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(1) scaling ofuseful form of worth fo
Two phase transitions: such thermo
Eq.ture constrains the facts are a that no
Three “pillars” of our view powers)
2 2,
The critical-point exponen
(a)critical Rushbrooke inequality
SCALING, UNIVERSALITY, and point, so this const
tial, near theobeystheRENORMALIZATION GROUP
(1)
(H, ) is the functional equatio
G s1: Scalingwhich exponent relations & “data collapse”
ply for quantities derived sca
the form by equality. of the th
implications agiven of anthe ratioDefiningfrom
PILLAR (implies
H H, a
follows thatT of state. H, . of t
such asG tion to the scaling s power
the equation
s G
which 1example, H critical point is appro
Consider, forthea M 1 athe M(H,T) e
are simply .
aH
a Exponent relations:a H scaling laws
of B. of
sforms uniaxial ferromagnet near the cr The a function
potentialsT Eliminating a Hexponent Eq. (5a)
0,T . On differentiating Eqs. (1)
arbitrary and a T from
c
so GHFs. obtain the Griffiths equality a path
The predictions of the scaling hyp
H, weasfind
essible
Thus one can 2.H
the itproperties of 1 1‘‘write (i) Legen
otential, aH aT GHFs: down by
a
M H, M and (7) .give
H,
GHFs are also Eqs. (5b), so exponent.
Similarly, GHFs, (5c),
odynamic for any critical-point all thermod
ity is valid for all positive val
(10)
Since Eq.erally, (ii) Derivatives of GHF
are GHFs. and proves useful in prac 2
hold for 1 . ferromagnet, cho
certainly of a uniaxialthe particularwe h
Since every thermodynamic function 2
Tw
Pillar 1 (continued): pone
same
and
Experimental test of mate
data collapse (Pillar class
1): Equation of
State for 5 different
C. R
magnets near their
respective critical Th
points. that
of a
Ham
Card
Pillar 2: Universality tion
FIG. 1. Experimental MHT data on five different magnetic for u
materials plotted in scaled form. The five materials are CrBr3 , (ii) p
First hint: all 5 EuO, Ni, YIG, and Pd3Fe. None of these materials is an ide-
tain
magnets have same alized ferromagnet: CrBr3 has considerable lattice anisotropy,
pone
EuO has significant second-neighbor interactions. Ni is an
scaled equation of itinerant-electron ferromagnet, YIG is a ferrimagnet, and
cepts
state. Pd3Fe is a ferromagnetic alloy. Nonetheless, the data for all On
materials collapse onto a single scaling function, which is that reno
3ˇ ´
calculated for the d 3 Heisenberg model [after Milosevic and dime
Stanley (1976)]. Niels
theo
3
M M
aH Pillar 2, Universality (11b)
/a
H
H 1/
,
(Universality classes):
H Experimental fact: (11c)
H 1/
A wide range of
scaled magnetization and scaled tempera-
function F (1) (x) materials
magnetic M(1,x) defined in Eq.
a scaling function.
belong to one M H is
he scaled magnetizationof two plotted
ed temperature H , and the entire family
families of “Universality
t,T) curves ‘‘collapse’’ onto a single func-
classes”: the Q-state
ng function F (1) (H) M(1, H ) evidently
zation function in fixed nonzero magnetic
Potts model (Potts 1952)
and the n-vector model
(HES 1968). The purely
geometric phase
IVERSALITY?
onetransition “percolation”be-
finds that all systems in nature
corresponds to the limit
f a comparatively small number of such
sses. Two specific microscopic interaction
Q=1, while the encompass the
appear almost sufficient to self-
sses necessary for static critical phenom-
avoiding random walk
FIG. 2. Schematic illustrations of the possible orientations of
corresponds to model
these is the Q-state Pottsn=0. (Potts, the spins in (a) the s-state Potts model, and (b) the n-vector
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2). One assumes that each spin i can be in model. Note that the two models coincide when Q 2 and n
possible discrete orientations i ( i 1. (c) North-south and east-west ‘‘Metro lines.’’
If two neighboring spins i and j are in the 4
Pillar 3: RENORMALIZATION
GROUP
Kadanoff site-to-cell coarse-
graining successively tames the
problem of an infinite
correlation length.
Example: Q=1 Potts model for
d=1 (1d percolation)
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Examples of open questions
• What matters in the interaction potential?
Ex: Lee-Yang lattice gas describes
EVERY fluid near its critical point
• What matters for dynamics? Ex: What
determines onset of spatially hetero-
geneous dynamics (dyn. heterogeinity)?
• What fixes limits of validity of classic
“laws”? Ex: Stokes-Einstein breakdown?
• What determines “switching” of a system?
