a maximum at ~3500 seconds, and a small
decrease (~8%) up to ~5400 seconds. Conversely, Ds started at a value of 2, and
rapidly increased to ~2.1 between 420 and ~900 seconds reaching a plateau at ~2300
seconds. After a slight increase between ~2300 and ~3800 seconds, a second plateau
was observed during which Ds reached a value of ~2.13. The evolution of Ds with
time clearly indicates a two-stage roughening of the scattering surfaces. The onset of
the second roughening stage derived from the fractal dimension, coincides with the
center of the exponential phase of the sigmoidal growth of Aʹ at ~1500 seconds. This
!
19!
point is important, as it also marks the onset of the growth, and crystallization, of the
primary species during stage IV (i.e., gypsum formation). The growth of the fractal
surfaces between 1500 and 5400 seconds is related both to the initial aggregation of
the primary species that formed in stage I and II and their further transformations
during the latter stage IV. This is also evidenced by the fact that after ~3500 seconds
the Aʹ value decreased and this is equivalent to a decrease in the total area of the
scattering features which can be explained by the coalescence and growth of the
individual primary scatterers building the aggregates in stage III, and ultimately the
larger internally homogeneous particles during stage IV.
Discussion
The analysis of our time-resolved and in situ scattering data revealed the
details of the structural evolution that lead to the formation of calcium sulfate
particles from aqueous solutions. The well-defined sub-3 nm primary species are the
first observable entities that formed in supersaturated solutions of CaSO4. These
primary units constitute the building blocks for larger structures throughout the early
stages of the CaSO4 formation process. The initial constant values of the φVpart(Δρ)2
and the normalized φ(Δρ)2 pre-factors suggest that: (i) the primary species formed
near-instantaneously after mixing of the Ca2+- and SO42--stock solutions (i.e., stage I),
(ii) although the length of these primary species grew by ~30%, the volume fraction
remained unchanged throughout stages I – III, which means that their number density
decreased, and hence the growth in length occurred by merging of short units into
longer ones. Stage III is characterized by aggregation of these primary species as
evidenced by the increase in intensity in the q < 1 nm-1 region (Figs. 1A & 2B)
originating from the evolution of the structure surface fractal factor, SSF(q). The onset
of stage IV is characterized by the growth of the elongated primary species within the
aggregates and their transformation into larger particles of less-defined geometries.
Noteworthy are the density fluctuation occurring in stage II of the reaction: our SAXS
data indicate that the primary species form a mixture of denser and less denser
domains in the bulk of the solution. In recent years, the pivotal role of density
!
20!
fluctuations in the nucleation process has been postulated40, 41 and observed in various
simulation studies on hard-spheres42.
Inexorably the question arises about the nature of these primary species and
their evolution through time. A partial answer can be derived through a detailed
analysis of the φVpart(Δρ)2 and the normalized φ(Δρ)2 pre-factors (Fig. 3B & C), which
allow us to monitor the reaction progress as a function of time. However, specific
information concerning the electron density, Δρ, and volume fraction, φ, of the
forming phase is not independently accessible because the pre-factors are the product
of these two components. Nevertheless, due to the fact that the scattered intensity is
expressed in absolute units, it is possible to estimate the changes in the φ(Δρ)2 prefactor in relation to the known and thus possible CaSO4 polymorphs and their
expected volume fractions. This way we can maybe correlate our scattering data with
thermodynamic solubility data and thus identify the formed phases based on their
electron densities.
We have calculated the electron densities of each of the three known CaSO4
phases with respect to their chemical formula and by taking into account the
corresponding bulk densities (see Supp. In.: S6). We evaluated the predicted volume
fraction, φ, for each phase by considering the original concentration of Ca2+ and SO42ions in solution and using the bulk solubility of each phase calculated with
PHREEQC19. We used the predicted values of φ (see Supp. In.: S6), to calculate the
electron densities of what we termed pseudo-phases, i.e., the electron densities of
phases with an assumed volume fractions for gypsum, bassanite and anhydrite. For
instance, if anhydrite was the phase contributing to the scattering of the primary
particles, the corresponding volume fraction value would yield the electron density of
anhydrite, and therefore the actual extracted (rather than expected) value is referred to
as a pseudo-anhydrite phase. In scattering one considers Δρ=|ρphase - ρsolvent|, and when
using the electron density of the solvent (water) of 334 e-/nm3, we could calculate the
density of the newly forming phase, ρphase. Fig. 4 shows the electron densities for the
first 1500 seconds of the reaction for all three pseudo-CaSO4 polymorphs: (A)
without accounting for their bulk solubility, (B) including the bulk solubility of
gypsum and anhydrite in pure water (values for pseudo-bassanite could not be
calculated as at these conditions its bulk solubility is below the concentration levels in
our experiments), and (C) including the bulk solubility of gypsum and anhydrite in
!
21!
the presence of Na+ and Cl- (from the initial stock solutions). The expected electron
densities for all three CaSO4 polymorphs are indicated by horizontal lines and bands.
For the first scenario (Fig. 4A), the volume fractions corresponding to gypsum
yielded an electron density of pseudo-gypsum that was ~20 e-/nm3 lower than the
expected value. Furthermore, again on average, the differences for pseudo-bassanite
and pseudo-anhydrite were ~75 e-/nm3 and respectively ~120 e-/nm3 lower than the
expected values. However, when the bulk solubilities in pure water were taken into
account (i.e., the volume fraction decreases; Fig. 4B) the pseudo-gypsum phase
emerged with an electron density on average ~20 e-/nm3 higher than gypsum, whereas
for pseudo-anhydrite a ~90 e-/nm3 higher than anhydrite value was obtained. Finally,
for the volume fractions derived, when the bulk solubility in the presence of dissolved
NaCl was taken into account (Fig. 4C), the pseudo-gypsum phase had an electron
density on average ~80 e-/nm3 higher than gypsum, whereas pseudo-anhydrite ~150 e/nm3 higher than anhydrite. Again no pseudo-bassanite values could be calculated due
to the fact that the known bulk solubility data indicate that bassanite should be
undersaturated and thus should not form in our experiments.
