|  Role of precursor composition in the polymorph transformations, morphology
control and ferromagnetic properties of nanosized TiO2
---|---
|  Dmitry Zablotsky,∗a Mikhail M. Maiorov,a Aija Krumina,a,b Marina Romanova,c
and Elmars Blumsa
|  Pure phase and mixed phase TiO2 nanoparticles have been produced using a
pyrolytic method from a non-aqueous carboxylate precursor. The precursor was
prepared by a multiphase cation exchange using pentanoic acid (C4H9COOH). The
thermal stability, polymorph content, morphology, size distribution and
surface region of the produced nanoparticles were studied by TGA/DSC, XRD,
FTIR and TEM. High quality monodisperse nanoparticles have been produced in
the size range from 7 to 27 nm. The nanoparticles showed room temperature
ferromagnetism revealed by VSM within bound polaron model. The carboxylate
precursor is a good alternative to standard sol-gel to produce nanoparticles
free from impurities.

##

††footnotetext: a University of Latvia, Jelgavas 3, 1004 Riga, Latvia; E-mail:
<EMAIL_ADDRESS>b Institute of Inorganic Chemistry,
Riga Technical University, Faculty of Material Science and Applied Chemistry,
P. Valdena 7, 1048 Riga, Latvia.††footnotetext: c Institute of Biomedical
Engineering and Nanotechnologies, Riga Technical University, Viskalu 36A, 1006
Riga, Latvia.

##  1 Introduction

Titanium dioxide (TiO2) is a multifunctional semiconducting metal oxide (SMO)
that is expected to play a significant role in environmental and energy
applications 1, 2. Amongst all SMOs (such as WO3, ZnO, Fe2O3, MnO2 etc.) TiO2
is the most intensively studied because its chemically inert, reusable,
abundant and inexpensive, non-toxic and biologically compatible and has a wide
band gap ($\sim$3 eV), which makes it an ideal photocatalyst that can be used
to break down and mineralize organic pollutants, inhibit bacterial growth,
self-sterilize surfaces or split water to hydrogen and oxygen by solar light3.
In the past decades a tremendous effort has been put into improving the
efficiency of TiO2 materials. Titania crystallizes in three distinct
polymorphs with different properties and structure: rutile (tetragonal, space
group P42/mmm), anatase (tetragonal, space group I41/amd) and brookite
(orthorhombic, space group Pcab). Bi-phasic anatase/rutile mixtures are most
wanted than pure phases because of superior properties 4, 5, 6, 7, 8. The
established standard - Aeroxide TiO2 P25 marketed by Evonik Industries
(Degussa) – a mixed phase photocatalyst consisting of about 70-80 wt.% anatase
and 20-30 wt.% rutile, has been extensively studied, benefited from
synergistic interaction of the phases 9, 10, 11, 12, 13. Fundamentally, a
considerable problem is a very limited range of precursors and process
chemistries to produce nanostructured TiO2. The solution-based sol–gel
chemistry is commonly used to produce TiO2 nanocrystallites. It starts from
either an organometallic titanium alkoxide (Ti ethoxide, butoxide or
isopropoxide) or inorganic titanium chloride (TiCl4 or TiCl3) and proceeds via
a two-step process: hydrolysis of the salt by added water resulting in the
formation of intermediate species (monomers) and their subsequent self-
assembly and polymerization into extended 3D network. The colloidal suspension
(sol) is precipitated, dried and calcined to complete the phase
transformation.

Previous studies have shown that during sol-gel the phase content, crystalline
structure, size and morphology of particles were found to be highly sensitive
to various parameters, such as the Ti:H2O ratio, anion content (e.g. Cl-), the
pH, and the choice of reaction modulator 14, 15, 16, 17, 18, 19, 20. The
hydrolysis of the Ti precursor is a critical step, which determines the
initial monomer species and strongly impacts the resultant phase 17, 21. For
instance, a change of molar ratios in the chemical environment (such as
decrease in instantaneous concentration of Ti precursor) results in different
hydrolysis speeds and time-dependent shifts of polymorph selectivity 22. The
complexity of the method and the lack of detailed understanding of the
chemical equilibrium of the species in solution and kinetics of nucleation and
growth of the different phases makes it difficult to achieve and reproduce a
substantial control over the relative phase content, usually obtaining a final
mixture of anatase, rutile and brookite. Secondly, there are safety concerns
over direct use of Ti alkoxides or chlorides in sol-gel as they are in general
very reactive and sensitive to moisture. The hydrolysis is vigorous (even at 0
∘C), hence, extensive precautions are necessary in handling these chemicals.
Likewise, the sol-gel generates highly corrosive acidic conditions, since the
alkoxide route requires acid catalysts (e.g. HCl, HNO3, H2SO4 17, 20, 22) to
control the steps in the reaction and phase selectivity, whereas chloride
precursors (TiCl4 or TiCl3) produce large amounts of corrosive HCl in situ
during hydrolysis, which results in chloride-rich solutions with very low pH.
Moreover, extensive dialysis/purification procedures are required to remove
the byproducts of sol-gel synthesis adsorbed on the particle surface and
strongly bound (e.g. Cl-, NO${}_{3}^{-}$, NH${}_{4}^{+}$) species obtained
from precursors cannot be totally removed 19, 23. The doping impurities,
secondary impurity phases and surface-adsorbed features definitely influence
the surface properties, which can hinder the application potential of the
material. For instancce, Aeroxide (Degussa) P25 - the most successful and used
photocatalyst - is instead produced by pyrogenic flame-hydrolysis, in which
vapourised TiCl4 is combusted in an oxy-hydrogen flame 24. The solid is then
separated and treated with steam at 450-550 ∘C to remove chlorine-containing
groups. The main advantage of pyrohydrolysis is that it is scalable to
industrial levels. Nevertheless, a toxic chlorine-rich flame presents
significant corrosive hazards and is unfriendly to the environment. Beyond the
obvious concerns with environmental pollution, Ti chlorides are not easy to
store and handle. Thus, it is a practically and scientifically important
challenge to develop alternative precursors to control the size, polymorph
content and morphology of TiO2 nanocrystals and, therefore, optimize the
properties of material. A metal-organic precursor is more favourable than
inorganic titanium chlorides, because it can be used in a chlorine-free system
without related hazards, however, the widely used Ti alkoxides suffer from
instability and poor cost-efficiency.

In this study we report the development of substitute precursors to produce
nanostructured TiO2 via pyrolysis or thermal decomposition using metal-organic
carboxylate-based extraction systems. Having the advantage of being
inexpensive and environmentally friendly, metal carboxylate complexes have
been used extensively in the production of nanoparticles of SMOs 25, 26, 27,
28, but the production of TiO2 nanoparticles from a carboxylate precursor has
not been reported. Herein, Ti chloride is used as the initial titanium source
preserving the key advantage of low cost, however, the chloride anions Cl- are
avoided with the extraction system via aqueous/organic phase segregation and
metal cation exchange resulting in in situ dialysis. The resulting precursors
are storable and the nanostructured TiO2 is produced by non-hydrolytic
pyrolysis. Thus, we are able to overcome many of the specific problems
described above to achieve better particle size and polymorph control than
hydrolytic routes by simply varying the temperature of pyrolysis.

##  2 Experimental

###  2.1 Preparation of Ti-containing organometallic precursor and
pyrolization treatment

Initially, an aqueous solution of TiCl3 was produced by dissolving fine-
grained c.p. titanium powder (1.2 g, particle size 63-100 $\mu$m, -140+230
mesh) in boiling hydrochloric acid solution (60 ml, 1:1 vol.). The metal
concentration was adjusted to 0.1M by dH2O addition (pH 0.5). Titanium-rich
carboxylate-based organometallic precursor was produced by liquid-liquid
extraction: 60 ml of TiCl3 aqueous solution and 20 ml (3:1 aqueous/organic
volume ratio) of pentanoic acid (without diluent) were placed in a separation
funnel; 1M NaOH soluion was added to the mixture in small portions, followed
by vigorous shaking after each addition until the pH of the aqueous phase
reached pH 1, after which the phases were allowed to separate for 0.5 h. After
complete stratification and removal of the aqueous phase followed by
filtration, a Ti-rich precursor E1 (0.15M Ti) was produced. To study the
effect of precursor concentration on produced TiO2 a more concentrated
precursor E2 (0.5M Ti) was prepared as described above, but with 5:1
aqueous/organic volume ratio. The precursors E1 and E2 were pyrolyzed for 1 h
at 350 ∘C (E1/E2-350), 400 ∘C (E1/E2-400), 450 ∘C (E1/E2-450), 550 ∘C
(E1/E2-550), 650 ∘C (E1/E2-650), 750 ∘C (E1/E2-750) in static air to sample
the phase content of derived TiO2 nanoparticles 29.

Fig. 1:  Schematic of preparation of carboxylate-based Ti organometallic
precursor: Left - phase contact of TiCl3 aqueous solution and pentanoic acid,
middle - extraction of Ti3+ from the aqueous solution into the organic phase
by cation exchange, right - produced non-aqueous Ti3+-rich carboxylate-based
organometallic precursor with characteristic violet color.

###  2.2 Preparation of reference TiO2 by sol-gel

A reference sample was prepared by standard sol-gel method, using 250 ml of
0.1M TiCl3 aqueous solution produced in the previous step. The gel-like
precipitate was obtained by dropwise addition of 0.5M NaOH aqueous solution at
a rate of 3 ml/min under vigorous stirring until pH 6 of the solution was
reached. The produced gel was aged for 24 h at ambient conditions, followed by
filtration and washing with dH2O. The absence of chlorine in the decantate was
checked by AgNO3 solution. The as-dried precipitate (P) was calcined at 450 ∘C
(P-450), 550 ∘C (P-550), 650 ∘C (P-650), 750 ∘C (P-750) in static air to
produce TiO2 nanoparticles.