Ex: in biology, and economics, there are
“tipping points” but no mechanical
switches!
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PUZZLE:
How does a
paramagnet “know”
when to
spontaneously order
itself?
ANSWER:
When the
exponential decay
along a 1-d path
balances the
exponential increase
in the number of
paths.
v = J/kT = n.n.
coupling strength
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Mark Kac: “How does a liquid know when to condense?”
How can a liquid exist in 1 phase?
ex: Lennard-Jones potential:
1 length scale implies 1 liquid phase
Can a liquid exist in 2 phases?
ex: 2-well Lennard-Jones potential:
2 length scales means 2 liquid phases
Why local tetrahedral interactions
lead to 2 different length scales
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O Si
Silica: SiO2
Silicon: Si
Three ubiquitous substances have in
common local tetrahedral structure and
hence more than one length scale in their
interaction potentials.
Do all 3 show a liquid-liquid transition?
Water: H2O
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H anomalies: 3
3 of the 64Eugene Stanley thermodynamic response functions singular
(at about 228K = -45 C)
(a)Compressibility (b)
Specific
Heat
(c) Coeff.
Thermal
Expansion
Figure 2. Schematic dependence on temperature of (a) the isothermal com-
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TRUE! L-L Ph.Tr. Hypothesis
Figure 1: Schematic illustration indicating the various Figure 2: Generalization of Fig. 1 to incorporate a second
phases of liquid water (color-coded) that are found at control parameter, the pressure. The colors are the same as
atmospheric pressure. Courtesy of Dr. O. Mishima. used in Fig. 1. Courtesy of Dr. O. Mishima.
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How to find a critical point:
Different answers for theorist and experimentalist!
(due to phenomenon of metastability)
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Anomalous Thermodynamic Properties of Supercooled Water
TS = 228 K
Specific Heat = - 45 ˚C Isothermal
Compressibility
C. A. Angell, et al, JPC 77, R. J. Speedy, et al, JCP 65,
3092 (1973) 851 (1976)
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MCM-41-S Structure
MCM-41-S is well ordered with
hexagonal symmetry.
Four MCM-41-S samples, fully hydrated:
MCM-41-S-10, with pore size <10 Å,
grH2O / grsilica ≈ 40%;
MCM-41-S-12, with pore size 12 Å,
grH2O / grsilica ≈ 48%;
MCM-41-S-14, with pore size 14 Å,
grH2O / grsilica ≈ 50%;
MCM-41-S-18, with pore size 18 Å,
grH2O / grsilica ≈ 55%. C.-Y. Mou, Taiwan
14
Using specific heat peak (response of entropy) to locate Widom line:
Widom line in MD simulations consistent with Oguni sp.heat experiments
Simulation P=0MPa Experiments
12
Sendai 2003
CP(Jg K )
10
-1
-1
8
6
4
220 240 260 280 300
T (K)
L. Xu, P. Kumar, S.V. Buldyrev, S.H. S. Maruyama, K. Wakabayashi, M. Oguni,
Chen, P.H. Poole, F. Sciortino, H.E. “Thermal Properties of Supercooled
Water Confined within Silica Gel Pores,”
Stanley, PNAS 102,16558 (2005).
AIP conferenceprodeedings 708, 67
(2004)).
15
16
Mallamace, Chen, et al 2008 PNAS
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Use a different response function to locate Widom line: coefficient thermal expansion
[Response of Density to infinitesimal increase of Temperature]
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F. Mallamace, Chen, et al, PNAS 2008
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Fragile-to-Strong Crossover for Hydrated MCM-41-S-14
Self Diffusion by NMR
F. Mallamace et al., JCP (2006)
α-relaxation time by
QENS
A. Faraone et al, JCP 121, 10843 (2004)
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Confirmation of a dynamic crossover in simulations upon crossing the Widom line
Dynamic crossover seems to be a consequence of structural changes of
water upon crossing the Widom line !
“Relation between the Widom Line and the Strong-Fragile Dynamic Crossover in Systems with a Liquid-liquid Phase
Transition”, L. Xu, P. Kumar, S.V. Buldyrev, S.H. Chen, P.H. Poole, F. Sciortino, H.E. Stanley, PNAS 102,16558 (2005).
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Q: Does Widom line depend on pore size or only on contents of pore?
Temp.
Depen
dence
of Ave.
Trans.
Relax.
Times
(P = 1
bar)
A. Faraone, L. Liu, C.-Y. Mou, C.-W. Yen, and S.-H. Chen, “Fragile-to-strong liquid
transition in deeply supercooled confined water”, J. Chem. Phys. Rapid Commu. 121,
10843-10846 (2004)
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THANK YOU!
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