molecular attractions - cuso
TRANSCRIPT
Molecular attractions: a.) van der Waals interactions b.) electrostatic correlation interactions c.) polyelectrolyte bridging interactions
Rudi Podgornik
Laboratory of Physical and Structural BiologyNational Institute of Child Health and Human Development
National Institutes of HealthBethesda, MD
Department of PhysicsFaculty of Mathematics and Physics, University of Ljubljana
Department of Theoretical PhysicsJ. Stefan Institute, Ljubljana Slovenia
TROISIÈME CYCLE DE LA PHYSIQUE EN SUISSE ROMANDE COURS DU SEMESTRE DE PRINTEMPS 2009
Three important sources of attractive interactions in soft matter systems. I will take time to introduce the general principles as well as introduce the audience to specific model systems:
1. van der Waals interactions, which are due to correlated electromagnetic field fluctuations in dielectrically inhomogeneous media, their role, magnitude and form in various systems, in particular in the interactions between carbon nanotubes
2. the electrostatic correlation interactions which are due to electrostatically strongly coupled mobile counterions between charged macromolecules, their role in DNA collapse in multivalent salts
3. polyelectrolyte bridging interactions that are due to elastic deformation of stretched polyelectrolyte chains between charged macromolecular interfaces, their role in organization of eucaryotic genome, specifically in the interactions between nucleosomal core particles.
Bonus: Packing of DNA in simple bacteriophages. I will present a few ideas on how DNA can be packed within small viruses and what is the energetics of this packing from the point of view of continuum theories.
Plan of lectures:
1.
van der Waals interactions, which are due to correlated electromagnetic field fluctuations in dielectrically inhomogeneous media, their role,
magnitude and form in various systems in particular in the interactions between carbon
nanotubes
Johannes Diderik van der Waals (1837–1923)
Hendrik Brugt Gerhard Casimir (1909-2000)
Evgeny Mikhailovich Lifshitz(1915 – 1985)
I mentioned my results to Niels Bohr, during a walk. That is nice, he said, that is something new... and he mumbled something about zero-point energy.
(Casimir, 1992)
His calculations were so cumbersome that they were not even reproduced in the relevant Landau and Lifshitz volume, where, as a rule, all important calculations are
given. (Ginzburg, 1979)
His equation of state was so successful that it stopped the developmentof liquid state theory for a hundred years.
(Lebowitz, 1985)
Dramatis personae
PhD thesis of P.N. Lebedev (1894):
Hidden in Hertz's research, in the interpretation of light oscillations as electromagnetic processes, is still another as yet undealt with question, that of the source of light emission of the processes which take place in the molecular
vibrator at the time when it gives up light energy to the surrounding space; such a problem leads us [...] to one of the most complicated problems of
modern physics -- the study of molecular forces.
[...] Adopting the point of view of the electromagnetic theory of light, we must state that between two radiating molecules, just as between two vibrators in which electromagnetic oscillations are excited, there exist ponderomotive forces: They are due to the electromagnetic interaction between the alternating electric current in the molecules [...] ; we must therefore state that there exist between the molecules in such a case molecular forces whose cause is inseparably linked with the radiation
processes.
Of greatest interest and of greatest difficulty is the case of a physical body in which many molecules act simultaneously on one another, the vibrations of
the latter not being independent owing to their close proximity.
1864 and 1873 J. C. Maxwell
1888 H. Hertz
Eigenfrequencies of the EM field :
H. B. G. Casimir , Proc. Nederl. Akad. Wetensch. 60:793(1948).
Let us investigate the EM field inside a cavity, of dimensions L X L X D. D is in the z direction. The corresponding eigenfrequencies have the form:
We have furthermore assumed that the transverse dimensions are large, L → ∞.In this case the transverse wave vector becomes continuous. Q is a 2D wave vector (two transverse
directions).
Schematic representation of the geometry of the problem.
We are solving the Maxwell’s equations between the two
bounding surfaces.
In empty space they are reduced to wave equations.Ideally polarizable (metal)
interfaces.
