Size-tunable germanium particles prepared by self-sustaining reduction of germanium oxide

Article abstract of DOI:10.1016/j.jssc.2018.11.005

Here, we report on a size-controlled synthesis of highly crystalline germanium particles, via the self-sustained reaction of GeO₂ with Zn, Mg, and NaBH₄. The thermodynamic analysis suggests that the GeO₂ + 2Zn and GeO₂ + NaBH₄ systems are characterized by moderate heats of reaction (?121 and ?204 kJ per mole of Ge, respectively), while GeO₂ + 2Mg exhibits a much higher reaction enthalpy (?623 kJ). Magnesium reduction has been found not to be suitable for the preparation of germanium crystals due to the high reaction temperature, which exceeds 2000 °C. The partial substitution of Zn by Mg, however, enables increasing the overall reaction temperature in the low caloric GeO₂ +Zn system. By using different reducers and their mixtures, the reaction temperature was optimized to be in the 600–1100 °C range. Such temperature modifications allow for control of the germanium particle sizes, ranging from 200 nm to 2 mm. Thermal analysis of the reacting mixtures and electron microscopy examination of the products indicates that dissolution-precipitation is the dominant formation mechanism of the germanium crystals in the GeO₂ + Zn system. The higher reaction temperatures in the GeO₂ + Zn + Mg and GeO₂ + NaBH₄ systems cause melting, and subsequent coalescence of the primarily precipitated germanium resulting in much larger particles.

Full text of DOI:10.1016/j.jssc.2018.11.005

Author’s Accepted Manuscript
Size-tunable Germanium Particles Prepared by
Self-sustaining Reduction of Germanium Oxide
Khachatur V. Manukyan, Ryan S. Schools,
Alexander S. Mukasyan
www.elsevier.com/locate/yjssc
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S0022-4596(18)30490-0
https://doi.org/10.1016/j.jssc.2018.11.005
YJSSC20461
DOI:
Reference:
To appear in: Journal of Solid State Chemistry
Received date: 18 August 2018
Revised date: 5 October 2018
Accepted date: 5 November 2018
Cite this article as: Khachatur V. Manukyan, Ryan S. Schools and Alexander S.
Mukasyan, Size-tunable Germanium Particles Prepared by Self-sustaining
Reduction of Germanium Oxide, Journal of Solid State Chemistry,
https://doi.org/10.1016/j.jssc.2018.11.005
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Size-tunable Germanium Particles Prepared by Self-sustaining Reduction
of Germanium Oxide
a*
b
b,c
Khachatur V. Manukyan , Ryan S. Schools and Alexander S. Mukasyan
a
Nuclear Science Laboratory, Department of Physics, University of Notre Dame, Notre Dame,
6556, IN, USA.
4
b
Department of Chemical and Biomolecular Engineering, University of Notre Dame, Notre
Dame, 46556, IN, USA.
c
National University of Science and Technology, ‘‘MISIS’’, Moscow 119049, Russia
*
Email: kmanukya@nd.edu, Phone: +1 574 631 5083
Abstract
Here, we report on a size-controlled synthesis of highly crystalline germanium particles, via the
self-sustained reaction of GeO with Zn, Mg, and NaBH . The thermodynamic analysis suggests
2
4
that the GeO +2Zn and GeO +NaBH systems are characterized by moderate heats of reaction (-
2
2
4
1
21 and -204 kJ per mole of Ge, respectively), while GeO +2Mg exhibits a much higher reaction
2
enthalpy (-623 kJ). Magnesium reduction has been found not to be suitable for the preparation of
o
germanium crystals due to the high reaction temperature, which exceeds 2000 C. The partial
substitution of Zn by Mg, however, enables increasing the overall reaction temperature in the
low caloric GeO +Zn system. By using different reducers and their mixtures, the reaction
2
o
temperature was optimized to be in the 600 - 1100 C range. Such temperature modifications
allow for control of the germanium particle sizes, ranging from 200 nm to 2 mm. Thermal
analysis of the reacting mixtures and electron microscopy examination of the products indicates
that dissolution-precipitation is the dominant formation mechanism of the germanium crystals in
the GeO +Zn system. The higher reaction temperatures in the GeO +Zn+Mg and GeO +NaBH
2
2
2
4
systems cause melting, and subsequent coalescence of the primarily precipitated germanium
resulting in much larger particles.
Graphical abstract:
1