###  2.3 Characterization

The thermal stability of the produced precursors was studied by thermal
gravimetric analysis (TGA) and differential scanning calorimetry (DSC) using
the STA PT1600 (LINSEIS) device. The sample was heated in static air from
ambient temperature to 700∘C at a rate of 10 ∘C/min.

IR spectra were recorded at r.t. using Bruker Tensor II FT-IR
spectrophotometer. For each spectrum 36 scans were performed in the range
4000-400 cm-1 with 7 mm KBr discs (TiO2/KBr mass ratio 1:100) prepared under a
load of 2000 kg.

Powder X-ray diffraction (XRD) patterns of dried samples were recorded in
$2\theta$ range between 20∘ and 75∘ using D8 Advance (Bruker Corporation)
diffractometer with CuK$\alpha$ radiation ($\lambda$ = 1.5418Å), accelerating
voltage 40 kV and current 40 mA; step size 0.02∘ at a scanning rate 2 s/step.
Zero-background Si sample holders were used. The quantitative phase
composition of the produced materials were determined by Rietvield refinement
(FullProf Suite ver. 7.30), a whole pattern fitting method that systematically
varies constraints in a simulated pattern to reproduce the experimental
pattern. The refinement was performed starting from the identified crystal
phases and known crystal structure data using isotropic size broadening (as
confirmed by HRTEM snapshots), Pseudo-Voigt reflection profile approximation
with Caglioti equation (FWHM${}^{2}=U\tan^{2}\theta+V\tan\theta+W$) and
experimental instrumental broadening parameters with standard reference
material. The background was modeled using a 6th-order shifted polynomial. The
parameters refined were zero shift, scale factors, unit cell and peak shape.

The morphology of the produced nanoparticles was checked by transmission
electron microscopy (TEM) (FEI Technai G2 F20 operating at 200 kV) in bright
field mode. Dynamic light scattering (DLS, Malvern Instruments, Zetasizer Nano
S90) was used to measure the hydrodynamic size distribution in particle
suspensions produced by electric ablation. For the measurement the initial
suspension was diluted with distilled water to achieve optimum count rate.

The ferromagnetic properties of the produced samples were investigated by
vibrating sample magnetometry (VSM). The magnetization curve was recorded at
r.t. (Lake Shore Cryotronics instrument 7404VSM) in a field range up to 1 T.
Several full magnetization cycles have been performed to detect potential
hysteretic behavior.

##  3 Results and discussion

The preparation of reference sample (P) within a standard sol-gel procedure
involves the hydrolysis of TiCl3, which generates highly acidic (low pH) and
chloride (Cl-) rich solution. The selectivity of formed TiO2 polymorphs is
highly dependent on instantaneous pH, Ti concentration and reaction time 23,
20, where small change in molar ratios would result in different phase
structure 30. At high Ti concentration ([Ti] > 0.25 M) the solution will
predominantly contain [TiO(OH2)5]2+ monomers and the condensation will proceed
by olation by sharing equatorial edges 31 with formation of hydroxo bridges
(Ti-OH-Ti) 22. In this case, rutile-rich crystallites are precipitated 31, 32,
33, 34, 35. To stabilize the production of anatase, Ti concentration was
decreased to 0.1 M 23, 31, 22. Moreover, the pH was raised to 6 to avoid the
formation of a substantial amount of brookite 23, 31, 35. Conversion of the
amorphous hydrated precipitate (P) to crystalline anatase is realized by
calcination treatment.

The organometallic Ti precursors E1 and more concentrated E2 were produced by
liquid-liquid extraction: the extraction of Ti3+ from aqueous phase (aq) into
the organic phase (o) by pentanoic acid is realized by cation exchange (Fig.
1), according to equation

$Ti^{3+}\left(aq\right)+3HOOCR\left(o\right)\leftrightarrow
Ti\left(OOCR\right)_{3}\left(o\right)+3H^{+}\left(aq\right)$

Pentanoic acid is an efficient cation exchanger due to its favourable
hydrophylic/hydrophobic balance. The organometallic extracts, which were
brightly violet immediately after extraction (Fig. 1), decolorated shortly
after production due to the change of the Ti oxidation state from Ti3+ to Ti4+
and were long-term storable.

###  3.1 Thermal analysis

The thermal behaviour of the precursors, phase transition and phase stability
of produced TiO2 nanoparticles were studied using TGA-DSC analysis. The
measured thermoanalytical curves in the range between 20 and 700 ∘C are shown
in Fig. 2. The thermogravimetric analysis of the as-dried reference
precipitate (P) shows active weight loss (ca. 20wt.%) when the sample is
heated from r.t. to 200 ∘C, which is accompanied by a pronounced endothermic
peak (peak max at 90 ∘C) in the DSC trace. This step is attributed to the loss
of weakly bound physisorbed water from the surface of the particles. The
second step in the range between 200 ∘C and 400 ∘C (peak max at 290 ∘C) is
identified by a broad endothermic process related to the liberation of
crystallization water and chemisorbed surface hydroxyls from hydrated
precipitates (titanium oxyhydrate). This is accompanied by a small mass loss
ca. 5 wt.%. According to the XRD results (see below) the crystallization of
anatase from the precipitates starts at about 450 ∘C, which does not associate
to any DSC event. Further temperature increase results in prolonged weight
loss of the precipitates at a reduced rate, associated with continuing
dehydration process up to 1000 ∘C.

Fig. 2:  TGA-DSC traces of carboxylate organometallic precursors E1 (0.15M Ti)
and E2 (0.5M Ti) produced by liquid-liquid extraction and sol-gel precipitates
P from monophasic precipitation. Symbols indicate sampling temperatures Ti,
∘C: 350, 400, 450, 550, 650, 750  Fig. 3:  X-ray diffraction patterns
(symbols) and corresponding Rietvield-refined profiles (black lines): A -
calcined precipitates produced by standard sol-gel method. B - TiO2 samples
produced by pyrolyzation of organometallic precursor. The residual lines show
the difference of the measured and simulated XRD profiles.

The organometallic carboxylate Ti precursors E1 and E2 produced by liquid-
liquid extraction show a strong endothermic peak in the range between 100 ∘C
and 186 ∘C related to the active evaporation of free extractant (pentanoic
acid) and is accompanied by a pronounced mass loss ca. 75 wt.%, which
terminates around the boiling point of pentanoic acid (Tboil=186 ∘C). At this
point the mass loss ends and amounts to 80-85 wt.%. Further temperature
increase initiates the thermal decomposition of non-reactive titanium
pentanoate contained in the precursor, which should proceed according to the
standard route for transition metal carboxylates 36, 37:
Ti(OOCR)${}_{4}\rightarrow$TiO2+2R2CO+2CO2. The decarboxylation of the organic
component is accompanied by the formation of ketones containing 3 to 9 carbon
atoms (i.e. dibutyl ketone, dipropyl ketone, diethyl ketone, acetone) 38, 39,
37. The ketones evolved during this process also undergo decomposition with
atmospheric oxygen to yield CO2 and H2O 36. The third step in the temperature
range between 200 ∘C and ca. 450 ∘C is marked by the broad endothermal process
associated with breakdown of organic groups and release of volatile species,
mainly CO2 and formed ketones 38, 39, 37. The signature of this process is
more pronounced in the DSC trace of precursor E2 having higher Ti
concentration. The liberation of the titanium ion from the organometallic
precursor leads to the formation of amorphous nuclei of TiO2 nanoparticles.
This process is accompanied by the gradual increase of the sample mass (by 5-6
wt.%) owing to the oxygen binding from the air during titanium oxidation. The
exothermic event at ca. 320 ∘C in the DSC trace indicates the energy release
from combustion/burning of carbonaceous residues. The more gradual removal of
organic groups continues up to 450 ∘C, as shown by infrared spectroscopy (see
below), which indicates that the carboxylate (COO-) functional group from
pentanoic acid is fully desorbed at this temperature. TGA analysis shows that
the conversion of titanium pentanoate to TiO2 is complete at 450 ∘C with no
further mass change. As confirmed by XRD analysis (see below) the produced
TiO2 is amorphous giving no XRD trace of crystalline structure. The formation
of high crystalline anatase starts at ca. 450 ∘C accompanied by a broad
exothermic process (peak max ca. 530 ∘C) associated with restructuring and
grain growth of anatase particles. The transformation is completed at ca. 560
∘C the product fully transforms into anatase phase.

Table 1: Summary of sample phase structure refinement. A - anatase, R - rutile, Imp. - impurity phase Na2Ti6O13 T, ∘C | Primary phase | Secondary phase
---|---|---
| Unit cell, Å | wt.% | Phase | Crystallite | Unit cell, Å | wt.% | Phase | Crystallite
| a, b | c |  |  |  size, dXRD | a, b | c |  |  |  size, dXRD
Sol-gel reference sample (P)
r.t. | \- amorphous - |  |  |  |  |
450 | 3.795 | 9.492 | 100% | A | 7 nm |  |  |  |  |
550 | 3.793 | 9.497 | 100% | A | 9 nm |  |  |  |  |
650 | 3.790 | 9.504 | 95% | A | 12 nm | 4.593 | 2.958 | 5% | R | 18 nm
750 | 4.596 | 2.960 | 79% | R | 64 nm |  | 15.109
---
3.749
9.181 | 21% | Imp. | 45 nm
Pyrolyzed carboxylate precursor E1 (0.15M)
350 | \- amorphous - |  |  |  |  |
400 | \- amorphous - |  |  |  |  |
450 | 3.794 | 9.481 | 100% | A | 8 nm |  |  |  |  |
550 | 3.790 | 9.504 | 100% | A | 15 nm |  |  |  |  |
650 | 3.788 | 9.513 | 65% | A | 24 nm | 4.596 | 2.961 | 35% | R | 27 nm
750 | 4.593 | 2.960 | 100% | R | 59 nm |  |  |  |  |
Pyrolyzed carboxylate precursor E2 (0.5M)
350 | \- amorphous - |  |  |  |  |
400 | \- amorphous - |  |  |  |  |
450 | 3.789 | 9.489 | 100% | A | 8 nm |  |  |  |  |
550 | 3.788 | 9.517 | 77% | A | 20 nm | 4.597 | 2.960 | 23% | R | 27 nm
650 | 4.595 | 2.961 | 81% | R | 46 nm | 3.785 | 9.520 | 19% | A | 33 nm
750 | 4.595 | 2.962 | 100% | R | 57 nm |  |  |  |  |

Gravimetric analysis confirmed 0.15M titanium loading in precursor E1 and 0.5M
in E2, correponding to the efficiency of cation exchange 45% and 100%
respectively. Higher ratio of aqueous to organic phase volumes (5:1 vs 3:1)
achieves complete extraction and efficient loading of metal cations in the
organic phase.