Fig. 4. Electron densities for pseudo-CaSO4 polymorph phases calculated based on the volume fractions of the
known crystalline phases during stage I to IV (in seconds): A) Not taking into account the bulk solubility;
B) taking into account bulk solubility in pure water; C) taking into account bulk solubility in a 100 mmol/L NaCl
solution. The colored horizontal bands indicate the range of electron densities expected for each given CaSO 4
polymorph.
A non-equivocal identification of the nature of the primary species through
their electron density is hindered by the fact that the actual solubilities of these nanosized primary species are not known; hence the evaluation of the volume fraction is
!
22!
likely biased. We cannot excluded that at the early stages of the reaction when these
nano-sized units are formed and the process is out of equilibrium, the solubilities of
any nano-particulate phase would be considerably different from its bulk counterpart
(e.g., the Ostwald–Freundlich relation43,
44
). In our previous study16 we have
suggested for the CaSO4 system that such a nano-solubility effect was the prime
reason for the stabilization of bassanite below its predicted bulk solubility. This
evaluation was based on enthaplic effects and surface free energy considerations and
matched similar observations in other nano-systems45-47.
Indeed the SAXS
measurements for the 50 mmol/L CaSO4 system at 12 °C, between 60 and 1500
seconds, indicate that the φ(Δρ)2 pre-factors remained constant, which indicates that
the well-defined, elongated primary scatterers formed near-instantaneously within the
first minute of the reaction, i.e., when mixing of the reagent solutions was completed.
Furthermore, the actual volume fractions representing these primary scatterers depend
on the early-stage equilibrium, which is changing fast and is unknown due to the lack
of non-bulk solubility data. On the other hand, the decrease of the surface area
contribution (change in Aʹ, Fig. 3F), the increase in roughness of the scattering
surfaces (observed as an increase in Ds), and the linear increase in the pre-factor,
φVpart(Δρ)2 (Fig. 3C) after 3000 seconds, likely indicate a progressive transformation
during the later stages, and an approach to equilibrium.
However, our considerations refer to the entire possible range of nano- to bulksolubility values. In all the cases the calculated volume fractions yield relatively high
electron density ranges considering each of the crystalline CaSO4 polymorphs. This
observation is true even for the unlikely case for which the bulk solubility was not
taken into account or assumed to be null (Fig. 4A). This implies that, due to the lack
of quantitative information about the solubilities of the CaSO4 phases at the
nanoscale, the primary species resemble structurally any of the crystalline polymorphs
of CaSO4,
Furthermore, one should also account for the actual geometry of the primary
scatterers with respect to the internal structure of three CaSO4 polymorphs. In
anhydrite, the Ca-SO4 units are relatively densely packed (Supp. In.: S7, Fig. S7.1A),
while for bassanite they are arranged in such a way that they form channels along the
c-axis and these are filled with H2O (Supp. In.: S7, Fig. S7.1B). This arrangement
contributes to the lower bulk electron density of the hemihydrate bassanite with
!
23!
respect to the dehydrated anhydrite (Fig. 4 and Supp. In. S6). In the dihydrate
gypsum, thermodynamically the most stable polymorphs in all our experiments, the
Ca-SO4 units are arranged into thin sheets separated by H2O double layers along the
a- and c-axes (Supp. In.: S7, Fig. S7.1C). Such an arrangement results in an even
lower average bulk electron density for gypsum. However, at the local scale of the
anhydrous Ca-SO4-cores all three calcium sulfate polymorphs are in fact very similar
in terms of the electron density (Supp. In.: S7, Figs. S7.1D – F), with in all
polymorphs the shortest Ca-Ca distance being ~0.4 nm
48-50
. This distance matches
well the average diameter (2R) derived for the elongated primary scatterers obtained
from our SAXS analysis (< 0.6 nm). In Supp. In.: S7, Figs. S7.1D-F, we have
proposed idealized elongated species with Ca-SO4-cores based on the three calcium
sulfate polymorphs. In all cases we obtain an ~2.8 nm long units (as compared to L
~2.7 nm obtained from SAXS). Therefore, our data suggests that in aqueous media
the formation and transformation to various CaSO4 polymorphs could proceed via the
nucleation of initial rods of anhydrous Ca-SO4-cores that latter rearrange (i.e., bricks
in a wall) via hydration into bassanite or gypsum.
The latter transformation observed in the SAXS patterns is complemented
through observations from our auxiliary WAXS data. The time-resolved in situ
WAXS diffraction patterns show that gypsum is formed during stage IV (Fig. 2F).
The temporal evolution of the gypsum peaks in the WAXS patterns in the 50 mmol/L
CaSO4 and 12 °C (Fig. 2F) experiment, were dominated exclusively by the signal
from the developing crystalline gypsum. No WAXS signal was observed prior to
~1500 seconds, which corresponds to the beginning of the transition from stage III to
stage IV in our SAXS data (Fig. 1A). The entire period prior to stage IV in the SAXS
data coincides with the induction period observed in WAXS, thus preceding the onset
of gypsum formation. Hence, the actual crystallization of gypsum corresponds to the
morphological change of the primary species (i.e., growth in all directions of the
primary scatterers). Thus, although stages I – III involve species of the structure
resembling that of the crystalline CaSO4 phases, the arrangement of those species
within the aggregates formed in stage III does not contribute to a coherent diffraction
signal until the beginning of stage IV. We attribute this phenomenon to the actual
mechanism of aggregation, in which the occurrence of surface fractals, scaling at lowq, suggests that envelopes of large surfaces must develop. (Fig. 5) These could be
either external surfaces of large aggregates or the internal surfaces if the aggregates
!