Size-tunable Germanium Particles
Key words: Germanium, crystal, nanoparticles. self-sustaining reaction, germanium oxide
1
. Introduction
Germanium exhibits high electron and hole mobility and consequently has been used in
electronic, optoelectronic devices, and radiation detectors.[1–4] Germanium-based materials are
also excellent candidates to be used in lithium-ion batteries, solar cells, catalysts in
polymerization reactions, and other fields.[5–11].
The conventional germanium preparation method involves the reduction of germanium
o
oxide (GeO ) by hydrogen at temperatures above 1000 C [12],[13]. The carbothermal reduction
2
of GeO produces lower grade germanium [12]. The volatile GeO intermediate forms by the
2
GeO + Ge = GeO reaction during these lengthy processes leads to significant losses in
2
germanium [12],[14]. Recently proposed preparation methods for Ge includes electrochemical
reduction of GeO [15], plasma synthesis [16],[17], solution-phase synthesis [18], laser ablation
2
[
19], solid-phase synthesis using organogermanium compounds [10], and thermal
disproportionation of GeO [20].
Several works have reported a magnesium reduction method for the preparation of
elemental germanium [7],[21]. Jia et al. mixed GeO and Mg powder, then the mixture was
2
2

slowly heated in a tube furnace to 650 °C for 7 h under the flow of Ar-5%H gas mixture [21].
2
Then the obtained powder was treated with a HCl solution to remove the MgO and Ge Mg by-
2
products. After purification, the sizes of porous germanium particles was ~ 500 nm. Cheng et al.
reported a similar procedure to produce germanium-carbon composites using GeO , Mg, and
2
glucose as reactants [7]. The carbon produced by glucose decomposition forms a thin layer on
the germanium particles, enhancing its electrochemical performance as an anode material in the
lithium-ion battery.
Magnesium reduction of oxides is highly exothermic and can proceed in a self-sustained
manner upon short local initiation [22–25]. For example, we reported the synthesis of silicon by
self-sustaining reaction in SiO +2Mg [24]. Compact cylindrical samples made of this mixture
2
were heated locally by a hot tungsten filament. After initiation, the reaction takes the form of a
moving wave and propagates along the sample. The maximum reaction temperature in this
o
system is ~1600 C, and the reduced silicon melts. However, by adding the desired quantity of
NaCl powder to the SiO +2Mg mixture, we were able to control reaction temperature and to
2
facilitate the formation of nanoscale silicon. Another approach to decrease the overall heat of the
process is the partial substitution of magnesium by carbon or zinc [23,26–28].
In this work, we investigate the self-sustained (combustion) reaction of GeO with Zn,
2
Mg or NaBH to produce germanium. We found that the type of reducer significantly influences
4
the reaction temperature and thus the resulting germanium microstructure. By using different
reducing agents, we were able to control the germanium particle size from 200 - 2,000,000 nm.
2
. Materials and Methods
GeO powder (Sigma Aldrich, 99.99%) was used as an initial precursor to fabricate the
2
germanium materials. Zinc (Alfa Aesar, 99.9%, particle size ranging from 5 to 7 m),
magnesium (Alfa Aesar, 99.8%, particle size below 30 m) and sodium borohydride (VWR,
BDH chemicals, ACS grade) powders were applied as reducing agents. The desired amounts of
reactants were thoroughly hand-mixed in a ceramic mortar for 15 minutes. In a typical synthesis
process, the reactant mixtures were compressed into cylindrical samples with 12 mm diameter
and 25 mm height (35 - 45% theoretical maximum density) by uniaxial pressing. Then samples
were inserted into a steel reactor. The reactor was sealed, evacuated and then purged with argon
gas (99.9998% purity, Airgas) and filled to 0.5MPa pressure.
3