As evident from thermal analysis the formation mechanism of nanoparticles from
produced organometallic precursors differ substantially from the standard
hydrolytic sol-gel process. The details of the phase content, stability and
surface functional groups of produced nanoparticles can be derived from the
XRD and FTIR analysis.

###  3.2 XRD analysis

The precipitate (P) from standard "one pot" sol-gel system was used as
precursor to produce TiO2 nanoparticles by calcination. The XRD patterns of
calcined nanopowders are shown in Fig. 3A and used as reference. The XRD
pattern of the as dried precipitate does not show clear signs of
crystallinity, indicating that the TiO2 precursor before calcination is in
amorphous state. The annealing temperature of 450 ∘C induces the
crystallization of precipitate and distinctive peaks emerge, attributable to
the tetragonal structure (space group: I41/amd) of crystalline anatase. All
the peaks are fully indexed within ICDD PDF Pattern 00-021-1272 indicating
that anatase titania has been formed at this temperature. The results of the
Rietvield refinement concerning phase content and crystallite size are
reported in Table 1.

Fig. 3B shows the XRD patterns of powders produced by pyrolysis of the
organometallic precursor (E1). The sample pyrolyzed at 400 ∘C shows only a
weak and diffuse peak at 2$\theta$ = 25.3∘ attributable to the [101]
reflection of anatase, which demonstrates that the sample still remains in
mostly amorphous state. Pyrolyzation at 450 ∘C induces the formation of pure
nano-crystalline anatase from this precursor. The corresponding diffractogram
in Fig. 3B closely reflects the XRD pattern of precipitate (P) calcined at 450
∘C (Fig. 3A) indicating identical phase state. The crystallite size of the
produced anatase nanoparticles is ca. 7-8 nm (Table 1).

When the processing temperature of the precursors is increased to 650 ∘C weak
[110] (27.4∘, 100% intensity) and [101] (36.1∘) reflections appear in the XRD
pattern of calcined precipitates corresponding to the tetragonal structure
(space group: P42/mnm) of crystalline titania rutile polymorph, which marks
the beginning of a high-temperature anatase-to-rutile transformation
accompanied by rapid grain growth (Table 1). The diffractograms of powders
obtained by pyrolyzation of organometallic precursor at this temperature show
additional reflections of rutile. All additional peaks are readily indexed
within standard ICCD PDF Pattern 00-021-1276 for rutile.

Fig. 4:  TEM images of samples produced by pyrolysis of organometallic
precursor at 450 ∘C (E1-450, top row) and 750 ∘C (E1-750, bottom row). Middle
- magnified high-resolution TEM image of few ca. 10 nm anatase nanocrystals
(top) and a secondary aggregate of crystallographically aligned and fused ca.
10 nm rutile grains (bottom). Right - lattice fringes of nanoparticles and
corresponding projections of anatase (A) and rutile (R) crystal structures.

At 750 ∘ both precursors complete the transformation from anatase to rutile
and rutile becomes the predominant crystalline phase. This course of
transformation is consistent with previous reports 40, 41. No traces of
brookite phase are found in any of the samples as confirmed by the absence of
indicative [121] (30.8∘, 90% intensity) reflection (ICDD PDF 00-029-1276).

At 750 ∘C simultaneously with the anatase to rutile transformation the
calcination of the precipitate obtained by the reference method finally leads
to the crystallization of the impurity phase in P-750 (Fig. 3A) identified as
sodium titanate Na2Ti6O${}_{1}3$ (ICDD PDF 00-014-0277, space group: C12/m1).
It is evident that an appreciable amount (ca. 20%, Table 1) of by-products of
the standard monophasic "one-pot" synthesis are remaining in the produced
materials despite extensive purification measures and the confirmed absence of
chloride in washwater effluent. In this case sodium from the precipitating
basic agent NaOH binds the chloride from the titanium chloride solution
forming a large amount of sodium chloride in the precipitate, which is
difficult to remove. The impurity phase becomes apparent in the XRD as the
product of the high-temperature reaction between the unremoved residues of the
sodium salt and titanium dioxide during calcination. In industrial pyro-
hydrolysis process used to produce commercial TiO2 the powder is treated with
steam at 450-550 ∘C to remove chlorine-containing groups. Our results show
that within the non-hydrolytic extraction-pyrolysis method the purification is
achieved already at the precursor preparation stage, employing a biphasic
system, in which one of the phases accumulates the target product, Ti-
containing organometallic compound, and the other accumulates by-products of
the synthesis, in this case chloride anions.

Moreover, non-hydrolytic pyrolysis of organometallic extract allows facile
phase control by just varying the temperature of the pyrolytic treatment. The
samples E1-550 and E2-550 produced by pyrolysis at 550 ∘C mimic the mixed
phase content of Aeroxide P25, consisting of ca. 70-80 wt.% anatase and 20-30
wt.% rutile with ca. 20-27 nm particle size in both phases (Table 1). The
phase evolution of the particles is independent of the concentration of
organometallic precursors, with just the start of the anatase-to-rutile
conversion beginning at ca. 100 ∘C lower temperature, which allows to use
concentrated precursors to produce phase-pure anatase.

###  3.3 TEM analysis

Fig. 4 shows characteristic TEM snapshots of TiO2 particles produced by
pyrolyzation of organometallic precursor E1 at 450 ∘C (E1-450). The particles
are roughly spheroidal, which is characteristic for nucleation directly from
the amorphous state. A tight size distribution is measured by particle
counting with an average size of 8.3 nm in agreement with XRD assessment based
on Rietvield modelling (Table 1). High resolution TEM shows lattice fringes
unambiguously identifiable as projections of anatase crystal structure.

As indicated by DSC analysis (Fig. 2), at ca. 560 ∘C the amorphous phase is
fully expended and the primary crystallite formation terminates. Pyrolyzation
at higher temperature is accompanied by primary crystallite fusion with
simultaneous anatase-to-rutile transition. Fig. 4 shows the particles
pyrolyzed at 750 ∘C (E1-750). The highly irregular-shaped particles are
constructed by joining of a number of ca. 10 nm primary crystalline grains
forming secondary aggregates. According to strict thermodynamic arguments,
single-phase anatase is only stable below ca. 11-14 nm size due to its
slightly lower surface free energy 42, with rutile being the macroscopically
(ca. >35 nm) stable polymorph 42, 43, 44. Epitaxial attachment of aggregated
phase-pure anatase nanoparticles leads to joining on their facets, where the
rutile nucleation is initiated assisted by intermediate phase, e.g. brookite
45, 46, 47 or high-pressure TiO${}_{2}-II$ phase 48, formation at twinned
interface between contacting anatase grains 45, 47, 49, 50. Rutile
transformation then propagates rapidly into the bulk of the crystallites. The
epitaxially fused nanoparticles with the rutile structure are noted in the
high resolution TEM snapshots. Close inspection of TEM microphotographs (Fig.
4) clearly reveals that the particles have many adjacent rutile crystallites
sintered together and crystallographically aligned with respect to each other.
The average size of the secondary aggregates is 70.3 nm, consistent with XRD
analysis (Table 1).

###  3.4 Infrared (IR) spectroscopy

While XRD provides the information about the bulk of the particles, FT-IR is a
surface-sensitive method. It probes the the surface region and chemistry of
the particles, which is not identified by XRD.

Fig. 5:  IR spectra of TiO2 samples produced by pyrolyzation of carboxylate-
based Ti precursor: (a) - E1-350, (b) - E1-400, (c) E1-450, (d) E1-550
(complete elimination of surface residual organics is noted at 450 ∘C); (e) -
electro-dispersed sample calcined at 850 ∘C (ED-850).

FT-IR spectra of nanoparticles produced by pyrolyzation of carboxylate
precursor (E1) are shown in Fig. 5. The broad IR absorption band at 3200-3600
cm-1 in all samples corresponds to the stretching vibration of the hydroxyl
groups ($\nu$OH) terminated on TiO2 surface (Ti-OH) and adsorbed molecular
water. The spectrum band centered at 1625 cm-1 is from the bending vibration
of chemisorbed water $\delta$H-O-H. Weak absorbance doublet at 2341 cm-1 and
2360 cm-1 is due to asymmetric C-O stretching vibration of carbon dioxide.
Since this band is present only in the pyrolyzed samples, we ascribe it to CO2
residues from the oxidation of pentanoic acid captured on the pores in the
nanopowder***cf. Fig. 5e, showing FT-IR of nanoparticles produced by
electrodispersion of initial metallic powder without any chemicals and
calcined at high temperature (see below). Peaks at 1375 cm-1 and 1520 cm-1
represent symmetric $\nu$COO${}^{-}_{sym}$ and asymmetric
$\nu$COO${}^{-}_{asym}$ stretching modes of the carboxylate (COO-) functional
group 51 coming from the organic component of the precursors used in the
synthesis process. The magnitude of the separation between carboxylate
stretches $\Delta$ = $\nu$COO${}^{-}_{asym}$ \-
$\nu$COO${}^{-}_{sym}$=$\sim$145 cm-1 is consistent with the value for ionic
carboxylate complexes ($\sim$164 cm-1 for ionic acetate 52). This typically
indicates that the carboxylate group is bound to the surface Ti-centers in
bidentate bridging configuration 53, where one metal cation is bound to one of
the oxygens of the COO- group and another metal cation to the other oxygen.
Carboxylate vibrations disappear with pyrolyzation temperature starting at 450
∘C, indicating complete elimination of organic residues when the anatase
crystallization starts. A remaining broad and strong absorption band between
1000 and 400 cm-1 envelops a set of peaks corresponding to the intrinsic Ti-O-
Ti, O-Ti-O and Ti-O lattice vibrations of nano-crystalline titanium oxides 54.