At zero temperature the EM field modes behave like harmonic oscillators.Each of them has a zero-point energy from QM. The total energy of EM field oscillators is thus:
This can be further reduced to:
Instead of this energy (which one can easily see is infinite!) we rather concentrate on the difference of the energies, for finite and for infinite separation (this is called regularisation of the interaction energy):
It is obtained in the following form (for infinite D the n sum also becomes continuous!):
1 cm2 areas 1 µm apart attract with 10-7 N - weight of water droplet 0.5 mm in diameter. At 10 nm the Casimir force is equivalent to 1 atm pressure.
Evaluating the integrals and the sums via the Poisson summation formula(plus taking into account the physical considerations about the response of any body at large
frequencies) one obtains Casimir’s result:
H. B. G. Casimir , Proc. Nederl. Akad. Wetensch. 60:793(1948). What is the physical meaning of this result? The Casimir force is the EM field depletion force!
Not all EM modes fit between the two ideally polarizable interfaces! Only those fit, with the
appropriate wavelength.
There are more modes outside then inside and each mode exerts Maxwell’s pressure on the
boundary thus - presto -
the Casimri effect is there!
In 1996 by Dutch scientist Sipko Boersma (A maritime analogy of the Casimir effect Am.J.Phys. 64. 539-541 (1996)) dug up the French nautical writer P. C. Caussée and his 1836 book The Album of the Mariner that two ships should not be moored too close together because they are attracted one towards the other by a certain force of attraction. Boersma suggested that this early observation could be described by a phenomenon analogous to the Casimir effect.
“Une force certaine d’attraction“P.C. Causee: L'Album du Marin, (Mantes, Charpentier, 1836)
In the age of great sailboats it was noted that at certain conditions of the sea the ships attract mysteriously, leading often to major damage.
G. Nolan: I had first hand experience of this in 1998, while waiting for our start in the sailing regatta for the New South Wales Hood championships on Sydney Harbour. We had ... a lot of waves caused by everything from power boat and ferry wakes to waves made by arriving and departing float planes. I made the prediction that, because of the conditions and the Casimir
effect, the waiting boats would drift together. Within minutes that's exactly what had happened and we had to continually fend off from boats that literally drifted together into clusters. A firm
push was enough to separate the boats but we'd soon drift into another cluster.
The strory is however more complicated. The original figure captions from P.C. Causee: The mariners’ album
Le calme plat(Flat calm)
Calme avec grosse houle(Calm with big swell)
Two figures presentedby Causee in his book.What do they actually
show?Is this really the Casimir
effect?
The ships are free floating, not moored. I was told that the effect was also reported in the world litterature: Herman Melville's "Moby Dick" and Philip Roth's "Rites of passage". It is not a myth, the
original paper Am.J.Phys. 64. 539-541 (1996) gives the quantitative theory to calculate the attractive force, given Ships rolling amplitude, weight, metacentric height, "Q" oscillator quality factor and
wave period. An example for two 700 ton clipper ships gives 2000 Newton, quite reasonable. The theory gives also another effect: Repulsion. An atom is attracted to a conducting plate but a ship in a wave field is repelled from a steep cliff. This is due to a difference in boundary condictions between Electromagnetic waves and Seawaves. This repulsion was already known to the Cape Horn sailors of the Cape Horn Society, Hoorn Holland. Caussé's error: Caussé put his ships in a "Flat Calm" without any waves. That won't work. However, already a small swell suffices if its period matches the natural
period of the ships and we have resonance magnification. A long light swell can easily have been overlooked by the mariners on board. The second possibility is that Caussé should have put his
ships In "Calm with Big Swell" which after all to me seems less likely.
S.L. Boersma Delft The Netherlands
Fabrizio Pinto thinks that the whole tale is symptomatic of physicists' approach to the history of their subject. "Physicists love lore about their own science," he says.
"There are other stories that are unfounded historically." (Nature, 4 may 2006).
You may read about this in Nature blog.http://blogs.nature.com/news/blog/2006/05/popular_physics_myth_is_all_at.html
“Une force certaine de confusion“
ε= 1 εε 12
D Thermodynamic average of the stress tensor at the boundary.
Maxwell stress tensor in vacuo:vacuum
Lifshitz in 1954 got the most prestigious state soviet science prize for this theory.
Enter Lifshitz (1954).