The combustion process was initiated by passing a direct current pulse through a tungsten
wire located close to the upper edge of the samples. The pulse was turned off immediately after
reaction initiation. A combustion wave propagated in a self-sustained manner along the sample
volume, converting the initial mixture into the final product. To measure temperature during the
combustion, a K-type thermocouple with 100 m wire diameter was inserted into the samples.
The processed materials that contained zinc or magnesium were treated with 5%
hydrochloric acid solution to remove metal impurities. After leaching the powders were vacuum-
filtered and dried overnight at room temperature under vacuum. The product of GeO +NaBH
2
4
mixtures were subjected to warm deionized water treatment to remove NaBO by-product.
2
A differential scanning calorimeter (DSC, Mettler-Toledo) was used to investigate the
behavior of precursors and initial mixtures under linear heating conditions. In these experiments,
1
0 mg of the powders were placed in aluminum crucibles and transferred into the DSC cell.
o
Then, argon gas was allowed to flow for 1 h before heating the samples with 50 C/min rate.
The crystallinity and phase composition of the materials were determined by X-ray
diffraction (XRD) analysis with Ni-filtered CuKa radiation (D8 Advance, Bruker), operated at 40

kV and 40 mA with the step-scan size of 0.02 and a counting time of 15 s for the angular range
of 10-90 (2). X-ray photoelectron spectroscopy (XPS) measurements were carried out by a PHI
VersaProbe II spectrometer with an Al Kα X-ray source operating at 1486.6 eV and a 90°takeoff
angle for near surface analysis of Ge 2p3/2 and O 1s electronic transitions. Binding energy
values were referenced to the C 1s peak (284.8 eV) that resulted from the adventitious
contamination layer. The product morphologies were examined by a field-emission scanning
electron microscope (SEM. Magellan 400, FEI). Atomic-level structure analysis was performed
on a Titan 80-300 (FEI) transmission electron microscope (TEM) at 300 kV in a high-resolution
transmission electron microscopy mode. BET specific surface area of samples was measured by
ASAP 2020 instrument (Micromeritics).
3
3
. Results
.1.Thermodynamic consideration
Thermodynamic analyses have been performed for the GeO +2Zn, GeO +2Mg, and
2
2
GeO +NaBH systems using the THERMO software package [29], which allows for the
2
4
calculation of adiabatic reaction temperature (T ) and product equilibrium compositions
ad
4

utilizing the Gibbs free energy minimization principle. Table 1 summarizes the results of
calculation along with the heat of reaction (H) for each system. The calculations indicate that
the GeO +2Zn and GeO +NaBH systems are characterized by moderate heats of reaction, while
2
2
4

H for the GeO +2Mg system is significantly higher.
2
The calculated adiabatic combustion temperatures for the GeO +2Zn and GeO +NaBH
4
2
2
o
o
systems are 938 and 1165 C, respectively. T for the GeO +2Mg system is 2748 C. The
ad
2
equilibrium products for GeO +2Zn and GeO +2Mg reactions are Ge as well as ZnO or MgO,
2
2
respectively. The product for the GeO +NaBH system consists of Ge, NaBO , and H . It should
2
4
2
2
be noted that hydrogen reduction of GeO is thermodynamically unfavorable at temperatures
2
o
below 1000 C.
It is known that the reduction degree and product microstructure of such reactive systems
depends on the combustion temperature. Based on above data, we can expect that the relatively
low caloric GeO +2Zn and GeO +NaBH systems could be used to prepare germanium powders
2
2
4
with fine particles. Significantly higher reaction temperatures in the GeO +2Mg system would
2
cause reactant as well as germanium vaporization, making magnesium an inappropriate reducer.
However, magnesium can be used as an additive to control the reaction temperature in a low
caloric system such as GeO +2Zn.
2
Figure 1 summarizes the results of detailed thermodynamic calculation for GeO +(2-
2

)Zn+Mg system depending on magnesium amount () and inert gas (argon) pressure. It can
be seen that from = 0 - 0.3 the reaction temperature gradually increases with Mg quantity
Figure 1A). The gas pressure does not influence the temperature in this region. At higher 
(
values Tad depends on both the magnesium amount and gas pressure (Figure 1A). Such
significant variation of reaction temperature influences the phase composition of the products
(Figure 1B-D). At higher  values, the high temperature causes partial vaporization of zinc, thus
decreasing the conversion degree of GeO to Ge. This analysis suggests that the  parameter
2
should be carefully optimized to obtain full conversion of the oxide to germanium. The
calculations also show that the optimum amount of magnesium depends on gas pressure (Figure
1
B-D). It is worth noting that more zinc can be substituted by magnesium at higher gas
pressures, which, in principle, allows for variation of the reaction temperature in a wide range
and for tailoring of the product microstructure.
3
.2. Reaction of GeO with different reducers
2
5