###  3.5 Defect-induced room-temperature ferromagnetism

Fig. 6:  Defect-induced room-temperature ferromagnetism in Ti-oxide
nanoparticles: A - schematic polaron model of mesoscale ferromagnetic coupling
between magnetic moments produced at cationic or anionic defect sites in
stoichiometric TiO2. M-H magnetization curves recorded from: B - precipitates
in a monophasic system calcined at (a) 550 ∘C (P-550, anatase phase), (b) 650
∘C (P-650), (c) 750 ∘C (P-750, rutile); B - pyrolyzed Ti-carboxylate precursor
at (a) 450 ∘C (E1-450, anatase), (b) 550 ∘C (E1-550, anatase), (c) 750 ∘C
(E1-750, rutile); C - electrically dispersed nanoparticles calcined at (a) 350
∘C (ED-350), (b) 450 ∘C (ED-450), (c) 550 ∘C (ED-550), (d) 750 ∘C (ED-750).

To obtain an estimation of the defect content in the produced materials we
characterize their ferromagnetic behaviour by VSM. The recorded magnetization
curves M-H are shown in Fig. 6 with the magnetic field varying in the range
between -10 kOe to +10 kOe. TiO2 precipitates produced by standard sol-gel are
predominantly paramagnetic $M\approx\chi H$ (Fig. 6B), irrespective of the
calcination temperature. The mass magnetic susceptibility $\chi$ of the
samples calcined at 550 ∘C (P-550) and 650 ∘C (P-650) having anatase as the
dominant phase is ca. $2\times 10^{-7}$ cm3 g-1. The anatase-to-rutile
transition in the calcined samples is accompanied just by a slight increase in
$\chi$ at 750 ∘C (P-750, Table 2).

Table 2: Summary of magnetic properties. The polaron dipole moment $\mu$ in units of Bohr magneton $\mu_{B}$ = 9.27 $\dots\times 10^{-21}$ erg/G. T, ∘C |  $\chi$ / $10^{-7}$ |  $M_{sat}$ / $10^{-3}$ |  $N$ / $10^{13}$ |  $\mu\mu_{B}^{-1}$ / $10^{3}$
---|---|---|---|---
|  cm3 g-1 |  emu g-1 |  g-1 |
Sol-gel reference sample (P)
550 | 1.8 | 0.3 | 0.3 | 12
650 | 2.1 | 0.1 | 0.1 | 15
750 | 2.4 | 0.3 | 0.4 | 9.7
Pyrolyzed carboxylate precursor (E1)
450 | 0.6 | 1.6 | 1.5 | 12
550 | 0.5 | 1.8 | 2.0 | 9.7
750 | 1.3 | 4.8 | 4.0 | 13
Electrodispersed sample (ED)
350 | 12.0 | 13 | 16 | 8.6
450 | 6.8 | 15.5 | 19 | 8.6
550 | 3.2 | 9.5 | 12 | 8.6
750 | 1.4 | 4.3 | 4.8 | 9.7

In contrast, the M-H curves of powders produced by pyrolyzation of
organometallic precursor (Fig. 6C) show distinct room-temperature
ferromagnetism superposed by the paramagnetic process. For all recorded
magnetization curves the hysteresis is negligible with small values of
coercitive field ranged within 50-90 Oe and the remanence factor varying
around 5-8%, indicating soft superparamagnetic-like behaviour characteristic
for d0 ferromagnetism 55.

Stoichiometric TiO2, having all Ti in 4+ oxidation state is non-ferromagnetic
due to the lack of unpaired electrons. Likewise, no magnetic impurities are
introduced during the preparation or handling of the samples. The staring
metallic Ti powder is chemically pure titanium as confirmed by X-Ray
fluorescence. The content of magnetic impurities, i.e. Fe, Ni or Co, is below
0.1 at.%. The doping of the produced particles by the trace elements
potentially present in the precursors is discarded as the origin of room-
temperature ferromagnetism, because all samples were produced from the same
metallic powder. Moreover, the absence of impurities in the sample produced by
pyrolyzation of organometallic precursor is assured, on account of high
selectivity of pentanoic acid extractant towards titanium ions and unfavorable
conditions for the concurrent bi-phasic extraction of other metallics in
precursor preparation. Hence, room temperature ferromagnetism (RTFM) in the
pyrolyzed samples is due to the intrinsic defects in the nanoparticles.

The origin of RTFM, known also as d0 magnetism, in undoped semiconducting
metal oxides (SMO) is ascribed to intrinsic defects in the crystal lattice,
generated during the production process 56. The emergence of ferromagnetism is
typically accompanied by a high number of oxygen vacancies VO in the anionic
sublattice of TiO2 films 57, 58, 59, 60, nanoparticles 61, 30 and single
crystals 62, 63. Likewise, the defect complex Ti3+-VO was noted, which forms
when oxygen is removed: the charge imbalance makes unpaired excess electrons
occupy 3d state of nearby Ti ions, generating reduced Ti3+ ions and providing
the local magnetic moments 64, 65. In oxygen-defficient TiO2-δ nanoparticles
the reduction mechanism Ti4+ $\rightarrow$ Ti3+ can also be initiated by Ti
interstitials, without involving oxygen vacancies 66. Moreover, cationic
vacancies VTi were shown to produce d0-ferromagnetism 67, 68. The balance of
these defects is strongly dependent on the preparation method.

In view of this, we have produced a reference sample with high defect loading
by physical condensation - direct electric ablation of the starting metallic
titanium precursor without additional chemicals, using a device shown
schematically in Fig. 7A 69.

Fig. 7:  Electric ablation experiment to produce highly defected Ti-oxide nanoparticles: A - schematic of the electric dispersion apparatus with the discharge ablation zone between a pair of electrodes pressed into the bed of titanium particles in distilled water. The repeated discharges ablate material from the surface of the particle bed; B - particle size distribution in as prepared colloid measured by DLS.  Fig. 8:  TEM snapshots of electrically dispersed nanoparticles calcined at 850 ∘C with varying magnification. HR-TEM image of lattice fringes and corresponding rutile lattice projection.  Fig. 9:  XRD patterns of electrodispersed nanoparticles (ED) and phase content evolution under calcination treatment. Table 3: Summary of electrodispersed sample phase structure refinement. A - anatase, R - rutile. T, ∘C | Unit cell, Å | wt.% | Phase | Crystallite
---|---|---|---|---
| a,b | c |  |  |  size, dXRD
r.t. | 4.190 | 4.190 | 62% | TiO | 26 nm
| 5.092 | 13.811 | 25% |  Ti2O3 | 12 nm
| 5.148 | 9.580 | 13% |  Ti3O | 35 nm
350 | 4.187 | 4.187 | 53% | TiO | 23 nm
| 5.066 | 13.803 | 19% |  Ti2O3 | 12 nm
| 5.147 | 9.548 | 16% |  Ti3O | 26 nm
| 3.787 | 9.428 | 12% | A | 24 nm
450 | 4.589 | 2.961 | 45% | R | 13 nm
| 3.785 | 9.526 | 23% | A | 52 nm
| 4.195 | 4.195 | 14% | TiO | 37 nm
| 5.134 | 9.626 | 9% |  Ti3O | 28 nm
| 5.149 | 13.642 | 9% |  Ti2O3 | 12 nm
550 | 4.593 | 2.958 | 71% | R | 27 nm
| 3.784 | 9.519 | 22% | A | 49 nm
| 5.124 | 9.659 | 7% |  Ti3O | 27 nm
650 | 4.596 | 2.961 | 81% | R | 33 nm
| 3.786 | 9.526 | 18% | A | 52 nm
| 5.137 | 9.695 | below 5% |  Ti3O | 10 nm
750 | 4.596 | 2.961 | 86% | R | 52 nm
| 3.786 | 9.527 | 14% | A | 62 nm
850 | 4.593 | 2.960 |  $\sim$100% | R | 73 nm
| 3.786 | 9.491 | below 5% | A | 28 nm

Briefly, a 20 g portion of metallic titanium powder was added to 200 ml of
distilled water forming a particle bed. Flat titanium electrodes connected to
a spectrometric discharge generator operating at 100 Hz and 2 kV were pressed
onto the particle bed. After turning on the generator a luminous network of
intermittent sparks appears within the particle bed. Reactive dispersion of
titanium using distilled water as the working dielectric is accompanied by
rapid ejection and oxidation of vaporized titanium and the formation of
titanium oxide nanoparticles. The rapid in situ quenching leads to a highly
defected oxygen-deficient structure.

The initially transparent liquid turned completely black after 30 min of
device operation, evidencing the condensation of nanodispersed phase. Dynamic
light scattering (DLS, Malvern Instruments, Zetasizer Nano S90) (Fig. 7B)
showed that the hydrodynamic particle size in the decanted supernatant varies
in a broad range between ca. 100 nm to 1 $\mu$m, which indicates that the
particles form associates in liquid medium due to the lack of stabilization
70, 71, 72, 73, 74. The produced suspension was dried, the electrodispersed
nanopowder (ED) was calcined at 350 ∘C (ED-350), 450 ∘C (ED-450), 550 ∘C
(ED-550), 650 ∘C (ED-650), 750 ∘C (ED-750), 850 ∘C (ED-850) in static air for
2 h to sample phase structure transformation and ferromagnetic properties.