24!
are internally porous and the pores are large in comparison with the primary
scatterers. Typically, for particles aggregating in solution one would expect massfractal-like aggregates, because such aggregates form at relatively low local particle
concentrations where the diffusion length is considerable51-55. However, Kolb &
Herrmann54 showed through Monte Carlo simulations of highly concentrated colloidal
aggregates, that if the “local” concentration is close to 1 (i.e., the particles have very
little space to move), than surface fractal aggregates are formed instead of their mass
(volume) counterparts54. Indeed in our case, before the onset of the surface fractal
growth (after 420 seconds), we observed a ~10-fold increase in the local volume
fraction v in comparison with the actual expected volume fraction ϕ (see Fig. 3E and
Supp. In. S6). We infer that surface fractal aggregates could form due to the collapse
of the particle-rich domains and thus leading to a high local concentration of primary
species. This process can be also enhanced by the elongated shape of the primary
scatterers allowing for the possibly denser packing along the direction of their longaxis. Nevertheless, even if such a process should occur it does not imply that
sufficient structural coherence, to the extent necessary for diffraction, could occur
within the aggregate. This is provided by further internal rearrangement and growth in
stage IV.
Fig. 5. Schematic representation of the four stages of CaSO4 precipitation from solution observed for our experimental
conditions: I) formation of well-defined, sub-3 nm primary species/scatterers; inset shows how these primary species increase
their local volume fraction; II) formation of denser domains of primary species; inset shows that scatterers are on average
separated by a distance of 2<ReHS>; III) aggregation of the primary species forming large surface fractal morphologies; IV)
growth and coalescence of the primary species within the aggregates; insets in C & D show the consequent stages of growth increase in length followed by increase in all dimensions eventually leading to larger morphologies.
!
25!
The process of ordering of structural building units within the crystalline
structure is also reflected in the 2D SAXS signal in the later stages after 5400 seconds
(Supp. In.: S5). Such a scattering pattern is expected for morphologies in which the
larger dimensions of anisotropic features are oriented along the flow (as well as the
long-axes of the particles containing these features), and smaller features are
perpendicular to the flow (and the long-axes of particles). This is in line with our
previous HR-TEM study where we showed that gypsum needles ~100 nm in crosssection and several µm in length were internally composed of oriented nano-sized
rods of bassanite with their long-axes parallel to the long-axes of the gypsum
crystals16. From our current data we predict that as the process of growth continues,
ultimately the surface fractal regime will only persist due to the coalescence of the
internal structures of particles and pores. Eventually, gypsum crystals would become
internally homogenous. Therefore we attribute the lack of observable crystallinity in
WAXS before 1500 seconds to the fact that the surface fractal aggregates are initially
built of primary species which are poorly aligned. Only upon further transformations,
forming better-ordered coherent structures, can a diffraction pattern arise.
Recent experimental results show that at room temperature the outcome of the
CaSO4 formation reaction in terms of phase selection (i.e., hydration) can be directed
by changing the available amount of water; i.e. by using different mixtures of
water – ethanol, the system can be guided to form either gypsum, bassanite, anhydrite,
or a mixture of these phases56, 57. In addition, at higher temperatures (> 80 ºC), the
CaSO4 formation outcome is dictated by salinity (i.e., ionic strength-water activity)
with solutions with a salinity of > 2.8 mol/L NaCl resulting in the first bulk phase to
form to be bassanite (instead of gypsum), which over time transforms into
anhydrite58. Based on these experimental observations, and those presented in this
current work, we propose that the aggregates formed in stage III were protostructures, formed with anhydrous Ca-SO4-core-based primary species, which are
common to all CaSO4 phases (see Supp. In. S7). In stage IV, the observed
crystallization to gypsum had to therefore in fact be driven by a hydration reaction. It
implies that in order to obtain well-ordered sheets of CaSO4-cores and H2O layers (as
found in gypsum – see Supp. In. S7), the primary species within the aggregates from
stage III, must radically transform and coalesce to larger particles in stage IV. We see
the evidence of this on-going process through the changes in the parameters
characterizing the surface fractal structure factor (Fig. 3F), and in the scattering pre!
26!
factor (Fig. 3C) after the induction time in WAXS (Fig. 2F). Thus the most likely path
towards the final gypsum polymorph, proceeds through an internal rearrangement of
the aggregates that assume the local structure of a not-yet-fully-hydrated phase
(e.g., bassanite) in the early part of stage IV. Subsequently the partially hydrated
phase transforms to the dihydrated gypsum in which the water structuring is more
complex, making bassanite a plausible intermediate nano-phase on the pathway to
gypsum formation in a water-abundant environment14-16,
18
. Consequently, if not
enough water is available the reaction will be halted at this intermediate point, and
bassanite will emerge as the bulk crystalline phase.
Conclusions
In situ and time-resolved SAXS data indicate that the formation of phases in
the CaSO4 system occurs in 4 stages (Fig. 5): (I) formation of well-defined, sub-3 nm
primary species, (II) grouping of these primary species into domains (i.e., density
fluctuations), (III) aggregation of the primary scatterer domains into large, denser,
structures and (IV) re-organization/evolution of the primary species within the
aggregates and their transformation to gypsum. The latter stage is also evidenced
through the appearance of a diffraction signal of gypsum in WAXS after the reorganization/evolution of the large aggregates. Although the aggregates in stage III do
not exhibit a bulk crystalline order, we assert that these phases are composed of
primary species of electron density comparable to those of known CaSO4
polymorphs, and being most likely anhydrous Ca-SO4-cores. Hence, these
observations imply that the aggregates formed in stage III could be considered as the
proto-structure of all CaSO4 phases, which transforms through hydration to gypsum in
stage IV.