Real dielectric as opposed to Casimir’s idealized interfaces. Casimir vs. Lifshitz
Hard boundary vs. soft boundary
E. M. Lifshitz, Dokl. Akad. Nauk. SSSR, (1954); (1955); Zh. Eksp.Teor. Fiz., (1955)
FD theorem:
Maxwell equations:
Constitutive relations and BC:
His calculations were so cumbersome that they were not even reproduced in the relevant Landau and Lifshitz volume, where, as a rule, all important calculations are given.
(Ginzburg, 1979)
The average of the stress tensor again on the vacuum side:
Theoretical constituents for a finite temperature, real dielectric interfaces in planar geometry.
Main ingredients of the Lifshitz calculation ...
All the frequency dependence is reduced to discrete sum over imaginary Matsubara frequencies:
Discrete frequencies are due to the poles of the coth function in the FT theorem in the complex plane.
Kramers-Kronig relations for the epsilons of imaginary frequencies - real and decaying!
Re z
Im z
10 n Hz15
Lifshitz result:
(why imaginary?)
The influence of temperature usually (but not always!!!) not very important. Summation over n turned into an integral.
For ideal metals ε(0) → ∞, obviously reduces to the Casimir result!
At small separations corresponds to the Hamaker formula:
Limiting forms:
Lifshitz result is a straightforward generalization of the Casimir result and contains it as a limit.An incredible tour de force!
Small separations
Large separations
Retardation:
Finite velocity of light has very important consequences for vdW interactions. Finite time is required for the fields in both interacting regions to correlate. This quenches the magnitude of the vdW
interaction. This effect is referred to as retardation.
I will later address the vdW interaction between point particles. For now let me just state that it has the familiar form:
Retardation effects. Finite velocity of light! n=0 terms is classical! The retardation effects arequantified by the retardation function R(x) that goes asymptoticallyas x-1. This is the Casimir -
Polder interaction.
where
Chapter VIII, E. M. Lifshitz & L. P. Pitaevskii, Statistical Physics, Part 2 in Landau & Lifshitz Course of Theoretical Physics, Volume 9.
Pressure is not necessarily monotonic!
Thickness from 10 Å to 250 Å. ε ~ 1.057 Sabisky and Anderson, 1970.
Complicated separation dependence because ε = ε(ω)
QED calculation.
Dzyaloshinskii, Lifshitz, Pitaevskii (1961).
Nice, but too difficult to use for anything!The theory just too complex to apply to any new problems, thus:
Van Kampen, Nijboer, Schramm (1968) - Parsegian, Ninham, Weiss (1972) - Barash, Ginzburg (1975). Based on the concept of EM mode eigenfrequencies and secular determinants.
In some respects a return to the original Casimir formulation! Take the eigenfrequencies of the EM fieldand get the corresponding free energy from quantum harmonic oscillators (which are not really
harmonic oscillators):
van Kampen et al. T= 0
Use the argument principle to do the summation over the modes:
Eigenmodes for a particulargeometry
.... and Parsegian et al. T ≠ 0.
The heuristic theory of vdW interactions
Secular determinant of the modes. It gives eigenfrequencies as a function of the separation.Much easier to calculate then Green functions!
An then the interaction free energy comes from the application of the argument principle:
EM modes and vdW interactionsThe brilliant idea of Niko van Kampen. Modes and energies.
Still looks complicated but can be cast into a variety of simplified forms and can be
easily generalized.
A lively subject to this day!
This is the interaction free energy between two planar dielectric interfaces. The following
definitions have been used:
Final result for the T-dependent vdW interactions
a
How does one derive the interactions between isolated atoms (molecules)?L.P. Pitaevskii, 1959.
ε(ω) ε(ω)ε(ω) ba
D
The Pitaevskii equation (1959):
Retardation effects. Finite velocity of light!
For rarefied dispersive media.
n=0 terms is classical!
Pair interactions and the Pitaevskii ansatz
London - van der Waals dispersion interactions
Lifshitz - Pitaevskii derivation thus CONTAINS the older London - van der Waals and Casimir-Polder dispersion interactions at zero temeperature.
At small separations we have:
D →∞
D → 0
Which is of course what was assumed by van der Waals in his equation of state, 1873:
At large separations we have the Casimir - Polder result, 1948:
Both these results were derived for T=0 but can be now straightforwardly generalized via the Lifshitz - Pitaevski theory to any temperature!