The phase content of electrodispersed nanoparticles is non-stoichiometric
oxygen-deficient oxides (Fig. 9, Table 3), predominantly titanium monoxide TiO
(space group: Fm-3m, indexed within ICDD PDF 01-086-2352), which is able to
accommodate up to 10-20% of O vacancies in the cubic structure 75, but also
trigonal Ti2O3 structure (space group: R3c, ICDD PDF 01-085-0868) and cubic
Ti3O structure (space group: P-31c, ICDD PDF 01-073-1117) have been found with
average crystallite size ca. 25-35 nm. Anatase appears after calcination at
350 ∘C and the content of non-stoichiometric oxides begins to decrease due to
a more complete oxidation. A mixed phase sample ED-750 of stoichiometric
anatase (14 wt.%) and rutile TiO2 (86 wt.%) is finally obtained calcining at
750 ∘C with grain size ca. 50-60 nm. Higher calcination temperature (850 ∘C)
completes the anatase-to-rutile transition with particles having slightly
irregular rounded shape (Fig. 8) and the lattice fringes are fully indexed to
projections of rutile structure.

As expected, due to non-stoichiometry and a high defect content generated by
the very rapid reactive quenching of electrically ablated vapourized material
in surrounding water, the produced nanoparticles show significantly enhanced
ferromagnetic properties (Fig. 6D).

Theoretical studies indicate that concentrated cationic 67 and anionic 76
defects couple ferromagnetically by exchange interaction, forming bound
magnetic polarons (BMPs), which are bubbles of ferromagnetially ordered spins
(Fig. 6A) 56. More quantitative understanding of the polaron-induced magnetism
can be obtained within the BMP-model by fitting the measured magnetization
with the equation of the type 56, 65, 77, 78

$M=M_{sat}\mathcal{L}\left(\xi\right)+\chi H$ (1)

where the first term describes the superparamagnetic contribution and the
second term describes to first order of magnetic field H the paramagnetic
processes in non-ferromagnetic matrix, which are not related to the
superparamagnetic properties of the sample. Here, $M_{sat}$ is the saturation
magnetization [emu g-1], which is the product of the number of bound magnetic
polarons involved in the polarization process per unit mass of the sample $N$
[g-1] and $\mu$ \- the spontaneous dipole moment [erg G-1] of a polaron;
$\mathcal{L}\left(x\right)=\coth\left(x\right)-\frac{1}{x}$ is the Langevin
function, $\xi=\frac{\mu H}{k_{B}T}$ \- Langevin parameter,
$k_{B}=1.38\ldots\times 10^{-16}$ erg K-1 is the Boltzmann constant, T -
temperature (ca. 300 K) and $\chi$ is paramagnetic susceptibility [cm3 g-1].
We have analyzed the recorded M-H curves for all samples in terms of the bound
polaron model Eq. (1) using polaron concentration $N$, their dipole moment
$\mu$ and magnetic susceptibility $\chi$ as fitting parameters. The
theoretical lines closely follow the measured M-H curves for all samples (Fig.
6). The calculated parameters are summarized in Table 2.

The superparamagnetic saturation magnetization of samples produced by
pyrolyzation of organometallic precursor is about an order of magnitude higher
than for reference sol-gel particles, indicating a more defected structure.
$M_{sat}$ consistently increases with the pyrolyzation temperature, reaching
4.8 memu/g at 750 ∘C (E1-750).

The highest superparamagnetic saturation $M_{sat}$ =15.5 memu/g is obtained
for electrodispersed sample ED-450, indicating highly defected structure was
produced. Calcination at high temperature >450 ∘C leads to the decrease of
ferromagnetic properties due to the gradually diminishing defect content, such
as the filling of oxygen vacancies in static air. The remaining
superparamagnetic saturation at the highest annealing temperature 750 ∘C is
$M_{sat}$ = 4.3 memu/g.

The phase composition, crystallite size (ca. 50-60 nm) and polaron content
($\sim$ 4 - 4.8 $\times 10^{13}$ g-1) in both rutile-rich samples E1-750 and
ED-750 (Table 2) produced respectively by pyrolyzation and electrodispersion
is similar. It is worth noting that according to the TGA traces (Fig. 2)
showing continuous mass gain, the pyrolyzation of organometallic precursor
relies on oxygen absorption from the environment, which might favor formation
of oxygen vacancies. This is in contrast to the precipitated sample, which
shows similar phase composition and crystallite size, but an order of
magnitude smaller polaron concentration, indicating a defect-free crystal
lattice, since the stoichiometry of the precipitates in aqueous sol-gel is
assured on account of the hydrolysis and condensation process.

Previously, 10 nm TiO2-δ nanoparticles with mostly anatase phase produced by
sol-gel showed saturation magnetization 9.5 memu/g and coercitivity of 96 Oe
after annealing under reducing H2/Ar atmosphere. The origin of the
ferromagnetism was attributed to a large content of VO. This was contested by
Parras et al. 66, where Magneli phase TiO1.84 was stabilized under similar
conditions. Rather than the influence of oxygen vacancies, the saturation
magnetization 10 memu/g was instead ascribed to the Ti3+ ions induced by
reduction, initiated by formation of Ti interstitials. Polycrystalline TiO2
treated under 4 MeV Ar5+ ion irradiation showed 0.4 memu/g saturation with 168
Oe coercitive field79. Oxygen vacancies VO were the dominant defects after
irradiation. Wang et al. 80 produced 10-20 nm anatase nanoparticles by sol-gel
and reported $\sim$1 memu/g saturation and 93 Oe coercitive field attributed
to VO content and charge transfer interactions between Ti3+ and Ti2+. Recently
81, 82, sol-gel produced 10 nm single-phase anatase TiO2 calcined at 400 ∘C
showed ca. 0.5-13 memu/g saturation and 11-305 Oe coercitivity, on account of
VO. Wang et al. 68 produced highly defected 10-20 nm anatase TiO2
nanoparticles with 25 memu/g superparamagnetic saturation and 18 Oe
coercitivity using solvothermal reaction. Moreover, defected TiO2 showed
remarkably higher photocatalytic activity vs. normal TiO2 because of the more
efficient charge transfer 68.

Our results show that the effective polaron magnetic moment $\mu\sim
10^{4}\mu_{B}$ ($\sim 10\times 10^{-17}$ emu, Table 2) does not distinctly
depend on the preparation method or thermal treatment. Remarkably, very
similar magnitudes of polaron magnetic moments were obtained in two previous
investigations, in which this quantity was estimated: $\mu\sim 6.5-8.9\times
10^{-17}$ emu was measured in large nanorods65 with diameter $\sim$100 nm and
length of few $\mu$m produced by NaOH-assisted solvothermal processing with
vacuum annealing and/or low-pressure calcination. The phase content varied
from pure brookite to anatase-rutile mix with large concentration of VO and
Ti3+ states. The same $\mu=7-7.5\times 10^{3}\mu_{B}$ polaron moment was
measured in sol-gel produced 10 nm single-phase anatase81 due to VO, despite
saturation magnetization being roughly 2 orders of magnitude smaller than in
the first study 65. Our results show that the polaron properties are roughly
unaffected neither by the preparation method, phase content of the produced
TiO2, nor annealing conditions, which lead just to the variability of polaron
concentration $N$ in the samples.

A unambiguous one-to-one linear correlation exists between the concentration
of bound magnetic polarons $N$ and the oxygen vacancy content65. Theoretical
calculations indicate that a single oxygen vacancy VO can produce magnetic
moment of about 1.4-2.4 $\mu_{B}$ 79, 62, 63, cation vacancy VTi – about 2.3-4
$\mu_{B}$ 67, 79, 62, 63 and di-vacancy – 2 $\mu_{B}$ 67, 55. For a
characteristic moment $\sim$ 2 $\mu_{B}$ per defect, ca. 0.4-0.7% of defective
cells are found in the electrodispersed powders with defect content decreasing
with higher calcination temperature. In contrast, in pyrolyzed samples the
defect content is ca. 0.01-0.04% without an obvious dependence on the
pyrolyzation temperature.

##  4 Conclusions

An organometallic precursor based on titanium carboxylate complex has been
developed to produce phase-pure anatase, rutile and mixed phase TiO2
nanoparticles with controllable polymorph content using a simple non-
hydrolytic thermal process. The new approach, based on a different reaction
mechanism, eliminates from consideration many reaction parameters, which are
difficult to control in the hydrolytic sol-gel processes, particularly, such
as instantaneous Ti:H2O ratio, hydrolysis rate, pH, reaction modulators, and
allows better control of the reaction process. A broad range of samples with
varying structures, sizes and phase compositions has been produced by
pyrolyzation of the carboxylate precursor. The temperature of the pyrolytic
treatment comes out as a single parameter determining the grain size and
crystal phase tuning of the final product in a simple, predictable and
reproducible way.

The carboxylate precursor produced by biphasic cation exchange is long-term
storable, hydrolytically stable and not sensitive to moisture contained in the
air, unlike inorganic Ti chlorides or organometallic alkoxides commonly
employed in the standard methods, which undergo vigorous hydrolysis. On this
account titanium carboxylates are also ideal precursors for film deposition.
The high chemical purity of the produced materials is assured on account of
the biphasic extraction involved in the preparation of the precursor. Common
impurities characteristic for the standard "one-pot" production methods have
not been found during the investigation. The results are correlated against
the traditional sol-gel method and confirm that the reported process is a good
technique to produce TiO2 nanoparticles having surfaces free from organics.

