Diapositiva 1

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PhD. Scienze e Tecnologie delle Mesofasi e dei
Materiali Molecolari (STM3) XVII ciclo
LXNMR_S.C.An. Structural and Conformational
Analysis by Nuclear Magnetic Resonance
spectroscopy of molecules dissolved in Liquid
Crystals
Relatore Dr. Giuseppe Pileio
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Program
Main purpose knowledge of LXNMR technique for
structural and/or conformational information on
molecules in liquid-like phases
Liquid crystals
1st hour
NMR Theory
2nd hour
The Conformational problem
3th hour
How to work in practice
Some examples
4th hour
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Liquid Crystals
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Liquids Liquid-Crystals solids
thermotropic
Lyotropic
nematics Smectics Colesterics etc
Calamitic Discotic Banana
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1D solid, 2D liquid
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Liquid Crystalline textures
NEMATICS
SMECTICS
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Nematic phase

• Absence of polarity n(r) -n(r)
• Point Group D8h
• ? const (translational symmetry T(3))
• Optically uniaxial (positive is nzn?? gt nx
  nyn?)

Smectic A phase

• Absence of polarity n(r) -n(r)
• Point Group D8h
• ?x, ?y const (translational symmetry T(2))
• The uniaxial order parameter is the same as
• in nematics, but its absolute value SAgtSN.

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Smectic C phase

• Absence of polarity n(r) -n(r)
• Point Group C2h
• Translational symmetry T(2) as in Smectic A
• (the position in layer is uncorrelated but the
  tilt it is)
• Optically biaxial (positive is nz? nx? ny)
• In SmC the director is free to rotate along the
  conical
• surface with an apex angle 2?

Smectic B phase

• Absence of polarity n(r) -n(r)
• The point group symmetry is D6h
• optical uniaxiality n?? ?n? and nz gt nx ny
• Three dimensional density wave

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Surprising things happen to physical properties
like Surface Tension, Osmotic Pressure and Light
Scattering when we add surfactant to water.

Osmotic Pressure
Surface Tension
Light Scattering
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Lyotropic phases
Lyotropic liquid crystalline phases form by water
solutions of amphiphilic molecules. The building
blocks of those phases are either bilayers or
micelles.

The form of the micelles can be spherical or
cylindrical and the micelles can be normal (tails
in the water, polar heads outside) or reversed
(water and polar heads inside, oil outside).
Examples of the structure of some typical
lyotropic phases (lamellar, cubic, haxagonal)
are shown

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NMR Theoretical Background
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NMR Hamiltonian
IRREDUCIBLE SPHERICAL SPATIAL TENSOR
INTERACTION CONSTANT FOR THE l-th SPIN
INTERACTION
IRREDUCIBLE SPHERICAL SPIN (or SPIN-FIELD) TENSOR
OPERATOR
m-th COMPONENT OF RANK L TENSOR
NMR INTERACTION
TENSOR RANK
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What is a tensor ?
A tensor is a mathematical object that transform
under rotation of the frame in a particular way.

• Any tensor may be expressed as a matrix of rank
  r. So
• a rank 0 tensor is a scalar
• a rank 1 tensor is a vector
• a rank 2 tensor is a matrix n x n
• And so on.

Main Property The trace of a tensor is
unvariant under any change of frame by rotation.
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Operations on tensors

• Addiction
• Aij Bij Cij
• Only for tensor of equal rank

• Outer product
• Aij x Bl Cijl
• Always possible (rank must be different)

• Inner product (contraction)
• Aij Bjl ?j (Aij Bjl) Cil
• Is possible when two or more indices are equal

Irreducible spherical tensors
Irreducible spherical tensors have component that
transform, under change of frame by rotation
defined by euler angles OF-F, as
? Wigner Rotation Matrix
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The Euler angles
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Some Wigner rotation matrices
Rank 1
Rank 0
Rank 2
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Why irriducible spherical tensors?
Symmetrical tensors
1/3 Tr(a)
1) axx ayy azz Tr(A)
2) aab aba
In the Principal Axis System PAS
3) axy axz ayz 0
Cylindrical symmetry D8h
4) axx ayy
It remains only L 0, 2 m 0
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Space, Spin (or Spin-Field) part of Hamiltonian
and pure irreducible spherical spin tensor
operators T
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Pure irreducible spherical space tensor A
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Two great hypoteses
High Field limit
The high magnetic field, make Z axis in the
laboratory frame the axis of quantization. For
just a 100 MHz spectrometer HZ gtgt HQ HD gtgt HJ
so that all off-diagonal matrix element of H can
be neglected ( m 1 and a part of m 2 are
removed )
Apolar nematic phase
Uniaxial phase (D8h symmetry), remove m 2 at
all. The apolarity of phase reduces to L even
the tensor elements (remember that NMR can give
only information up to rank 2)
then
L 0 isotropic contributions L 2 anisotropic
contributions
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Molecular motions and NMR interaction
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From PAS to LAB
Since T is usually expressed in the laboratory
frame (LAB) while A in the Principal Axis System
(PAS) it is necessary to transform the spatial
tensor A from PAS to LAB using the Wigner
rotation matrices D