The inveerse: Hamaker - de Boer summation
Historically the path taken was different from the course described here. A reversed course via Hamaker - de Boer
summation, 1937.
Thus we get for the London - van der Waals pair interaction:
.. and for the Casimir - Polder pair interaction:
Which are exactly the two limits that also fall out of the Lifshitz theory!
So...are the two approaches equivalent?There is a fundamental difference:
single molecule properties vs. collective properties!
vs.
Lifshitz description is a collective description and Hamaker - de Boer description is an effective single particle description.
As soon as there are interactions between the constituents of matter Lifshitz description is the only appropriate one (Lebedev!). Relevant also for
simulations!
α(ω) α(ω)α(ω) aa bε(ω) ε(ω)ε(ω) ba
D D
vs.
Dense dielectric media vs. rarefied media at vanishing density!
Condensed medium Single molecules
The dielectric spectrum of water.
Connecting the strength of van der Waals interaction with spectra. Lebedev’s dream fulfilled.Parsegian - Ninham calculations, 1970-80.
Calculating vdW interactions - Lebedev’s dream fulfilled
At small separations corresponds to the Hamaker (pairwise summation) formula:
Let us investigate the limit of small separations:
In order to evaluate the Hamaker coefficient one needs the dielectric spectrum ε(iω).This spectrum can be sometimes measured directly or can be calculated from models.
10−21 zepto is Sextillionth Trilliardth 0.000 000 000 000 000 000 001
The Hamaker coefficient
The Hamaker coefficient quantifies the magnitude of the vdW interaction.Different cultures within the physics community.
SOFT
HARD
28
(Ice Skating, by Hy Sandham, 1885, courtesy of the Library of Congress.)Sonja Heine
Fridtjof Nansen (1994), crossing Greenland:
The severe cold we experienced made things, in this respect, unusually bad; the snow, as we were fond of saying, was as heavy as sand to pull upon.
Donald MacMillan (1925), living among Eskimos in Alaska:
At 36 below zero, the sledges (with steel runners) dragged hard over young ice covered with an inch of granular snow, Sand could hardly have been worse.
Why is ice slippery? Very unusual. Try skating on iron crystals.Why is ice slippery?
29
On September 8, 1842 Michael Faraday recorded his thoughts about snow and ice, and he then began a series of investigations that were to last twenty years.
“When wet snow is squeezed together, it freezes into a lump (with water between) and does not fall asunder as so much wetted sand or other kind of matter would do.”
“In a warm day, if two pieces of ice be laid one on the other and wrapped up in flannel, they will freeze into one piece.” “Al l this seems to indicate that water at 32◦ will not continue as water, if it be between two surfaces of ice touching or very near to each other.” “The ice probably acts as a nucleus, but it appears that the effect on surface of ice on water is not equal to the joint effect of two.”
Yakov Frenkel was sure that the phenomenon must exist; all one has to do is observe. He wrote (1946): It is wel l known that under ordinary circumstances an overheating (of a crystal), similar to the overheating of a liquid, is impossible. This peculiarity is connected with the fact that the melting of a crystal, which is kept at a homogeneous temperature, always begins on its free surface. The role of the latter must, accordingly, consist in lowering the activation energy, which is necessary for the formation of the liquid phase, i.e. of a thin liquid layer, down to zero.
24
Michael Faraday and the Faraday layer
Michael Faraday (1791-1867)
30
Faraday vs. Kelvin
Faraday: equilibrium layer of water on top of ice. On contact with another surfaces this layer freezes.
J. Tyndall (1824-1907): regelation.
Not favoured by the Thomson brothers, J. Thomson in W. Thomson (Lord Kelvin).
Pressure melting theory of the slipperiness of ice. Faraday made a number of experiments and proved that it can not be correct. An enormous pressure required away from the
melting temperature. But this theory found its way into curricula and textbooks.
W. Thomson - Lord Kelvin(1824 –1907)
Both points of view are incompatible. It took a long time until the Faraday layer came back again
with a twist of van der Waals interactions.
31
Water and ice - coincidence or intelligent design?
There is an interval of frequencies where the two L-vdW dispersion spectra cross. This has fundamental consequences on the vdW interaction and on the
existence of the Faraday layer. (Elbaum in Schick, 1991).