Based on the thermodynamic calculation results the combustion of the stoichiometric
GeO +2Zn mixture at 0.5 MPa argon gas pressures has been investigated. A thermocouple
2
o
inserted in the reacting samples indicated that the maximum reaction temperature is ~600 C
o
(
Figure 2, curve a), which is lower than the thermodynamically predicted Tad (937 C). This
effect is related to the heat losses due to the thermal exchange with the gas environment.
XRD of reaction product (Supplementary Figure S1) shows intense diffraction peaks for
Ge and ZnO, as well as traces of GeO . The acid-treated product contains Ge and a small
2
quantity of non-reduced GeO (Figure 3a). The results imply that the relatively short combustion
2
process and low reaction temperatures resulted in incomplete germanium reduction. To verify
o
this hypothesis, we placed the same initial GeO +2Zn mixture in a furnace and heated at 600 C
2
for 30 min. The XRD pattern of this product proves the hypothesis showing that GeO was
2
entirely reduced to Ge (Figure S2).
Thermodynamic calculations predicted that a possible way to increase reaction
temperature and conversion degree is a partial substation of Zn by Mg. To verify this prediction,
we investigated combustion in the GeO +1.75Zn+0.25Mg mixture at 0.5 MPa argon gas
2
pressure. The thermocouple measurements suggest that the maximum combustion temperature in
o
this mixture is ~1000 C. Thus substituting 0.25 moles of zinc with magnesium increases the
o
reaction from temperature from 610 to 1000 C (Figure 2, curve b), which is still below the
o
thermodynamically predicted T (~1350 C) for this mixture. The XRD pattern displays strong
ad
diffraction peaks for Ge, ZnO, and a small quantity of MgO (Figure S1). The XRD pattern of the
acid-purified powder consists of only diffraction peaks for germanium, implying that GeO was
2
reduced under the combustion conditions (Figure 3b). The results of the XPS analysis (Figure
S3) confirm that the surface of the purified Ge particles is partially oxidized, showing a broad
peak with the Ge(II) component at ~1221 eV. XPS analysis also suggests that the estimated
oxygen content on surface is about ~13 at%.
We also investigated the combustion reaction in the GeO +NaBH mixture. However, our
2
4
attempts to measure the maximum reaction temperature were unsuccessful due to the melting
and intensive mass transfer processes during the reaction. It should be noted that the initial
cylindrical samples, with a diameter of 12 mm, became tubes with 6-7 mm openings after the
reaction wave propagation (Figure S4). We can assume that reaction temperature in the center of
o
the samples exceeds the melting points of both reaction products, i.e., Ge (938 C) and NaBO
2
6

o
(
966 C). The molten products tend to flow toward colder, not molten, edge of the sample thus
forming open space. Closer inspection of broken reacted samples shows double-phase structure
Figure S4). The XRD pattern of the ground product mixture indicates diffraction peaks for both
(
Ge and NaBO (Figure S1), which agrees with the thermodynamically predicted equilibrium
2
phase composition (Table 1). The treatment of the reacted samples with warm water allows
dissolution of the NaBO and obtaining course Ge powder as evidenced by XRD analysis (Figure
2
3
c).
SEM images of the initial GeO powder and the materials produced by three different
2
reducers are presented in Figure 4. Although some large (~200 nm) particles can be seen in SEM
image of GeO powder, it mainly consists of particles with sizes ranging from 20 to 80 nm
2
(
Figure 4A). The combustion products from the GeO +2Zn mixture involves of two types of
2
particles with different sizes (Figure 4B); porous agglomerates of small particles (20-80 nm) and
larger ones (500-1000 nm) with smooth surfaces. SEM images (Figure 4C) for the product of the
GeO +1.75Zn+0.25Mg mixture shows large germanium particles (1,000 - 3,000 nm) with near-
2
spherical shape. These particles are sintered together forming porous aggregates. Course
germanium powder (Figure 4D) prepared by combustion of the GeO +NaBH mixture consists
2
4
of much larger (200,000 to 2,000,000 nm) particles. Some of these particles exhibit near-
spherical shape, while others have a distinctive conical “tails” (Figure S5).
4
. Discussion
The results indicate that the self-sustaining reduction reactions of GeO offer a rapid and
2
energy saving preparation approach for germanium. It is more important that it allows for control
over the produced particle size. Another advantage of this approach over existing high-
temperature methods, whether hydrogen or carbon reduction, is negligible germanium loses due
to the formation of volatile GeO intermediate.
SEM images shown in Figure 4 provide some insight on the product microstructure
formation mechanism during the combustion reactions. We also used TEM and surface area
measurements to investigate the product microstructure as well as DSC analysis of the initial
mixture to understand the structure formation mechanism in more detail.
Bright-field TEM shows the existence of small (20-50 nm) and larger (200-500 nm)
particles in the acid-purified product of the GeO +2Zn reaction (Figure 5A). The selected area
2
electron diffraction pattern (inset in Figure 5A) implies both fine diffraction rings, correspond to
7