!
27!
Supplementary Information
S1. Saturation indices (SI) of gypsum, bassanite and anhydrite
We followed the CaSO4·xH2O formation reactions in experiments with final
mixed solution concentrations between 50 and 150 mmol/L (Table S1.1). Based on
the following reaction:
CaCl2·2H2O + Na2SO4
CaSO4·xH2O↓ + 2NaCl
the supersaturation indices (SI) of CaSO4·xH2O polymorphs in aqueous solutions
were calculated with the geochemical code PHREEQC19 using the LLNL database. In
all cases it is noteworthy to mention that only bulk solubility data are available for all
three polymorphs.
Table S1.1: Final mixed solution concentration, salinity, temperature and saturation indices for the three
CaSO4·xH2O polymorphs.
[CaSO4]
[mmol/L]
[NaCl]
[mmol/L]
T
[°C]
SIGypsum
SIBassanite
SIAnhydrite
50
100
12
0.50
-0.45
0.20
50
100
21
0.50
-0.37
0.28
50
100
30
0.49
-0.38
0.36
50
100
40
0.49
-0.19
0.45
75
150
21
0.70
-0.16
0.48
100
200
21
0.63
-0.02
0.84
150
300
21
1.04
0.18
0.83
The SI values show that under the indicated physicochemical conditions all mixed
solutions were supersaturated with respect to gypsum and anhydrite (SIGypsum > 0,
SIAnhydrite > 0), whereas bassanite was in all cases, except for 150 mmol/L (SIBassanite >
0), undersaturated and should dissolve (SIBassanite < 0).
S2. Mathematical expressions for SAXS data modeling
The scattering intensity from the collection of non-interacting homogenous
cylindrical objects of length L and radius R is given by Eq. S2.1 (see ref. 26):
I(q, R, L,φ) = φVpart ( Δρ) Pcyl (q, R, L) →
2
→ Pcyl (q, R, L) = ∫
π/2
0
!
F (q, R, L,α)sinα dα = ∫
2
π/2
0
2
⎡ 2J1 (qRsinα) sin( 12 qL cosα) ⎤
⎢ qRsinα ⋅ 1 qL cosα ⎥ sinα dα
⎣
⎦
2
28!
(S2.1)
where ϕ is the volume fraction of scatterers, Δρ the scattering length density
difference between a scatterer and the matrix (a solvent), Vpart is the volume of a
single particle (scatterer) and Pcyl(q,R,L) is the form factor of the cylinder.
Furthermore, J1(x) is the first-order Bessel function, and α is the angle between the
long axis of the cylindrical object and the primary beam. In S2.1, the form factor
Pcyl(q,R,L) is equal to the square of the amplitude F2(q,R,L, α) averaged over all
angles α, so that Pcyl (q, R, L) = F 2 (q, R, L,α α .
For interacting cylindrical objects, an expression for the scattering intensity
can be derived by combining a form factor Pcyl(q,R.L) with an interparticle structure
factor S(q) using a decoupling approximation29:
2
⎡
⎤
F(q, R, L,α α
I(q) = φVpart (Δρ) Pcyl (q, R, L) ⎢1+
S(q) − 1) ⎥
(
⎢
⎥
F 2 (q, R, L,α α
⎣
⎦
2
(S2.2)
In Eq. S2.2, the amplitude F(q,R,L,α) is defined for cylinders as in Eq. S2.1. It is also
assumed that interactions are dependent only on particle average sizes, and are
independent of particle orientations.
We also introduce the hard-sphere structure factor, SHS(q), which depends on
the local volume fraction of interacting particles, v, and the effective hard-sphere
radius, ReHS, and is expressed as follows30:
1
1+ 24v {G(q,v, ReHS ) / (2qReHS )}
1
S(q,v, R ) =
1
1+ 24vG(q,v, R )
S(q,v, R
)=
where
the
G(q,v,ReHS) has the following form:
eHSfunction
1+ 24vG(q,v, ReHS )
SHS (q,v, ReHS ) =
(S2.3a)
eHS
eHS
G(q,v, ReHS ) = A
2(2qReHS )sin(2qReHS ) + ⎡⎣ 2 − (2qReHS )2 ⎤⎦ cos(2qReHS ) − 2
sin(2qReHS ) − (2qReHS )cos(2qReHS )
+B
+
2
(2qReHS )
(2qReHS )3
−(2qReHS )4 cos(2qReHS ) + 4 ⎡⎣ 3(2qReHS )2 − 6 ⎤⎦ cos(2qReHS ) + 4 ⎡⎣(2qReHS )3 − 6(2qReHS ) ⎤⎦ si(2qReHS ) + 24
2(2qReHS )sin(2qReHS ) + ⎡⎣ 2 − (2qReHS )2 ⎤⎦ cos(2qReHS )
sin(2qReHS ) − (2qReHS )cos(2qR
eHS )
(2qR
)
G(q,v, ReHS ) = A
+B
(S2.3b) 3
(2qReHS )2
(2qReHS )
(1+ 2v)
−6v(1+ v / 2)
A=
, B=
, C = vA / 2
(1− v) S2.3b three
(1− v) parameters A, B, C are dependent on the local volume fraction v,
In Eq.
−(2qReHS )4 cos(2qReHS ) + 4 ⎡⎣ 3(2qReHS )2 − 6 ⎤⎦ cos(2qReHS ) + 4 ⎡⎣(2qReHS )3 − 6(2qReHS ) ⎤⎦ si(2
+
C
through the following expressions:
(2qReHS )5
+C
5
eHS
2
4
A=
2
4
(1+ 2v)2
−6v(1+ v / 2)2
,
B
=
, C = vA / 2
(1− v)4
(1− v)4
(S2.3c)
The actual physical dimensions of interacting cylindrical particles could be
compared to the typical distance between neighboring particles, through the
comparison of effective hard-sphere radii, ReHS, (from Eqs. S2.3a-c) with the
!