An advantage of the the carboxylate precursor in pyrogenic synthesis is that
it allows to avoid the commonly used titanium chlorides and the related
corrosive hazards in chlorine rich flames. Flame synthesis 83, 84 is a widely
employed to produce high quality metal oxide nanoparticles through a one-step
process. Hence, the carboxylate precursor is a favourable alternative in the
production of TiO2 nanoparticles.

## Acknowledgements

We acknowledge technical staff from the University of Latvia, which was
involved in the preparation of some of the samples under grant agreement No.
1.1.1.1/16/A/085 (2017-2018).

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# Attractive interactions in the microstructures of asymptotically flat black
holes

Deyou Chen<EMAIL_ADDRESS>School of Science, Xihua University, Chengdu
610039, China  Jun Tao<EMAIL_ADDRESS>Center for Theoretical Physics,
College of Physics, Sichuan University, Chengdu, 610065, China  Xuetao Yang
<EMAIL_ADDRESS>Center for Theoretical Physics, College of Physics,
Sichuan University, Chengdu, 610065, China

###### Abstract

In this work, we investigate the microstructure of asymptotically flat black
holes with Ruppeiner curvature. Specially, the cosmological constant is
considered to have a fluctuation around $0$. Under such consideration, both
repulsive and attractive interactions are found in the Reissner-Nordström and
Kerr black holes, while the Schwarzschild black hole has dominant attractive
interaction. The result obtained is quite different from that of excluding the
fluctuation of cosmological constant, where these black holes are found to be
always characterised by repulsive interaction.

##  I Introduction

The microstructure of black holes has posed a significant challenge in
theoretical physics since the establishment of the four laws of black hole
thermodynamics. Alternative avenues of research include exploring macroscopic
thermodynamic properties of black holes, which provide valuable insights into
the underlying coarse-grained structure. Recently, there has been considerable
interest in studying the phase transition of anti de Sitter (AdS) black holes
within an extended phase space, where the cosmological constant is treated as
a thermodynamic pressure $P$ Kastor:2009wy ; Dolan:2010ha ; Dolan:2011xt ;
Cvetic:2010jb . This perspective has revealed intriguing similarities between
the phase transition of certain AdS black holes and that of a van der Waals
(VdW) system Kubiznak:2012wp ; Gunasekaran:2012dq ; Kubiznak:2016qmn , hinting
at a fluid-like microstructures for AdS black holes. Various phase transition
phenomena observed in AdS black holes further support this notion, including
reentrant phase transitions Altamirano:2013ane , triple points
Altamirano:2013uqa ; Wei:2014hba ; Frassino:2014pha , and superfluid phase
transitions Hennigar:2016xwd . Other interesting results are found in GWWY1 ;
GWWY2 ; GWWY3 ; GWWY4 ; GWWY5 ; RGC ; LMLX ; CKM ; GM ; BLT ; LW1 ; LW2 ; LW3
; AMYV ; CY ; WZ ; ACMR1 ; ACMR2 ; ACMR3 ; ACMR4 ; CHPZ ; XCH1 ; XCH2 ; XCH3 ;
XCH4 ; XCH5 ; XCH6 ; XCH7 ; XCH8 .

To further explore microscopic nature of black holes, Ruppeiner geometry has
been included. Unlike conventional thermodynamics, this approach reverses the
order by utilizing macroscopic thermodynamic details as the foundation for
microscopic investigations. In standard thermodynamics, it has been
established that the scalar curvature associated with this thermodynamic
geometry reflects microscopic interactions, where positive or negative
curvature indicates repulsive or attractive interactions, respectively
ruppeiner1979thermodynamics ; ruppeiner1981application ;
janyszek1989riemannian ; janyszek1990riemannian ; oshima1999riemann ;
mirza2009nonperturbative ; may2013thermodynamic . Applying this framework to
black hole systems is straightforward, as the entropy serves as the
thermodynamic potential in Ruppeiner geometry Ruppeiner:2008kd1 ;
Ruppeiner:2008kd2 ; Wei:2015iwa ; Wei:2019uqg .

An initial application of this method focused on the Reissner-Nordström (RN)
black hole, revealing a vanishing scalar curvature and suggesting no
interaction Aman:2003ug . However, considering that the spacetime is curved,
it was anticipated that a non-vanishing scalar curvature would be present.
This issue was re-evaluated by relating the results of the RN black hole to
those of the Kerr-Newman-AdS (KN-AdS) black hole Mirza:2007ev . By considering
entropy ($S$), angular momentum ($J$), and charge ($Q$) as the complete set of
parameter space and appropriately taking the limit for the KN-AdS black hole,
a non-vanishing scalar curvature was obtained for the RN black hole. Notably,
this scalar curvature is always positive, indicating a dominance of repulsive
interaction within the black hole. Additionally, the scalar curvature
calculated for the RN black hole in an alternative phase space
Dehyadegari:2016nkd  yielded similar results Xu:2019nnp .

In these studies, the cosmological constant is considered as a constant, which
can be tuned to 0. Nevertheless, in a more “fundamental theory”, one may
consider the fluctuation of $P$ around $0$, instead of a constant $P=0$ for
the asymptotically flat black hole. Equivalently, we can first investigate the
microstructure of AdS black holes considering the the fluctuation of $P$, and
take the special case of $P\rightarrow 0$ to be the microstructure of
asymptotically flat black holes. Indeed, a form of the transition from AdS to
asymptotically flat space with a black hole has been proposed in Gauss-Bonnet
gravity Camanho:2013uda ; Hennigar:2015mco . By considering a dynamical
variable cosmological constant, i.e. the thermodynamic pressure $P$, it was
demonstrated that a phase transition can occur between thermal AdS space and
an asymptotically flat geometry with a black hole, regardless of the
temperature range. This transition implies that considering the fluctuation of
$P$ around $0$ seems to be reasonable in some “fundamental theory”. Recently,
a higher-dimensional origin for the dynamical $P$ was also exactly proposed in
the context of holographic braneworlds Frassino:2022zaz . It would be
interesting to investigate the microstructure of asymptotically flat black
hole with the inclusion of the fluctuation of $P$.

In this work, we will regard the cosmological constant to be thermodynamic
pressure $P$ that can vary, and investigate the microstructure for RN, Kerr
and Schwarzschild black holes by taking appropriate limits for the Ruppeiner
curvature of KN-AdS black hole. As we will see soon, the inclusion of the
fluctuation of $P$ in the parameter space will have a significant influence on
the microstructure of these black holes.

The paper is structured as follows. Section II provides a concise overview of
the thermodynamics of the Kerr-Newmann-AdS black hole in the parameter space
of $(S,Q,J,P)$. In Sec. III, the Ruppeiner geometry is constructed for
microstructure study. By taking appropriate limits, the microstructure of RN,
Kerr and Schwarzschild black holes are discussed separately. Finally, we
present our results in Sec. IV.

##  II Thermodynamics for KN-AdS black holes

For the bulk action of four-dimensional Einstein-Maxwell theory with
cosmological constant $\Lambda=-3/l^{2}$ ($l$ is the AdS radius)

$S_{EM}=\frac{1}{16\pi}\int
d^{4}x\sqrt{-g}\left(R-F_{\mu\nu}F^{\mu\nu}-2\Lambda\right),$ (1)

the most general black hole solution is the KN-AdS solution that describes a
charged, rotating black hole in AdS space. In Boyer-Lindquist-like
coordinates, it is given by

$\displaystyle ds^{2}=$
$\displaystyle-\frac{\Delta_{r}}{\rho^{2}}\left(\mathrm{~{}d}t-\frac{a\sin^{2}\theta}{\Xi}\mathrm{d}\phi\right)^{2}+\frac{\rho^{2}}{\Delta_{r}}\mathrm{~{}d}r^{2}+\frac{\rho^{2}}{\Delta_{\theta}}\mathrm{d}\theta^{2}$
(2)
$\displaystyle+\frac{\sin^{2}\theta\Delta_{\theta}}{\rho^{2}}\left(a\mathrm{~{}d}t-\frac{r^{2}+a^{2}}{\Xi}\mathrm{d}\phi\right)^{2},$

where

$\displaystyle\rho^{2}$ $\displaystyle=r^{2}+a^{2}\cos^{2}\theta,$ (3)
$\displaystyle\Delta_{r}$
$\displaystyle=\left(r^{2}+a^{2}\right)\left(1+\frac{r^{2}}{l^{2}}\right)-2mr+q^{2},$
$\displaystyle\Delta_{\theta}$
$\displaystyle=1-\frac{a^{2}}{l^{2}}\cos^{2}\theta,$ $\displaystyle\Xi$
$\displaystyle=1-\frac{a^{2}}{l^{2}}.$

The associated gauge potential $A$ is given as

$A=-\frac{qr}{\rho^{2}}\left(\mathrm{~{}d}t-\frac{a\sin^{2}\theta}{\Xi}\mathrm{d}\phi\right).$
(4)

From these equations, the entropy $S$, temperature $T$, thermodynamic volume
$V$, electric potential $\Phi$ and angular velocity of the horizon
$\Omega_{H}$ can be calculated as Caldarelli:1999xj

$\displaystyle S$ $\displaystyle=\frac{\pi\left(r_{+}^{2}+a^{2}\right)}{\Xi},$
(5) $\displaystyle T$
$\displaystyle=\frac{r_{+}\left(1+\frac{a^{2}}{l^{2}}+3\frac{r_{+}^{2}}{l^{2}}-\frac{a^{2}+q^{2}}{r_{+}^{2}}\right)}{4\pi\left(r_{+}^{2}+a^{2}\right)},$
$\displaystyle V$
$\displaystyle=\frac{2\pi}{3}\frac{\left(r_{+}^{2}+a^{2}\right)\left(2r_{+}^{2}l^{2}+a^{2}l^{2}-r_{+}^{2}a^{2}\right)+l^{2}q^{2}a^{2}}{l^{2}\Xi^{2}r_{+}},$
$\displaystyle\Phi$ $\displaystyle=\frac{qr_{+}}{r_{+}^{2}+a^{2}},$
$\displaystyle\Omega_{H}$ $\displaystyle=\frac{a\Xi}{r_{+}^{2}+a^{2}},$

where $r_{+}$ represents the event horizon radius of the black hole. Treating
the cosmological constant $\Lambda$ as the thermodynamic pressure $P$
Kastor:2009wy ; Dolan:2010ha ; Dolan:2011xt ; Cvetic:2010jb , one can find