GeO , and brighter diffraction spots representing the Ge phase (see the interpretation of the
2
electron diffraction pattern in Table S1). This result is in good agreement with the XRD pattern
for the GeO +2Zn product (Figure 3a). High-resolution TEM image of the same material
2
indicates that small nanoparticles are anchored on the surface of larger particles (Figure 5B).
EDS analysis (not shown) suggest that the smaller particles contain germanium and oxygen,
while the larger particles are essentially oxygen free. These findings suggest that a significant
portion of small GeO particles transformed into the much larger Ge particles during zinc
2
reduction. BET surface area measurements also confirm this assumption. The initial GeO has 7
2
2
2
m /g surface area, while the acid-purified powder produced exhibit only 0.5 m /g area. Such a
significant modification in the microstructure can be attributed to the formation of an
intermediate liquid phase during the process.
The DSC curve of the GeO +2Zn mixture (Figure 6a) contains an endothermic peak
2
o
o
centered at the 405-445 C interval. We assigned this peak to the zinc melting (m.p.~420 C). A
o
broad peak at 470–570 C corresponds to the exothermic reaction between GeO and Zn. Such a
2
sequence of peaks suggests that zinc melts within the combustion wave and the reaction starts
between solid GeO particles and molten zinc. We can suggest that the germanium nuclei form at
2
o
the melt(Zn)/solid(GeO ) interface. The Ge-Zn phase diagram exhibits a eutectic point at 395 C
2
and 94.2 atom % Zn [30], suggesting that the reduced germanium is gradually dissolved in
molten zinc. Increasing the amount of dissolved germanium and conversion of molten zinc to
solid ZnO would cause precipitation of germanium nuclei followed by their growth (Figures 4b
o
and 5a). The relatively low reaction temperature (~600 C) makes the interaction of residual Zn
and GeO challenging under the relatively short combustion process. Longer (30 min) heating of
2
o
the same mixture at 600 C provides conversion of GeO to Ge.
2
Substituting part of zinc with magnesium in the GeO +1.75Zn+0.25Mg mixture increases
2
o
the maximum reaction temperature to ~1000 C. The DSC curve for this mixture exhibits an
o
additional endothermic peak at ~640 C for magnesium melting followed by a sharp exothermic
peak corresponding to GeO +Mg interaction (Figure 6b). The higher temperature results in
2
o
melting of the precipitated germanium (m.p. ~940 C) particles, and their coalescence leads to
the formation of porous agglomerates (Figure 4c). The presence of a large quantity of solid ZnO
(and some MgO) prevents further coalescence of the germanium droplets, thus limiting the size
of agglomerates to ~5-10 m.
8