Dielectric spectra of water and ice are very similar but not identical. They both show Debye relaxation and an optical window. Different mechanisms of relaxation: Bjerrum and ionic defects vs. dipolar
rotation.
32
Elbaum and Schick, 1991.
27
Why is there a liquid layer on ice?
van der Waals - Lifshitz interactions and the thickness of
the surface layer on ice. An interesting answer to the puzzle.
The dielectric spectrum of water.
Parsegian - Ninham calculations, 1970-80.Connecting the strength of van der Waals interaction with spectra. Lebedev’s dream fulfilled.
Modern approachSrTiO3 vdW interaction across grain boundaries. R. French, 2003.
Kienzle, 1999.
Calculated Hamaker coefficients from measured dielectric spectra:
R. French, 2003.
R. French, 2003.
Measuring the dielectric response of interacting media
Dispersion interaction will not be reproduced or estimated properly if pairwise additivity is assumed for a general condensed medium.
The Hamaker - de Boer summation (which is also what could be used in simulation) would thus always be questionable. Especially for grain boundaries in Xtals!
In the case of GB (Sigma 5 and Sigma13 van Benthem et al.) it seems that we are dealing with a pretty condensed medium!
Realistic models of dielectric response: non-locality
Now we need to make a reality check. The dielectric response in GBis not given by Lifshitz piecewise assumption.
Dielectric response in SrTiO3 across grain boundaries. Kienzle, 1999. R. French, 2003.
Inhomogeneous response!
Non-locality in the Lifshitz theory
How does one deal with a continuous profile instead of piecewise constant?
This turns out to be a non-trivial endevour. Two solutions (very technical details):
Rydberg, Lundqvist, Langreth, Dion (2000)Based on the DFT but giving a non-local
contribution to the energy.MATTER BASED.
Podgornik, Parsegian (2004)Based directly on the Lifshitz theory
and also giving non-local terms.FIELD BASED.
Formally in both approaches the interaction free energy looks the same,but the definitions of D(L) are very different.
GV showed that nevertheless the interaction (free) energy is in fact THE SAME!
T → 0
Continuous spatial dispersion profile
Continuously varying dispersion properties of the GB gap have some very important consequences
for dispersion interactions.
R. Podgornik and V.A. Parsegian, J. Chem. Phys. (2004).
There is no small separation infinity that plagues the Lifshitz
theory.
But this is as far as any continuum theory can be pushed ...The scale here is atomic! Valid until atomic overlap.
Experimental confirmation of the van der Waals interaction.Deryagin and Abrikosova (1953), Spaarnay, 1958. 100% error!
"did not contradict Casimir's theoretical prediction"
Shih and Parsegian, 1975.
Atomic beam of alkali metals above gold.
Almost quantitative correspondence ...
r(n) = 2 ξ(n) R/cComputation based upon Lifshitz theory.
Lamoreaux, 1997.
Mohideen and Roy, 1998.
Sensitive sphere. This 200-µm-diameter sphere mounted on a
cantilever was brought to within 100 nm of a flat surface (not shown) to detect the elusive
Casimir force.
Modern developments show that also the Casimir effect proper can be exactly measured.
Experimental confirmation of the Casimir interaction.
Experimental confirmation of the Casimir interaction.
The setup used by Capasso in the experimental study of the vdW
interactions. Based on the MEMS and NEMS ( Micro and Nano
electromechanical systems).
Federico Capasso, Fellow, IEEE, Jeremy N. Munday, Davide Iannuzzi, Member,
IEEE, and H. B. Chan, IEEE Journal (2007).
Experimental confirmation of the van der Waals interaction.Again various methods are used here. From Israelachvili’s book. Interaction of two cleaved mica
surfaces across water:
A. Larraza and B. Denardo 1998.
Acoustic “Casimir effect”
Not a thermal accoustic noise.Results depend on the nature of
the noise spectrum.
This variant of the “Casimir effect” is not driven by thermal fluctuations!It is driven by the artificially generated accoustic noise.
Flat white-noise spectrum vs.frequency dependent spectrum.
Non-monotonic interactions!
Hydrodynamic Casimir effect invoked in a cryptic remark at the end of the Dzyaloshinskii et al. 1961 paper.
K. Autumn, W.-P. Chang, R. Fearing, T. Hsieh, T. Kenny, L. Liang, W. Zesch, R.J. Full. Nature 2000. Adhesive force of a single gecko foot-hair.