$m=\frac{64\pi^{2}a^{4}P^{2}q^{2}-48\pi a^{2}Pq^{2}+24\pi
a^{2}Pr_{+}^{2}+9a^{2}+24\pi Pr_{+}^{4}+9q^{2}+9r_{+}^{2}}{18r_{+}}.$ (6)

The physical mass $M$, charge $Q$, and angular momentum $J$ can be expressed
by

$M=\frac{m}{\Xi^{2}},\quad Q=\frac{q}{\Xi},\quad J=\frac{am}{\Xi^{2}}.$ (7)

After a simple calculation, one can express $T$, $M$ and $V$ in terms of $S$,
$J$ and $P$:

$\displaystyle T$
$\displaystyle=\frac{-12\pi^{2}J^{2}+S^{2}\left(64P^{2}S^{2}+32PS+3\right)+16\pi
PQ^{2}S^{2}-3\pi^{2}Q^{4}}{4\sqrt{\pi}S^{3/2}\sqrt{12\pi^{2}J^{2}(8PS+3)+\Sigma^{2}}},$
(8) $\displaystyle M$
$\displaystyle=\frac{\sqrt{12\pi^{2}J^{2}(8PS+3)+\Sigma^{2}}}{6\sqrt{\pi}\sqrt{S}},$
$\displaystyle V$
$\displaystyle=4\sqrt{S}\frac{2592\pi^{6}J^{6}(8PS+3)\left(\pi
Q^{2}(8PS+3)+3S\right)}{3\sqrt{\pi}\Sigma\left(36\pi^{2}J^{2}+\Sigma^{2}\right)^{2}\sqrt{12\pi^{2}J^{2}(8PS+3)+\Sigma^{2}}}$
$\displaystyle+4\sqrt{S}\frac{432\pi^{4}J^{4}\left(\pi
Q^{2}(8PS+3)+2S(4PS+3)\right)\Sigma^{2}}{3\sqrt{\pi}\Sigma\left(36\pi^{2}J^{2}+\Sigma^{2}\right)^{2}\sqrt{12\pi^{2}J^{2}(8PS+3)+\Sigma^{2}}}$
$\displaystyle+4\sqrt{S}\frac{6\pi^{2}J^{2}\Sigma^{4}\left(S(8PS+15)+3\pi
Q^{2}\right)+S\Sigma^{6}}{3\sqrt{\pi}\Sigma\left(36\pi^{2}J^{2}+\Sigma^{2}\right)^{2}\sqrt{12\pi^{2}J^{2}(8PS+3)+\Sigma^{2}}},$

where

$\Sigma=S(8PS+3)+3\pi Q^{2}.$ (9)

By identifying the black hole mass as the enthalpy, the first law of
thermodynamics reads

$dM=TdS+\Phi dQ+\Omega dJ+VdP.$ (10)

It will reduce to the RN-AdS and Kerr-AdS black hole cases with $J\rightarrow
0$ and $Q\rightarrow 0$, respectively. Sequentially, by taking $P\rightarrow
0$, the thermodynamics discussed above will reduce to the RN and Kerr black
hole cases. If we simultaneously set $J$, $Q$ and $P\rightarrow 0$, the KN-AdS
black hole will finally degrade into a Schwarzschild black hole.

##  III Ruppeiner geometry

In this section we explore the microstructure of the RN and Kerr black holes
using Ruppeiner geometry. In this thermodynamic geometry, the entropy $S$ is
treated as the thermodynamic potential and its fluctuation $\Delta S$ is
connected to the line element in the Riemannian geometry Ruppeiner:1995zz

$\Delta l^{2}=-\frac{\partial^{2}S}{\partial x^{\mu}\partial x^{\nu}}\Delta
x^{\mu}\Delta x^{\nu},$ (11)

where $x^{\mu}$ are the independent thermodynamic quantities.

Within this framework, an empirical observation suggests that the scalar
curvature $R$ calculated using Eq. (11) is associated with the microscopic
interactions occurring in the thermodynamic system under consideration
ruppeiner1979thermodynamics ; ruppeiner1981application ;
janyszek1989riemannian ; janyszek1990riemannian ; oshima1999riemann ;
mirza2009nonperturbative ; may2013thermodynamic . Specifically, the positive
(negative) sign of $R$ indicates a repulsive (attractive) interaction, while
$R=0$ corresponds to the absence of interaction.

Considering the first law of the black hole Eq. (10), the basic thermodynamic
differential relation between entropy and other thermodynamic quantities can
be cast into

$dS=\frac{1}{T}dM-\sum_{i}\frac{y_{i}}{T}dx^{i},$ (12)

where

$y_{i}=(\Phi,\Omega,V),\quad x^{i}=(Q,J,P).$ (13)

With these symbols, the line element Eq. (11) can be expressed as

$\Delta l^{2}=-\Delta z_{\mu}\Delta x^{\mu},$ (14)

where $z_{\mu}=(1/T,-y_{i}/T)$ and $x^{\mu}=(M,x^{i})$, and we have the
following relationships

$\begin{gathered}\Delta
z_{0}=\Delta\left(\frac{1}{T}\right)=-\frac{1}{T^{2}}\Delta T,\\\ \Delta
z_{i}=\Delta\left(-\frac{y_{i}}{T}\right)=\frac{y_{i}}{T^{2}}\Delta
T-\frac{1}{T}\Delta y_{i}.\end{gathered}$ (15)

The line element can be then obtained as

$\Delta l^{2}=\frac{1}{T}\Delta T\Delta S+\frac{1}{T}\Delta y_{i}\Delta
x^{i}.$ (16)

In this paper, we focus on the fluctuation of $S$ and $P$, thereby the above
line element finally becomes

$\Delta l^{2}=\frac{1}{C_{P}}\Delta S^{2}+\frac{2}{T}\left(\frac{\partial
T}{\partial P}\right)_{S}\Delta S\Delta P+\frac{1}{T}\left(\frac{\partial
V}{\partial P}\right)_{S}\Delta P^{2}.$ (17)

where we have expanded $\Delta T$ and $\Delta P$ by use of $S$ and $P$, and
utilized the Maxwell equation $(\partial T/\partial P)_{S}=(\partial
V/\partial S)_{P}$ deduced by the first law of thermodynamics Eq. (10). Here
$C_{P}$ is the heat capacity at constant pressure

$C_{P}=T\left(\frac{\partial S}{\partial T}\right)_{P}.$ (18)

The line element (17) was first presented in Xu:2020gud ; Ghosh:2019pwy  and
used to study the thermodynamic geometry of Schwarzschild and charged Gauss-
Bonnet black holes in AdS space. In the following, we adopt the approach from
Dolan:2011xt ; Wei:2019uqg  and order $P\to 0$ in Eq. (17) to investigate the
microstructure of RN, Kerr and Schwarzschild black holes. It is worth noting
that the line element (17) is not guaranteed to be positive definite, thus the
standard thermodynamical stability criterions Eqs. (22)-(24) in
Ruppeiner:2013yca  may not be satisfied, and the thermodynamical stability
will be uniquely examined by using the sign of $C_{P}$ in this paper.

###  III.1 Microstructure of RN black holes

Figure 1:  The isobaric curves ($P=0$) (red solid curves), spinodal curve
(blue dashed line), and sign-changing curve (black dashed line) for the RN
black hole in the $T-S$ diagram. The charge for isobaric curves are set as
$Q=1$, $2$, $3$ and $4$ from left to right.

If we consider the RN systems as the limit of KN-AdS black holes by setting
$P\rightarrow 0$ and $J\rightarrow 0$, the associated thermodynamic curvature
will be given by

$R_{RN}=-\frac{S-2\pi Q^{2}}{S\left(S-\pi Q^{2}\right)}.$ (19)

It diverges at $S=\pi Q^{2}$. Setting the numerator equals zero, we get one
zero point

$S=2\pi Q^{2}.$ (20)

Obviously, $R_{RN}$ takes positive values when $\pi Q^{2}<S<2\pi Q^{2}$,
otherwise takes negative values for $S<\pi Q^{2}$ or $S>2\pi Q^{2}$. To assess
the thermodynamic permissibility of these regions, we plot the isobaric curves
($P=0$) (red solid lines), spinodal curve (black dashed line) and sign-
changing curve (blue dashed line) in the $T-S$ diagram, as shown in Fig. 1.
The isobaric curve for a RN black hole is expressed as

$T_{P=0}=\frac{S-\pi Q^{2}}{4\sqrt{\pi}S^{3/2}}.$ (21)

By using of $\left(\partial_{S}T\right)_{P=0}=0$, one can easily get the
spinodal curve

$T_{sp}=\frac{1}{6\sqrt{\pi S}}.$ (22)

As shown in Fig. 1, the spinodal curve separate the black hole phases into
stable ($C_{P=0}>0$) and unstable ($C_{P=0}<0$) ones. The unstable phases
corresponds to a negative heat capacity at constant pressure
$C_{P=0}=T\left(\partial_{T}S\right)_{P=0}$, which is above the spinodal
curve.