29!
particles’ radii of gyration (Rg). The gyration radius, Rg, of a cylinder of length L and
radius R can be expressed as:
Rg =
R 2 L2
+
2 12
(S2.4)
For a polydisperse hard-sphere structure factor, <SHS(q)> (main text, Eq. 3) we
used a Zimm-Schulz-Flory distribution35, and D(r) is expressed as follows:
rz ⎛ z +1 ⎞
D ( < ReHS >, z,r ) =
Γ(z + 1) ⎜⎝ < ReHS > ⎟⎠
z+1
⎛
z +1 ⎞
exp ⎜ −
r
⎝ < ReHS > ⎟⎠
(S2.5)
Here, ! denotes a gamma function, z is related to the width of the distribution, and the
standard deviation σ is expressed by Eq. S2.6:
σ=
< ReHS >
z +1
(S2.6)
S3. Monodisperse versus polydisperse hard-sphere structure factor
Fitting with the monodisperse SHS(q) structure factor from Eqs. S2.3a-c,
introduces oscillations to the fitting curve, which are not present in the original data.
This is illustrated by a fit to the pattern collected at 330 seconds (Fig. S3.1), based on
combining Eqs. S2.1-S2.3a-c, and resulting in the blue curve. The red curve in Fig.
S3.1 presents the improved fit to the experimental data for a selected scattering
pattern with an expression for the scattering intensity including a polydisperse
<SHS(q)>
structure factor (main text, Eq. 3). For both the monodisperse and
polydisperse variants of the hard-sphere structure factor, the curve fittings yielded
very similar values of ReHS (monodisperse) and <ReHS> (polydisperse) parameters.
But, when <SHS(q)> was used the size distribution function (Eq. S2.5) led to smearing
out and dampening of the oscillations (arrows in Fig. S3.1).
!
30!
Fig. S3.1. SAXS pattern at 330 seconds together with fits (solid lines) in which Pcyl(q) is expressed by Eq. S2.1,
and two expression for the structure factor are compared: monodisperse SHS(q) (blue), and polydisperse < SHS(q)>
(red). Arrows mark the oscillations in the fitted curve caused by the monodisperse SHS(q). Inset presents a part of
the scattering pattern relevant for the structure factor.
S4. Structure factor during the transition between stages II and III
In order to explain the scattering contributions in the patterns collected
between 420 and 510 seconds (i.e., the transition between stages II and III), we
considered a collection of primary species, in which some of the units would be
grouped into domains of scatterers that interact via a hard-sphere structure factor
<SHS(q)>, but at the same time some of those units would start forming larger,
densely-packed aggregates. These new aggregates would be composed, at short
length-scales, of primary species, but arranged in such a way that at larger lengthscales the structure would appear to be a solid coarse surface. In such a case, the lowq regime of the scattering curves, where by definition Pcyl(q → 0) = 1, could be
characterized by the linear combination of scattering from co-existing hard-sphere
structures (domains of primary species) and surface fractals (internally denser
aggregates of primary species). This is in contrast to the high-q regime which is
dominated by scattering from the form factor and where the structure factor is
assumed as S(q → ∞) = 1. The effective structure factor of such a collection of
primary species can be expressed by the following semi-empirical expression:
!
31!
Seff (q) =< SHS (q,< ReHS >, z) > +A'⋅ SF(q, Ds )
(S4.1)
Here, the components of the sum are defined by Eqs. 3 and 4b from the main text, and
Aʹ is proportional to the relative surface of all scattering features and normalized
against the form factor in order to fulfill the condition that S(q → ∞) = 1. In its form
Eq. S4.1 is based on the similar expression for droplet-like clusters59. Between 420
and 510 seconds both contributions followed the fit as expressed by Eq. S4.1, but due
to the decreasing local volume fraction v (obtained from the fits), the relative
contribution from <SHS(q)> became small relatively fast compared to the quicklyevolving surface fractal term. After 510 seconds the scattering model contained
exclusively the latter term. Hence, we assumed that after 510 seconds <SHS(q)> → 1,
and the expression for the resulting structure factor yielded SSF(q) (Eq. 5 in the main
text). Fig. S4.1 shows a scattering curve at 450 seconds with a fit to the data including
Eq. S4.1 to represent the structure factor. For the sake of comparison the respective
contributions from <SHS(q)> and SSF(q) are also plotted.
!
Fig. S4.1. SAXS pattern at 450 seconds together with a fit (red solid line) in which Pcyl(q) is expressed by Eq.
S2.1, and the structure factor Seff(q) is expressed as a linear combination of < SHS(q)> and Aʹ·SF(q) (Eq. S4.1). The
green dashed and blue dotted lines represent fits of the corresponding expressions considering <SHS(q)> and SSF(q)
respectively. The inset shows the detail of the scattering pattern relevant for the structure factor,
!
32!