o
The reaction onset temperature for GeO +NaBH was determined to be ~500 C (Figure
2
4
6
c). The DSC curve shows that the GeO +NaBH system is characterized by a single-step highly
2 4
exothermic process, which coincides with the endothermic decomposition of pure NaBH4
Figure S6). We can suggest that the decomposition products of NaBH react with GeO .
(
4
2
Martelli et al showed earlier that NaBH forms elemental Na and B as well as Na-H and Na-B
4
binary compounds as residual products upon decomposition [31]. Our thermodynamic
calculation (not shown) demonstrated that both the GeO +Na and GeO +B systems are highly
2
2
exothermic, indicating that after NaBH decomposition both elements can readily react with
4
GeO . The molten sodium would accelerate the GeO reduction kinetics. The produced
2
2
germanium can partially dissolve in molten sodium. The Ge-Na phase diagram [32], however,
indicates the presence of a GeNa compound which was not identified in our product.
3
Thermodynamically the formation of NaBO is favorable, indicating that the dissolved
2
germanium in the melt would precipitate rather than form GeNa . The large particles (Figure 4D
3
and Figure S4) of purified product suggest that the higher reaction temperatures cause both
NaBO (m.p. 966°C) and Ge to melt. Molten germanium coalescence to large droplets since two
2
mutually insoluble fluids separate from each other (Figure S5).
We can suggest that the dissolution-precipitation mechanism of germanium is prevailing
in all system at the initial stages. Due to the low reaction temperatures in the GeO +2Zn system,
2
germanium particles form solely by this mechanism. Substituting magnesium into the
o
GeO +1.75Zn+0.25Mg mixture increases the reaction temperature to ~1000 C thus causing
2
melting and partial coalescence of the precipitate particles. A significant quantity of solid oxide
(ZnO, MgO) limits coalescence of molten germanium droplets. The higher reaction temperatures
and melting of both NaBO and Ge create conditions for Ge droplets to coalesce and form much
2
larger particles upon solidification.
5
. Conclusions
This work demonstrates that self-sustaining reactions of GeO with different reducers Zn, Mg,
2
and NaBH offer a rapid and energy saving preparation approach of size-tunable germanium
4
particles. The flexibility of this method to adjust the self-generating synthesis temperature allows
for control of the germanium particles sizes from 200 nm to 2 mm. The results of this work
suggest that the dissolution-precipitation mechanism of germanium formation prevails in the
GeO +2Zn system. Substituting a part of Mg in the GeO +1.75Zn+0.25Mg mixture increases the
2
2
9

reaction temperature, causing melting and partial coalescence of the precipitated particles. A
significant quantity of solid zinc and magnesium oxides limits the coalescence of the molten
germanium, leading to the formation of micron size agglomerates. Formation of NaBO and Ge
2
molten phases during the GeO +NaBH reaction creates conditions for Ge droplets to coalesce
2
4
and form large (mm) droplets upon solidification.
Acknowledgments
ASM acknowledges the support by the Ministry of Education and Science of the Russian
Federation (Increase Competitiveness Program of NUST «MISiS» К2-2017-083.The authors are
grateful to Dr. C.E. Shuck and Mr. J. Pauls for useful discussions. We also acknowledge the
support of the ND Materials Characterization Faculty for performing XPS analysis.
Supplementary information
X-ray diffraction (XRD) patterns for reaction produces, additional scanning electron microscopy
(SEM) images of products and differential scanning calorimetry (DSC) analysis for NaBH4.
Tables for structure and microstructural characteristics.
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1
3

Table 1: The result of thermodynamic calculations
System
Heat of reaction,
kJ/mole Ge
-121
Adiabatic
Equilibrium product, moles
o
temperature, C
GeO +2Zn
938
2748
1165
Ge, 2ZnO
Ge, 2MgO
2
GeO +2Mg
-623
2
GeO +NaBH
-204
Ge, NaBO , 2H2
2
4
2
1
4

1
5

Figure 1: The results of thermodynamic analysis for GeO +(2-)Zn+Mg system depending on
2
magnesium amount () and inert gas pressure. Calculated adiabatic temperature (A) and
composition of equilibrium products (B-D).
1
6

Figure 2: Time – temperature profile for combustion of the GeO + 2Zn (a) and GeO + 1.75Zn +
2
2
0
.25Ge (b) mixtures.
1
7

Figure 3: XRD patterns of purified products for GeO + 2Zn (a), GeO + 1.75Zn + 0.25Ge (b)
2
2
and GeO + NaBH (c) mixtures.
2
4
1
8

Figure 4: SEM images of initial GeO powder (A) and purified products for GeO + 2Zn (B),
2
2
GeO + 1.75Zn + 0.25Mg (C) and GeO + NaBH (D) mixtures.
2
2
4
1
9

Figure 5: A bright-field TEM (A) and high-resolution TEM (B) images of purified products
for GeO + 2Zn. Electron diffraction pattern (inset in A). Interpretation of the diffraction
2
pattern summarized in Table S1.
2
0

Figure 6: DSC curves for GeO + 2Zn (a), GeO + 1.75Zn + 0.25Ge (b) and GeO + NaBH (c)
2
2
2
4
mixtures.
2
1