Nature | Vol 584 | 20 August 2020 | 385from the LLCP, and then decreases, in contrast to the monotonic
increase of density along the liquid–gas transition. Such a behaviour
was recently predicted by ab initio molecular dynamics calculations
along the LLT line of phosphorus^34 , which suggested that the order
parameter describing the LLT contains contributions from both the
density and the entropy and that at least at low temperatures, entropy—
rather than density—governs the transition. This is at odds with the
liquid–gas transition, for which density is the sole order parameter.
A two-order-parameter model including density and a bond-order
parameter describing locally favoured structures has been proposed^13
to explain the existence of LLTs. The putative LLT in water has also been
described as entropy-driven, on the basis of a model in which water
is considered as an ‘athermal solution’ of two molecular structures
with different entropies and densities^35. We note, however, that the
present LLT in sulfur is different from that in water and phosphorus,
in the sense that the transition line has a positive slope in sulfur but
a negative one in water and phosphorus. This may signal that sulfur
belongs to a different class of LLT.
This work also provides the first, to the best of our knowledge, experi-
mental evidence for a critical point terminating the line of an LLT. Such
an LLCP was proposed in phosphorus at about 3,500 K and 0.02 GPa
(ref.^34 ), conditions that have not yet been achieved experimentally.
In supercooled liquid silicon, classical empirical calculations have
predicted an LLCP at negative pressures (−0.6 GPa, 1,120 K)^36 but ab
initio calculations recently determined that the HDL– and HDL–vapour
spinodals form a continuous reentrant curve, making supecooled Si
a critical-point-free system^37. In water, the existence of an LLCP was
proposed^1 in 1992 to explain the many anomalies in the thermody-
namic properties of water, such as the heat capacity, compressibility
and thermal expansion coefficients, and evidence for this LLCP has
been explored and debated ever since. The experimental observa-
tion of this hypothetical LLCP in water may never be possible as it
is located in the ‘no man’s land’, that is, the P−T domain below the
homogeneous-nucleation temperature. Finally, an LLCP has also been
predicted in simple molecular systems, such as H 2 (ref.^6 ) and N 2 (ref.(^7) ), but their experimental observation remains extremely challenging.
The LLCP in sulfur, being in a P–T range easily accessible by experiment,
provides a unique opportunity for the study of critical phenomena
associated with LLTs. We thus expect that the present work will generate
a new interest in LLTs that will provide a solid basis for understanding
the principles that govern LLTs in general. Future studies should also
focus on deciphering the microscopic processes at the origin of the
LLT in sulfur.
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Raman intensity100 200 300 400 500 600
Raman shift (cm–1)1 bar, 394 K (I)1 bar, 595 K (II)1.2 GPa, 600 K (III)a d
43210PDF0.1 0.2 0.3 0.4 0.5 0.6
r (nm)0.56 GPa 571 K (E)0.11 GPa 428 K (A)0.21 GPa 487 K (C)0.36 GPa 487 K (D)0.17 GPa 442 K (B)PDFr (nm)1.4 423 K
1.3
1.2
1.1
1.0473 K
573 Kb3. 5
3. 4
3. 3Distance (Å)0.1 0.2 0.3 0.4 0.5 0.6
Pressure (GPa)4. 6
4. 5
4. 4
4. 3
4. 2
4. 1
4. 0
3. 9
3. 5
3. 4
3. 34. 6
4. 5
4. 4
4. 3
4. 2
4. 1
4. 0
3. 9
Second neighboursThird/fourth neighboursc1.31.21.11.0PDF0.40 0.45 0.500.40 0.420.44 0.460.48 0.50
r (nm)ABEC
DFig. 3 | Local order in the LDL and HDL sulfur. a, The PDF, g(r), of liquid
sulfur at selected pressure and temperature conditions along path P11 of
Fig. 1. Curves A–E correspond to the P, T conditions in Fig. 1 at which the
measurements were performed. The curves are vertically offset by 0.3 for
clarity. The inset shows a magnification of the PDF in the region 0.4–0.5 nm,
which contains contributions from third and fourth neighbours.
b, Magnification of the third peak of g(r) from the ambient-pressure neutron
data of ref.^28 at 423 K, 473 K and 573 K. The curves were reproduced with
permission from ref.^28 (American Physical Society, 1990). c, Positions of the
second, third and fourth neighbours as a function of pressure, as deduced from
the peak positions of the PDF along path P11. d, Selected Raman spectra of
liquid sulfur collected at the (P, T) points I, II and III of Fig. 1 : I, LDL below the
λ-transition; II, LDL above the λ-transition; III, HDL. Error bars indicate 1 s.d.