S5. Evolution in the morphology of CaSO4 precipitates after 5400 seconds
In the main text, we presented the development in the SAXS patterns up to
5400 seconds for the case of 50 mmol/L CaSO4 at 12 °C. This period contains the
most significant changes in SAXS and is therefore the most relevant part for revealing
the mechanisms of CaSO4 precipitation under the aforementioned physicochemical
conditions. Nonetheless, the actual measurements were performed further up to ~4
hours (14370 seconds). In this later period very limited changes were observed in the
SAXS signal and therefore only the final scattering pattern is compared to the last
scattering pattern from stage IV at 5400 seconds (Fig. S5.1). Throughout most of the
reaction we dealt with isotropic scattering as indicated by the circular shape of
scattered intensity in 2D (presented in Fig. S5.1A for 5400 seconds). However,
during the last stages of CaSO4 growth the 2D SAXS patterns became anisotropic, as
shown in Fig. S5.1A for 14370 seconds. If the particles are sufficiently large and
elongated, they will become aligned with respect to their long-axes in the horizontally
mounted capillary of the flow-through cell, Furthermore, if within the accessible qrange there are any orientation-dependent internal variations in the microstructure of
particles with respect to their long-axis, the resulting 2D SAXS pattern would appear
to be anisotropic. In Fig. S5.1A for the pattern collected at 14370 seconds, stronger
scattering is observed in the direction almost parallel to the Y axis of the detector
(equatorial direction), and thus normal to X axis (meridional direction). Such a
scattering pattern is expected from morphologies in which larger dimensions of
anisotropic features are oriented along the flow (and long-axes of the particles
containing these features), and smaller features are perpendicular to the flow (and
long-axis of particles). Fig. S5.1B compares 1D scattering curves from the isotropic
case at 5400 seconds, with those from the anisotropic case representing scattering in
equatorial and meridional directions at 14370 seconds. It can be seen that only minor
changes in SAXS intensity occurred between 5400 and 14370 seconds, but
characteristically at high-q the scattering pattern shifted slightly further towards lower
q-values during the more advanced stage of the process. This shows that beyond 5400
seconds, at the nano-scale, internal building units continued to grow, and more
pronounced morphologies were observed for the equatorial direction (parallel to the
flow).
!
33!
Fig. S5.1 A) 2D SAXS patterns for CaSO4 50 mmol/L at 12 °C at 5400 and 14370 seconds; B) 1D SAXS curves
for the same conditions. Shown also is the change in the I(q) dependence of the scattering exponent to emphasize
the differences in the high-q part of the data (dashed lines).
!
34!
S6. Selected physicochemical properties of CaSO4·xH2O polymorphs
In Table S6.1 we present the relevant physicochemical information regarding
the three calcium sulfate polymorphs. The solubility of CaSO4·xH2O in aqueous
solutions was calculated with the geochemical code PHREEQC19 using the LLNL
database, and is based on bulk solubility data.
Tables S6.1. Collected values of molar mass, density, electron density and solubilities for the three polymorphs
at 12 °C.
Density
[g/cm3]
Electron
density
[e-/cm3]
Solubility
in pure
H 2O
[mmol/L]
Solubility in
100 mmol/L
NaCl
solution
[mmol/L]
172.17
2.31 - 2.33
711 - 717
15.48
22.75
CaSO4·0.5H2O
(bassanite)
145.15
2.69 - 2.76
815 - 836
93.09
107.50
CaSO4 (anhydrite)
136.14
2.97
893
27.10
36.42
Molar
mass
[g/mol]
CaSO4·2H2O (gypsum)
Phase
Based on Table S6.1, we calculated that for a 50 mmol/L CaSO4 solution at 12 °C:
•
Without taking into account the solubility, one would expect an approximate
volume fractions, φ, of 0.369% - 0.373% for gypsum, 0.263% - 0.270% for
bassanite, and 0.229% for anhydrite.
•
By taking into account the bulk solubilities in pure water at 12 °C, the
expected φ is 0.256% - 0.258% for gypsum and 0.104% for anhydrite, whereas
for bassanite no solid phase is expected (i.e., undersaturated).
•
Finally when the solubility in an actual solution containing the dissolved Na+
and Cl- is also considered (which are the counter-ions to the Ca2+ in CaCl2 and
to SO42- in Na2SO4 in our original stock solutions; see methods main text), the
expected φ is 0.201% - 0.203% for gypsum and 0.062% for anhydrite, whereas
again for bassanite no solid phase is expected.
The calculated volume fraction values do not take into account the change in volume
between a solution containing only a fully dissolved phase and one with a solid phase.
However, this change is negligible because only low concentrations of reagents are
dissolved, and thus, the calculated volume fractions based on the above data can be
regarded as a very close approximation.
!
35!
S7. Selected structural aspects of the various CaSO4·xH2O polymorphs
Fig. S7.1 presents the selected structural aspects of the three CaSO4·xH2O
polymorphs.
Fig. S7.1. Visualizations of CaSO4 phases60. Projections along c-axis for A) CaSO4, anhydrite48 (AMCSD
0005117); B) CaSO4·0.5H2O, bassanite49 (AMCSD 0006909); C) CaSO4·2H2O, gypsum50 (COD 2300259);
Proposed structures of CaSO4 elongated primary species based on D) anhydrite; E) bassanite; F) gypsum.
!
36!
References
1.
Freyer, D.; Voigt, W. Crystallization and phase stability of CaSO4 and
CaSO4–based salts. Monatshefte für Chemie/Chemical Monthly 2003, 134, 693-719.
2.
Warren, J. K. Evaporites: sediments, resources and hydrocarbons. Springer
Science & Business Media: 2006.
3.
Akpokodje, E. G. The occurrence of bassanite in some Australian arid-zone
soils. Chemical Geology 1984, 47, 361-364.
4.
Allen, R.; Kramer, H. Occurence of bassanite in two desert basins in
southeastern California. American Mineralogist 1953, 38, 1266-1268.
5.
Gendrin, A.; Mangold, N.; Bibring, J.-P.; Langevin, Y.; Gondet, B.; Poulet, F.;
Bonello, G.; Quantin, C.; Mustard, J.; Arvidson, R. Sulfates in Martian layered
terrains: the OMEGA/Mars Express view. Science 2005, 307, 1587-1591.
6.
Wray, J. J.; Squyres, S. W.; Roach, L. H.; Bishop, J. L.; Mustard, J. F.;
Dobrea, E. Z. N. Identification of the Ca-sulfate bassanite in Mawrth Vallis, Mars.
