willem kegel & jan groenewoldstatic.sif.it/sif/resources/public/files/va2015/kegel_iv.pdfbulk...
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Colloids as nucleons
Finite-size equilibrium structures <-> macroscopic phase separation Equilibrium clusters & periodic structures in systems with long-range repulsion and short-range attraction
Willem Kegel & Jan Groenewold
Van ‘t Hoff Laboratory
Utrecht University The Netherlands
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Finite size –micro- versus macroscopic phase separation
Free energy of dispersed matter in capillary approximation:
0F F S
Bulk Surface
Minimize F by minimizing surface area S macroscopic phase separation
‘Micro’ phase separation: internal constraints Micelles, microemulsions, vesicles, diblock copolymers…
Amphiphilic building blocks
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…virus capsids ~ ‘coats’ of viruses 30 nm
few nm
(Un)coating regulated by: * Hydrophobic interactions between apolar patches on protein surface •Screened-Coulomb interactions
[WKK & P. vd Schoot, BPJ 2004; 2006]
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Patchy ‘colloidal molecules’ [DJ Kraft ea, PNAS 2012] (with Ran Ni, Frank Smallenburg, Marjolein Dijkstra)
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Atomic nuclei – internal constraint = long-range Coulomb repulsion
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Outline
• Weakly charged colloids, low screening: colloids as
nucleons (High screening has been done: DLVO)
• Cluster phases in colloids & proteins – controversies
• Strong attraction : non-equilibrium clusters
• Higher densities – link with dense nuclear matter
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Colloids & nucleons
[J. Groenewold & WKK. J. Phys. Chem. B 105, 11702 (2001); J. Phys.: Condens. Matter 16 S4877 (2004) ]
Colloidal cluster Atom nucleus
Short-range attraction +
long-range repulsion
Van der Waals / Strong nuclear force depletion (Screened) Coulomb Coulomb
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Mass formula (nuclei of atomic number A, nuclear charge Z):
2
1/3 2 4/32/ 1vol sym surf Coul
ZF A a a a A a Z A
A
Nucleus charge vs mass
0,3
0,4
0,5
0,6
0 100 200 300 400
AZ
/A
Binding energy / nucleon
-10
-8
-6
-4
0 50 100 150 200
A
E/A
Me
V
56Fe
(neglect pairing term)
electron capture: p + e- n + ν
~ A+2/3
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Colloidal equivalent:
Attractive potential of mean force by (e.g.,) depletion of polymers Quiz: equiv of charge generating term?
Charged colloids in solvents with small dielectric constant: long-range repulsion (long screening length)
0 2 4 6 8 10
0
u(r)
r/d
Ionic dissociation at low dielectric constants: dissociation energy kTλB/b – ionization due to increased translational entropy of counter ions.
bond length “ion capture”: C(n+1)+ + i- Cn+ Bjerrum length
/2
0 3
B be
rb
Free energy density of spherical colloidal cluster of radius R:
1 2 2
0 0
43 2 ln( / ) 1
5Bf f R R
Charges/ volume Surface tension Entropy
(ions + combinatorial)
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Intermezzo: site-binding model
Ions can be bound to colloid surface with energy
2
0
/
4
B
B
b
e
kT
or translate freely in bulk
*Free energy
max( )
/max
max
3
ln ln
!
!( )!
!
B
ads bulk
Z Zb
ads
Z
bulk Z
F Q Q Q
ZQ e
Z Z Z
VQ
Z b
2
max / 4
colloid
Z r
nv
V
/
2
0 3
B be
rb
02 ln( / ) 1cluster
Ff
V
*Minimum ρ * (without Coulomb term) r ≡ colloid radius
Take Z<<Zmax
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Expand around ρ0:
‘Entropic’ term ≡ charge - generating
Similar role as symmetry term in mass formula:
2
0 0 0
0
12 ln( / ) 1 2 ( )
Now cluster free energy isomorphic to ‘mass formula’ !