Suction? (Salamander). Capillary adhesion? (Small frogs). Interlocking? (Cockroach)
It’s van der Waals interactions!
How does Gecko manage to walk on vertical smooth waals?
Casimir effect in vivo
TextText
A single seta can lift the weight of an ant 200 µN = 20 mg. A million setae (1 square cm) could lift the weight of a child (20kg, 45lbs). Maximum potential force of 2,000,000 setae on 4 feet of a gecko = 2,000,000 x 200 micronewton = 400 newton = 40788 grams force, or
about 90 lbs! Weight of a Tokay gecko is approx. 50 to 150 grams.
T. Emig, 2003.
Effects of boundary conditions.
Li and Kardar, 1991.
BC can qualitatively change vdW interactions.
Some “modern” developments ...
Boundary conditions for the EM field and the form of the boundary are very important for the magnitude and scaling ofthe van der Waals interactions or the Casimir effect.
Sometimes even the sign is difficult to guess.
Spherical geometry. Boyer, Davies, Balian and Duplantier, Milton, DeRaad and Schwinger (1978)
Z < 0!
Von Guericke (1602-1686) and the Magdeburg sphere.
Casimir model (1956) of the electron.Electrostatic repulsion and Casimir attraction
have to balance!
Scattering of EM waves!
The pseudo-Casimir effect (1)n=0 (classical) term . Casimir effect exists also for non-EM fields described with similar equations.
Critical fluids (Fisher and de Gennes, 1978).
Superfluid films (Li and Kardar, 1991).
Critical Hamiltonian
Critical density fluctuations. Superfluid He Goldstone (massles) bosons associatedwith the phase of the condensate.
Nematic LC
Nematic film with stiff boundaries (Mikheev, 1989).
Smectic LC
Smectic films (Li and Kardar, 1992).
Nematic wetting (Ziherl, Podgornik and Zumer, 1998).
The pseudo-Casimir effect (2)
Charged fluids(Podgornik and Zeks, 1989).
(Podgornik and Zeks, 1989). (Golestanian and Kardar, 1998).
Pseudo-Casimir interaction coincides with the n=0 Lifshitz result exactly.
Fluctuation (pseudo-Casimir) interactions are non-pairwise additive.
The total energy of an assembly is difficult to calculate.
(Podgornik and Parsegian, 2001).
Non-pairwise additive effects are essential in all fluctuation driven interactions.
The pseudo-Casimir effect (3)
Membrane inclusions
(Goulian, Bruinsma, Pincus . 1993). (Golestanian, Goulian and Kardar, 1996).
Interaction between (lipid) membrane inclusions such as proteins.
Important in understanding aggregation of membrane proteins.
The pseudo-Casimir effect (4)
Complexes of DNA and cationic lipidsbut also neutral lipids!
DOTAP + helper lipid. Raedler et al. 1997.Dan Lasic et al. 1996.
Multilamellar arrays are ubiquitous in biophysics.Are interactins between a pair of bilayers different from interactions in a stack?
Interactions between neutral molecules.
vdW interactions in lipid arrays
Interactions in multilamellar arrays are characterized by short range repulsive part and a long range van der Waals part that eventually leads to a stable interlamellar spacing.
Measurements of vdW interactions in multilamellar lipid arrays
The position of the minimum in the interaction energy determines the value of
the Hamaker coefficient.
Specific features of vdW interactions in multilamellar arrays
Discontinuity & Propagation
Reduction to an algebra of 2X2 matrices.
Dispersion interaction ~ log transfer matrix!
A formal tour-de-force that enormously simplifies the calculations of DI in multi slab geometries.
R. Podgornik, P.L. Hansen and V.A. Parsegian, J. Chem. Phys. (2002).
Strong non-pairwise additive effectsand also retardation effects (?).
The effective Hamaker coefficient is 4.3 zJ.
The best value currently available in the literature.
R.Podgornik, R.H. French and V.A. Parsegian:, J, Chem. Phys. Vol. 124, 044709 (2006).
vdW interactions in multilamellar lipid arrays
Repulsive vdW interactionsThe sign of the Lifshitz free energy depends on the differences of dielectric response
functions at thermal frequencies.