Icarus 2010, 209, 416-421.
7.
Sharpe, R.; Cork, G. Gypsum and anhydrite. Industrial Minerals & Rocks:
Commodities, Markets, and Uses 2006.
8.
Weiss, H.; Bräu, M. F. How much water does calcined gypsum contain?
Angewandte Chemie International Edition 2009, 48, 3520-3524.
9.
Mineral commodity summaries 2014: U.S. Geological Survey; U.S.
Geological Survey: 2014; pp 1-195.
10.
Cruft, E. F.; Chao, P.-C. In Nucleation kinetics of the gypsum-anhydrite
system, IO9-118. 3rd Symp. on salt, 1970.
11.
Packter, A. The precipitation of calcium sulphate dihydrate from aqueous
solution: Induction periods, crystal numbers and final size. Journal of Crystal Growth
1974, 21, 191-194.
12.
Amathieu, L.; Boistelle, R. Crystallization kinetics of gypsum from dense
suspension of hemihydrate in water. Journal of Crystal Growth 1988, 88, 183-192.
13.
He, S.; Oddo, J. E.; Tomson, M. B. The nucleation kinetics of calcium sulfate
dihydrate in NaCl solutions up to 6 m and 90 C. Journal of Colloid and Interface
Science 1994, 162, 297-303.
!
37!
14.
Wang, Y.-W.; Kim, Y.-Y.; Christenson, H. K.; Meldrum, F. C. A new
precipitation pathway for calcium sulfate dihydrate (gypsum) via amorphous and
hemihydrate intermediates. Chemical Communications 2012, 48, 504-506.
15.
Wang, Y.-W.; Meldrum, F. C. Additives stabilize calcium sulfate hemihydrate
(bassanite) in solution. Journal of Materials Chemistry 2012, 22, 22055-22062.
16.
Van Driessche, A.; Benning, L.; Rodriguez-Blanco, J.; Ossorio, M.; Bots, P.;
García-Ruiz, J. The role and implications of bassanite as a stable precursor phase to
gypsum precipitation. Science 2012, 336, 69-72.
17.
Saha, A.; Lee, J.; Pancera, S. M.; Bräeu, M. F.; Kempter, A.; Tripathi, A.;
Bose, A. New Insights into the transformation of calcium sulfate hemihydrate to
gypsum using time-resolved cryogenic transmission electron microscopy. Langmuir
2012, 28, 11182-11187.
18.
Jones, F. Infrared investigation of barite and gypsum crystallization: Evidence
for an amorphous to crystalline transition. CrystEngComm 2012, 14, 8374-8381.
19.
Parkhurst, D. L.; Appelo, C. Description of input and examples for PHREEQC
version 3: a computer program for speciation, batch-reaction, one-dimensional
transport, and inverse geochemical calculations; 2328-7055; US Geological Survey:
2013.
20.
Mueller, M.; Wang, M.; Schulze-Briese, C. Optimal fine φ-slicing for single-
photon-counting pixel detectors. Acta Crystallographica Section D 2012, 68, 42-56.
21.
Huang, T. C.; Toraya, H.; Blanton, T. N.; Wu, Y. X-ray powder diffraction
analysis of silver behenate, a possible low-angle diffraction standard. Journal of
Applied Crystallography 1993, 26, 180-184.
22.
Fratzl, P.; Misof, K.; Zizak, I.; Rapp, G.; Amenitsch, H.; Bernstorff, S.
Fibrillar Structure and Mechanical Properties of Collagen. Journal of Structural
Biology 1998, 122, 119-122.
23.
Bateman, J.; Derbyshire, G.; Diakun, G.; Duxbury, D.; Fairclough, J.; Harvey,
I.; Helsby, W.; Mai, S.-M.; Mykhaylyk, O.; Sankar, G. First user results from the
HOTWAXS detector system. Nuclear Science, IEEE Transactions on 2008, 55, 729733.
24.
Smith, A. Software Manuals for SAXS.
http://confluence.diamond.ac.uk/display/SCATTERWEB/Software+Manuals+for+SAXS
(accessed November 2013).
!
38!
25.
Cheary, R.; Coelho, A. Programs XFIT and FOURYA, deposited in CCP14
powder diffraction library; Engineering and Physical Sciences Research Council,
Daresbury Laboratory: Warrington, England, 1996.
26.
Guinier, A.; Fournet, G. Small angle scattering of X-rays. J. Wiley & Sons,
New York 1955.
27.
Beaucage, G. Approximations Leading to a Unified Exponential/Power-Law
Approach to Small-Angle Scattering. Journal of Applied Crystallography 1995, 28,
717-728.
28.
Stawski, T. M.; Besselink, R.; Veldhuis, S. A.; Castricum, H. L.; Blank, D. H.
A.; ten Elshof, J. E. Time-resolved small angle X-ray scattering study of sol-gel
precursor solutions of lead zirconate titanate and zirconia. Journal of Colloid and
Interface Science 2012, 369, 184-192.
29.
Kotlarchyk, M.; Chen, S. H. Analysis of small angle neutron scattering spectra
from polydisperse interacting colloids. The Journal of Chemical Physics 1983, 79,
2461-2469.
30.
Kinning, D. J.; Thomas, E. L. Hard-sphere interactions between spherical
domains in diblock copolymers. Macromolecules 1984, 17, 1712-1718.
31.
Beaucage, G.; Schaefer, D. W. Structural studies of complex systems using
small-angle scattering: a unified Guinier/power-law approach. Journal of NonCrystalline Solids 1994, 172–174, Part 2, 797-805.
32.