From Eqs. (19) and (21), we can obtain two sign-changing curves in the $T-S$
diagram

$T_{1}=0,\quad T_{2}=\frac{1}{8\sqrt{\pi S}}.$ (23)

Considering the non-negative requirement of temperature, there is only one
physical sign-changing curve $T_{sc}\equiv T_{2}$. The positive $R_{RN}$
region has been labeled in lightyellow color, which is below the sign-changing
curve $T_{sc}$, indicating a possible dominant of repulsive interaction among
microscopic ingredients of the black hole. When $T_{sc}<T<T_{sp}$, there is a
stable phase for negative $R_{RN}$, which may correspond to a attractive
interaction. For $T>T_{sp}$, there is another region with negative $R_{RN}$,
but such region is thermodynamic unstable.

This result is interesting. Early study Aman:2003ug  suggested that the RN
black hole has a zero thermodynamic curvature. While in Ref. Mirza:2007ev ,
$(S,J,Q)$ was regarded as the complete set of parameter space. By setting
cosmological constant $\Lambda\rightarrow 0$ and $J\rightarrow 0$ in the
Ruppeiner curvature calculated for KN-AdS case, the sign-changing behaviour
was found only at the extremal limits ($T=0$), where a black hole changes its
nature to a new phase, i.e., naked singularity. When $T>0$, the corresponding
scalar curvature is always positive, indicating a repulsive-dominanted RN
black hole. A similar result was also obtained in Ref. Xu:2019nnp , where
$(S,q^{2})$ was taken as complete parameter space of a RN black hole.

The difference in these results seems arise from the choice of parameter
space. In our case, the $(S,J,Q,P)$ are together taken as complete set of
parameter space. Especially, the pressure $P$, i.e., the cosmology constant
$\Lambda$ is treated as a parameter that can vary. Then the Ruppeiner
curvature for a RN black hole is calculated by taking the limits $P\rightarrow
0$ and $J\rightarrow 0$ in the Ruppeiner curvature calculated for the KN-AdS
case. While in Refs. Mirza:2007ev ; Xu:2019nnp , the pressure $P$ is
considered as a constant, although it can also be tuned to an infinitesimal
value.

###  III.2 Microstructure of Kerr black holes

Using the the line element for the complete phase space of parameters
($S,J,Q,P$), and taking the limits $P\rightarrow 0$ and $Q\rightarrow 0$, the
Ruppeiner curvature for Kerr black hole can be calculated as

$R_{Kerr}=\frac{A}{S\left(S^{2}-4\pi^{2}J^{2}\right)B^{2}},$ (24)

where

$\displaystyle A$
$\displaystyle=110592\pi^{12}J^{12}S^{2}+118272\pi^{10}J^{10}S^{4}+112000\pi^{8}J^{8}S^{6}$
(25)
$\displaystyle\quad+65952\pi^{6}J^{6}S^{8}+10368\pi^{4}J^{4}S^{10}+8192\pi^{14}J^{14}-81S^{14},$
$\displaystyle B$
$\displaystyle=80\pi^{4}J^{4}S^{2}+72\pi^{2}J^{2}S^{4}+64\pi^{6}J^{6}+9S^{6}.$

The scalar curvature $R_{Kerr}$ diverges at $S=2\pi J$. Setting $A=0$, we can
get one zero point

$S\approx 11.7267J.$ (26)

Similar to the discussion in the RN black hole case, we plot the isobaric
curve ($P=0$) (red solid lines), spinodal curves (black dashed line) and sign-
changing curve (blue dashed line) in the $T-S$ diagram, as shown in Fig. 2.
The isobaric curve and spinodal curve for Kerr black holes are calculated as

$\displaystyle T_{P=0}$
$\displaystyle=\frac{S^{2}-4\pi^{2}J^{2}}{4\sqrt{\pi}S^{3/2}\sqrt{4\pi^{2}J^{2}+S^{2}}},$
(27) $\displaystyle T_{sp}$
$\displaystyle=\frac{\sqrt{3}+1}{2\sqrt{2\left(7\sqrt{3}+12\right)\pi S}},$

which are displayed as red solid line and blue dashed line in Fig. 2,
respectively. From Eqs. (24) and (27), we can obtain two sign-changing curves
in the $T-S$ diagram

$T_{1}=0,\quad T_{2}\approx\frac{0.0886341}{\sqrt{S}}.$ (28)

Figure 2:  The isobaric curves ($P=0$) (red solid curves), spinodal curve
(blue dashed line), and sign-changing curve (black dashed line) for Kerr black
hole in the $T-S$ diagram. The charge for isobaric curves are set as $Q=1$,
$4$, $7$ and $10$ from left to right.

Similar to the RN black hole, there is only one physical sign-changing curve
$T_{sc}\equiv T_{2}$. The positive $R_{Kerr}$ region is below the sign-
changing curve $T_{sc}$, indicating a possible dominant of repulsive
interaction among microscopic ingredients of the black hole. When
$T_{sc}<T<T_{sp}$, there is a stable phase for negative $R_{Kerr}$, which may
correspond to an attractive interaction. In Ref. Mirza:2007ev , a scalar
curvature for Kerr black hole was also calculated in the $(S,J,Q)$ parameter
space. The sign-changing behaviour was found only at the extremal limits
($T=0$), and the corresponding scalar curvature is always positive, implying a
repulsive-dominanted Kerr black hole. The difference in results is from the
choice of parameter space, as explained before.

###  III.3 Microstructure of Schwarzschild black holes

If we simultaneously set $P$, $J$ and $Q\rightarrow 0$ in the Ruppeiner
curvature calculated for the KN-AdS case, the curvature for a Schwarzschild
black hole will be given as

$R_{Schwarzschild}=-\frac{1}{S},$ (29)

which is always negative. This result suggests an attractive-dominated black
hole. The isobaric curve for Schwarzschild black holes is calculated as

$T_{P=0}=\frac{1}{4\sqrt{\pi S}},$ (30)

which always has negative slope, indicating there is no stable black hole
phase with attractive interaction. Interestingly, if the pressure $P$ is not
included in the parameter space, one will get an opposite conclusion, where
the Ruppeiner curvature is shown to be always positive Mirza:2007ev ;
Xu:2019nnp .

Figure 3:  The temperature curves for Schwarzschild black hole $(P=Q=J=0)$, RN
black hole $(P=J=0,Q=1)$ and Schwarzschild-AdS black hole
$(Q=J=0,P=3/(800\pi))$.

On the other hand, we notice that there is another interesting conclusion on
the microstructure of Schwarzschild black hole: the black hole may be
characterised by dominant attraction interaction at low temperature, whereas
by dominant repulsive interaction at high temperature Xu:2020gud . This
conclusion comes from the comparison between temperature curves of
Schwarzschild, RN and Schwarzschild-AdS (SAdS) black holes, as we showed in
Fig. 3. The temperature behavior of Schwarzschild black hole is close to that
of SAdS black hole at high temperature (or small scale). The Ruppeiner
curvature for SAdS black hole can be calculated by taking the limits
$Q\rightarrow 0$ and $J\rightarrow 0$ in KN-AdS case, which turns out to be

$R_{SAdS}=-\frac{1}{S(1+8PS)}.$ (31)

It always takes negative values. Hence, one can get a conclusion that the
Schwarzschild black hole may be characterised by dominant repulsive
interaction at high temperature. While at low temperature (or large scale),
the temperature behavior of Schwarzschild black hole is close to that of RN
black hole. If starting off from a parameter space excluding the pressure $P$,
one can say that the Schwarzschild black hole has dominant attraction
interaction at low temperature Mirza:2007ev ; Xu:2019nnp .

Such interaction pattern leaves us an interesting question: which internal
mechanism leads to the change in the interaction? However, if the pressure $P$
is included in the parameter space, we can avoid this question, for a simple
conclusion that the Schwarzschild black hole is always characterised by
dominant attractive interaction at all scale ranges. This conclusion is also
self-consistent, since the RN black hole has also dominant attractive
interaction at large scale, as discussed before.

##  IV Conclusion

In this paper, we have used the Ruppeiner geometry for KN-AdS black holes to
probe the microstructure of RN, Kerr and Schwarzschild black holes, by taking
appropriate limits. We first review the thermodynamics for KN-AdS black holes,
where the $(S,J,Q,P)$ are together taken as complete set of parameter space.
The Ruppeiner geometry is then constructed to investigate the microscopic
interactions. By setting $P\rightarrow 0$ and $J\rightarrow 0$ in the
Ruppeiner curvature calculated for KN-AdS case, the Ruppeiner curvature for a
RN black hole is obtained. In the stable black hole phase, both attractive
($R_{RN}<0$) and repulsive ($R_{RN}>0$) interactions are found. Analogously,
by setting $P\rightarrow 0$ and $Q\rightarrow 0$, the Ruppeiner curvature for
Kerr black hole is also calculated. Similar to the RN black hole, both
attractive and repulsive interactions may exist in the system. If we
simultaneously set $P$, $J$ and $Q\rightarrow 0$, the Ruppeiner curvature for
a Schwarzschild black hole can be easily calculated, which is found to be
always negative, indicating an attractive-dominated black hole.

These results are different from the ones obtained from a parameter space
excluding the thermodynamic pressure $P$. Specifically, the RN, Kerr and
Schwarzschild black holes were found to be repulsive-dominated in the
literatures Mirza:2007ev ; Xu:2019nnp . To conclude, whether regarding the
cosmological constant as a dynamical variable, i.e., the pressure $P$, gives
quite different microstructures of RN, Kerr and Schwarzschild black holes. In
Refs. Camanho:2013uda ; Hennigar:2015mco , a form of the transition from AdS
to asymptotically flat space with a black hole has been proposed. It seems to
be reasonable to consider a fluctuation of $P$ around $0$.

###### Acknowledgements.

We would like to thank Dr. Ningcheng Bai for his useful discussions. This work
is supported by NSFC (Grant No. 12275183, 12275184, 12105191).

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