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Map cluster free energy onto mass formula. Result:
34 /3A R v3
04 / 3Z R v
0 02vola f v v 2/34.84surfa v
0syma kT v
2 5/3
00.48Coul Ba kT v
Numbers comparable for: ~1 μm 10 and sufficient charge density
…experimentally observed? First indications: Segre ea, PRL 86, 6042, (2001)
colloids in solvent
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[Segre ea, PRL 86, 6042, (2001)]
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[Sedgwick ea , J. Phys.: Condens. Matt. 16, S4913, (2004) Stradner ea Nature 432, 492, (2004)]
Model prediction: optimum cluster radius 3
*R
[M. van Schooneveld ea JPCB 2009]
3
*.#Aggr R
[Stradner ea Nature 432, 492, (2004)] Other cluster shapes see, e.g, S. Mossa ea, Langmuir 2004
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Origin of attraction: depletion interaction
Overlap volume
Volume inaccessible to
depletant com Depletant
Pairwise:
3
34 3 11
3 4 16p
r rU kT
gR r (Asakura-Oosawa (AO) potential)
r
rg
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Why 3
*R
Quiz:
?
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Why 3
*R
/2
0 3
B be
rb
?
Minimize free energy density - result
As long as ρ ≈ ρ0, and
3
* 2
15
8 B
R
QED
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Stable clusters also observed in aqueous protein solutions (without added salt)
[Stradner ea Nature 432, 492, (2004)]
3
*.#Aggr R
Numbers (small R, large ε) make sense Cluster size cannot (much) exceed Debye length
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[PNAS 105, 5075, (2008)]
In case of clusters: expect constant peak with lysozyme concentration
…but ‘critical cluster concentration’ ≈ 200 g/l !
Controversy
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Evidence for equilibium protein clusters in vivo: [KP Johnston ea ACSNano 2012]
Cluster size >> Debye length -> expect unstable to further growth … but clusters hardly contain water -> low local dielectric constant, ionic strength. Low screening.
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3
* 2
15
8 B
R
/2
0 3
B be
rb
24 r
3
*R predict
Cluster size versus attraction strength ε
... Opposite trend [Zhang ea Soft Matter2012]
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Explain trend by classical nucleation theory:
Cluster formation free energy (out of colloidal gas state)
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Result
Nucleation rate
J (ε = -8.8 kT) ~ 104 J (ε = -5.5 kT)
Initially more & smaller clusters with more attraction
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Large nuclear densities: neutron star interiors Several scenarios; first attempt: [Baym, Bethe & Pethick, Nucl. Phys. A175, 225, (1971)]
Core: Density 1014 gcm-3
How far can we push the analogy with nuclear matter?
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Nuclear matter at high density : several predictions, e.g.,
“Spaghetti” “Lasagna” “anti – spaghetti”
[Watanabe, Sato, Yasuoka, Ebisuzaki, Phys. Rev. C66, 012801, (2002); 68, 035806, (2003)]
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Periodic structures in MD simulation of MONODISPERSE ‘colloidal’ system
Model potential
[A. De Candia ea, PRE 74, 010403(R), (2006)]
‘disordered’
columnar lamellar
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Experiments at higher colloid volume fractions : gelation. Force gel into columnar – like state by E-field and see what happens [Zhang ea, Soft Matter 2015 ]
After preparation: gel Field on : columnar
Field off – after 4 days
Field off – after 7 days
MD simulations of polydisperse systems: periodic structures unstable beyond 1% polydispersity [Zhang ea, Soft Matter 2015 ]
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Conclusions and further work
Improve theory:
• beyond spherical clusters • inhomogeneous charge distributions • interactions between clusters • allow for dielectric constant variation
• Clusters are stable (colloidal) state of matter under conditions of long range repulsion (relative to the size of a colloid) and short-range attraction
• Non-equilibrium clusters appear at strong attractions
• At high colloid concentrations, colloidal gel is the stable state
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Thank You!
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Controversies related to equilibrium protein clusters
•1 Zero-Q peak: Long-range attraction in protein solutions
[Y. Liu, E. Fratini, P. Baglioni, R-R Chen, S.H. Chen, PRL 95, 118102, (2005)]
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Zero Q peak appears several days after sample preparation,
related to impurities
[A. Stradner, F. Cardinaux, P. Schurtenberger, PRL 96, 219801, (2006)]