Stawski, T. M.; Veldhuis, S. A.; Besselink, R.; Castricum, H. L.; Portale, G.;
Blank, D. H. A.; ten Elshof, J. E. Nanostructure Development in AlkoxideCarboxylate-Derived Precursor Films of Barium Titanate. The Journal of Physical
Chemistry C 2012, 116, 425-434.
33.
Besselink, R.; Stawski, T. M.; Castricum, H. L.; ten Elshof, J. E. Evolution of
microstructure in mixed niobia-hybrid silica thin films from sol-gel precursors.
Journal of Colloid and Interface Science 2013.
34.
Pedersen, J. Determination of size distribution from small-angle scattering
data for systems with effective hard-sphere interactions. Journal of Applied
Crystallography 1994, 27, 595-608.
35.
Flory, P. J. Molecular Size Distribution in Three Dimensional Polymers. III.
Tetrafunctional Branching Units. Journal of the American Chemical Society 1941, 63,
3096-3100.
!
39!
36.
Bale, H. D.; Schmidt, P. W. Small-Angle X-Ray-Scattering Investigation of
Submicroscopic Porosity with Fractal Properties. Physical Review Letters 1984, 53,
596.
37.
Wong, P.-z.; Bray, A. J. Small-angle scattering by rough and fractal surfaces.
Journal of Applied Crystallography 1988, 21, 786-794.
38.
Cahill, C.; Benning, L.; Barnes, H.; Parise, J. In situ time-resolved X-ray
diffraction of iron sulfides during hydrothermal pyrite growth. Chemical Geology
2000, 167, 53-63.
39.
Davidson, L. E.; Shaw, S.; Benning, L. G. The kinetics and mechanisms of
schwertmannite transformation to goethite and hematite under alkaline conditions.
American Mineralogist 2008, 93, 1326-1337.
40.
Lutsko, J. F. A dynamical theory of nucleation for colloids and
macromolecules. The Journal of Chemical Physics 2012, 136, 034509.
41.
Lutsko, J. F.; Nicolis, G. Theoretical evidence for a dense fluid precursor to
crystallization. Physical Review Letters 2006, 96, 046102.
42.
Anwar, M.; Dorosz, S.; Schilling, T.; Berryman, J. T. On the importance of
density fluctuations to the process of crystal nucleation from the hard sphere fluid.
arXiv preprint arXiv:1503.07732 2015.
43.
Ostwald, W. Über die vermeintliche Isomerie des roten und gelben
Quecksilberoxyds und die Oberflächenspannung fester Körper. Z. phys. Chem 1900,
34, 495-503.
44.
Freundlich, H. Kapillarchemie. Akademische Verlagsgesellschaft: 1922.
45.
McHale, J.; Auroux, A.; Perrotta, A.; Navrotsky, A. Surface energies and
thermodynamic phase stability in nanocrystalline aluminas. Science 1997, 277, 788791.
46.
Navrotsky, A. Energetic clues to pathways to biomineralization: Precursors,
clusters, and nanoparticles. Proceedings of the National Academy of Sciences of the
United States of America 2004, 101, 12096-12101.
47.
Lin, Z.; Gilbert, B.; Liu, Q.; Ren, G.; Huang, F. A thermodynamically stable
nanophase material. Journal of the American Chemical Society 2006, 128, 6126-6131.
48.
Hawthorne, F.; Ferguson, R. Anhydrous sulphates. II : Refinement of the
crystal structure of anhydrite. The Canadian Mineralogist 1975, 13, 289-292.
!
40!
49.
Ballirano, P.; Maras, A.; Meloni, S.; Caminiti, R. The monoclinic I2 structure
of bassanite, calcium sulphate hemihydrate (CaSO4· 0.5 H2O). European Journal of
Mineralogy 2001, 13, 985-993.
50.
Henry, P. F.; Weller, M. T.; Wilson, C. C. Neutron powder diffraction in
materials with incoherent scattering: an illustration of Rietveld refinement quality
from nondeuterated gypsum. Journal of Applied Crystallography 2009, 42, 11761188.
51.
Keefer, K.; Schaefer, D. Growth of fractally rough colloids. Physical Review
Letters 1986, 56, 2376.
52.
Teixeira, J. Small-angle scattering by fractal systems. Journal of Applied
Crystallography 1988, 21, 781-785.
53.
Sorensen, C. M. Light Scattering by Fractal Aggregates: A Review. Aerosol
Science and Technology 2001, 35, 648 - 687.
54.
Kolb, M.; Herrmann, H. Surface fractals in irreversible aggregation. Physical
review letters 1987, 59, 454.
55.
Sorensen, C. M.; Cai, J.; Lu, N. Test of static structure factors for describing
light scattering from fractal soot aggregates. Langmuir 1992, 8, 2064-2069.
56.
Tritschler, U.; Van Driessche, A. E.; Kempter, A.; Kellermeier, M.; Cölfen, H.
Controlling the Selective Formation of Calcium Sulfate Polymorphs at Room
Temperature. Angewandte Chemie International Edition 2015, 54, 4083-4086.
57.
Tritschler, U.; Kellermeier, M.; Debus, C.; Kempter, A.; Coelfen, H. A simple
strategy for the synthesis of well-defined bassanite nanorods. CrystEngComm 2015.
58.
Ossorio, M.; Van Driessche, A.; Pérez, P.; García-Ruiz, J. The gypsum–
anhydrite paradox revisited. Chemical Geology 2014, 386, 16-21.
59.
Narayanan, T. Synchrotron Small-Angle X-Ray Scattering Studies of
Colloidal Suspensions. In Applications of Synchrotron Light to Scattering and
Diffraction in Materials and Life Sciences, Springer: 2009; pp 133-156.
60.
Momma, K.; Izumi, F. VESTA 3 for three-dimensional visualization of
crystal, volumetric and morphology data. Journal of Applied Crystallography 2011,
44, 1272-1276.
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