The interactions can be attractive as well as repulsive. First measured by Capasso, Munday
and Parsegian, Nature (2008).
vdW interactions in the nano-world
In his famous musing, Plenty of Room at the Bottom, Feynman (1959) noted, “...as we go down in size, there are a number of interesting problems that arise. All things do not simply scale down in proportion. There is the problem that materials stick together by the molecular (van der Waals) attractions. It would be like this: After you have made a part and you unscrew the nut from a bolt, it isn't going to fall down because the gravity isn't appreciable; it would even be hard to get it off the bolt. It would be like those old movies of a man with his hands full of molasses, trying to get rid of a glass of water. There will be several problems of this nature that we will have to be ready to design for.”
Single walled carbon nanotubesRolling up a graphene sheet produces different variety of carbon nanotubes.
(n,0)
(n,n)
Ch
Armchair(n,n)
metallic
Zigzag (n,0)
Chiral(n,m)
CNT Metallic & Semiconducting
Chiral vector Ch defines a CNT
Metallic CNT: n - m is multiple of 3
(n,m)
Metallic and Semiconducting SWCNTs 50
[9,3 Metallic] [6,5 Semiconducting]
Build CNT’s By Rolling Graphene With m,n Chirality• Graphite Interlayer Spacing Is 0.167 nm• Used to Define Cylinder Diameters
Armchair [n,n] CNTs are n=m and all metallic. • [5,5,m], [6,6,m] and [7,7,m].
Zig Zags [n,0] CNTs are [5,0,s], [6,0,m], [7,0,s]. • Metallic When (n-m)/3 = Integer.
Chiral CNTs [n,m]
Do the interactions between SWCNTs mirror their complicated structural details?
51
Loose helical wrappingAvg. spacing = 11nmangle=42ht of CNT=1.08nmwidth of CNT=1.9nmht of CNT/DNA= 2.16nm
"DNA-Assisted Dispersion and Separation of Carbon Nanotubes" , M. Zheng, A. Jagota, E. D. Semke, B. A. Diner, R. E. Richardson, N. G. Tassi, Nature Materials, 2 338-342 (2003)."Structure-Based Carbon Nanotube Sorting by Sequence-Dependent DNA Assembly", M. Zheng, A. Jagota, M. S. Strano, A. P. Santos, P. Barone, S. G. Chou, B. A. Diner, M. S. Dresselhaus, R. S. Mclean, G. B. Onoa, G. G. Samsonidze, E. D. Semke, M. Usrey, D. J. Walls, Science, 302, 1545-1548 Nov (2003). "Theory of Structure-Based Carbon Nanotube Separations by Ion-Exchange Chromatography of DNA/CNT Hybrids", S. R. Lustig, A. Jagota, C. Khripin, M. Zheng, J. Phys. Chem B 109 2559-2566 (2005).
Dispersion, Sorting, Deposition: DNA-CNT Hybrids
52
Anion Ion Exchange Chromatography: Stick/Strip Via Screening Electrostatics
Initially (2003) Separation Only Between Semiconductor And Metallic Classes•M. Zheng And Et. Al., Science, 302, 1545 (2003)
Years Later (2006), Separate Semiconductors Of Identical Diameter/Band Gap •M. Zheng And E. D. Semke, J. Am. Chem. Soc., 129, 6084 (2007)
Question: Do Known SWCNT Band Gap Trends Also Contribute A Chirality Dependent Vdw Interaction To Assist This Process?
The most ubiquitous interaction is the vdW interaction. What can we learn from investigating its properties for the various types of CNTs?
Separating CNTs by means of their interactions
53Formulating and calculating the vdW interactions
Formulate the problem for two composite media containg cylinders with anisotropic dielectric properties. Take the limit of zero cylinder density and use the Pitaevskii ansatz to obtain the effective interactions between two cylinders, i.e. two carbon nanotubes.
The Lifshitz non-retarded interaction free energy. Matsubara summation.
Anisotropy: Composite: D:
54The Pitaevskii ansatzLet us just concentrate on two cylinders at arbitrary angles.
In the Pitaevski ansatz one expands the Lifshitz interaction free energy to the second order in the density v of the composite media. It then follows that:
for two parallel cylinders for two skewed cylinders
From the second order expansion one then obtains the pair interaction.
This method works as long as the interaction scales linearly with the length, or does not scale with the length of the cylinders. All other cases can not be treated by this method.
It also assumes all epsilons are finite!
55Results:Explicit forms of the interaction for small and large separations.
Skewed cylinders:
Parallel cylinders:
Effective Hamaker coefficients in terms of dielectric properties of the anisotropic cylinders.
Interaction free energy as a function of separation, radius of the cylinders and mutual angle of orientation.
Hamaker coefficients are defined in exactly the same way.
In order to calculate them we need the frequency spectra of the dielectric functions. They can not be obtained experimentally, so they need to be calculated ab initio.
56Summary of the calculation:
Gecko Hamaker: Open Source Hamaker ProgramFull Spectral, Retarded Hamaker, Coefficients http://geckoproj.sourceforge.net/
London dispersion spectrum of CNTs
(n,m)
LD spectrum
Hamaker coefficients for CNTs
Near limit is < 2 Diameters, far limitis > 2 Diameters.1. Semiconducting CNT’s Hamaker Coefficient Larger Than Metallic2. Far Limit Hamaker Coefficients Larger Than Near Limit
Hamaker coefficient for twoparallel cylinders and a cylinders close to a wall.
van der Waals torques
vdW Torques Arise When There Is A Preferred Interaction Direction. First computed by Parsegian and Weiss (1972). Detected by Capasso et al. (2005).
van der Waals torques between CNTsvdW Torques Arise When There Is A Preferred Interaction Direction. Again, Any Rod-substrate Alignment Is Purely Due To Optical Properties.
This opens up new possibilities for nanoelectromechanical devices that would operate on the basis of vdW torques. The magnitude of the torque depends drastically on the dielectric spectral properties.
Consider The 5 CNT Types
• “Metals”: Armchair, Zigzag, Chiral• Semiconductors: Zigzag, Chiral
How To Understand Trends In vdW-Ld Interactions?
Explore Trends With CNT Radius & Chirality• Dielectrophoresis Experiments Have Already Shown A Correlation Of Interaction Energy With SWCNT Diameter
What Is A “Metal”?
From Band Structure Or Transport Perspective• Want Electron Transport At 0 eV
From Optical Property, Drude Metal Perspective• Want Large Drude Metal Peak, Near 0 eV, In Axial Direction
The vdW-Lds Classification = Conduction (Metal Vs. Semiconductor) + Structure (Armchair, Zigzag, Or Chiral).
ab initio Optical Properties Of 60 CNTs
Different types of CNTs show extremely varied behavior in terms of their optical properties as well as their correspodning Hamaker coefficients of the vdW interaction.
ZigZag vs. Armchair SWCNTsNotable Trends Both Within And Between Classes From 0-5 eV and from 5-30 eV. vdW-Ld Interactions Equally Dominated By 5-30+ eV Range.
0-5 eV Range Is Highly Correlated To Electronic Conduction (i. e. Metal, Semiconductor, Or “Small-gap” Semiconductor)5-30 eV Range Is Highly Correlated To Structure (i. e. Zigzag Vs. Armchair Vs. Chiral)
ZigZag
Chiral
Armchair& ZigZag
“Drude” Metal Peaks vs. Classification Type
Classification Of CNT Types: Optical Perspective66
DOS
InterbandOptical Properties
Band Crossing Band Touching Band Gap No Drude Metal Peak Drude Metal Peak? Drude Metal Peak Semi-metal Metal or Semi-metal Small Gap Metal
Electronic Conduction And Optical Perspectives: Confusing Terminology & Classifications
Armchair CNT Example of System DesignSystem: Bare SWCNT + Polystyrene Substrate + High Index Medium
Several tunable features of vdW interactions between different types of CNTs:
1. A Critical Radius Of Attractive/Repulsive Divide2. Changing The Index of Refraction Can Change The Critical Radius Location3. Multi Stage Experiments Could Create Mono-disperse Populations4. More Parameters To Explore (Surfactants, Mwcnts, Etc.)
68Conclusions
A fascinating and multifaceted force.
Fundamental in its origin because it is due to quantum and thermal fluctuations.
Ubiquitous.
Universal.
Difficult to figure out even the sign.
Experimentally detectable.
Essential on the nano level.