• This operation will be carried out in two step
• From DIR to LAB
• From PAS to DIR

1)
? Wigner Rotation Matrix
2)
? Microscopic order parameters
Averaged by molecular motions
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Molecule and Phase symmetry
m refers to PAS
HOW MANY TERMS
m refers to DIR
m all or m 0, 2 if the director system is
principal m all or m 0, 2 if the
molecular system is principal
m 0, 2 if the director system is
principal m 0
m 0 m all
m 0 m 0
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Order Parameters
Molecular Order Parameters are averages of the
Wigners Rotation Matrix elements
They are appropriate for all phases and molecular
symmetry
Motional Constants are averages of Spherical
Harmonics
They are usually used in case of cylindrical
symmetry about the director
Saupe Order Parameters are averages of
transformation matrix elements in Cartesian frame
They are usually used for uniaxial phases
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Dipolar couplings Dij
m 0 if phase is uniaxial (nematic)
Angle between ZLAB and ZDIR
m 0 if molecule is uniaxial
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The Conformational Problem
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What is a model (1) ?
Observations
Model A
Model B
? Model A
Model B ?
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What is a model (2) ?
If no information on the error are available, the
two model are resonable.
If some information on the error are available,
one may discriminate between them.
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Rigid and flexible molecules a simple
classification
Internal motions

• small amplitude - high frequency motions
  (vibrations)
• large amplitude - low frequency motions and
  puckering of unstrained rings
• puckering of strained rings and rotations along
  double bonds

? Rigid molecules have only small amplitude
motions
Flexible molecules have also large amplitude
motions ?

• Being f the set of internal angles describing
  all internal motions and f the set of angles
  the Order Parameters are depending on
• Order Parameters are independent (in first
  approximation) from small amplitude motions
  (vibrations
• averaged order parameters) ? lt Dlmm(O)gtv
• Order Parameters depend on large amplitude
  motions (average over orientations) ?
  Dlmm(O,f)

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Orientational Distribution Function
The average of any single-molecule property
X(O) over the orientation of all molecules is
defined by introducing a singlet distribution
function f(O) as
? Microscopic order parameters
Supposing f(O) originating from an orientational
pseudo-potential V(O), then
O are the Euler y, q, f angles

a b g
Nematics must be approssimated by cilindrical
rods so V(W) is independent of f (phase is
unaxial) and y (molecule have cylindrical
symmetry). Then, as m is restricted to 0
? Mean Torque Potential
For furher informations R. Y. Dong, in NMR of
Lyquid Crystals, Springer-Verlag, New York,
1994.
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Conformational distribution function
In order to perform the average on different
conformational states we must introduce the
equilibrium probability function P(a,b,g,f) for
finding the molecule in the n-th conformation
defined by the set of internal angles f and to
sum the NMR interaction on the N available
conformations
In isotropic phases
The shape anisotropy of molecules influence the
intermolecular potential
In anisotropic phases
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1 J. W. Emsley, G. R. Luckhurst and C. P.
Stockley, Proc. R. Soc., London Ser. A,
(1982), 381, 117.
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Another Two Different Approaches
ME1 (Maximum Entropy)
The distribution compatible with the set of
observable may be derived with the unbiased
Maximum Entropy method making use of the Lagrange
Multipliers techniqe
Lagrange Multipliers
Dipolar couplings irreducible spherical tensor
RIS2 (Rotamer Isomeric State)
Supposing that only a finite set of N conformers
are populated (fn) with the probability pn for
each of them, the internal potential function cam
be written
Dirac Delta function
Number of minima in The potential function
1 D. Catalano, L.Di Bari, C. A. Veracini, G. N.
Shilstone and C. Zannoni, J. Chem. Phys., (1991),
94, 3928 2 J. W. Emsley, in Enciclopedia of
NMR, ed. D. M. Grant and R. K. Harris, Wiley,
New York, 1996.
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The Vibrational Problem
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A model to treat vibrational motions
Using cartesian notation and supposing no
vibro-rotational coupling, dipolar coupling may
be expressed by
r
r
re
Being re the equilibrium distance and r the
istantaneous excursion of r from re we may
approximate
Where e, a, h mean equilibrium, anharmonic and
harmonic
and dipolar couplings expanded as
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Covariance Matrix
Covariance Matrix is a matrix whose elements are
the products of Cartesian displacements
In order to calculate these, we need the molecule
Force Field (F) and to extract from it IR
frequencies and normal modes of vibrations.
Having
Where G is diagonal containing the mass of atom
i, and Z the amplitude of the n-th vibrational
coordinate. Finally,
For furher informations S. S?kora, J. Vogt, H.
Bösiger and P. Diehl, J. Magn. Reson., (1979),
36, 53-60
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Some resolved samples
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What we have to do ?
prepare the sample (distillation, dissolution in
LC, etc)
prepare the experimental conditions (T, homog.,
etc)
Record the spectrum
record the FID
Searching for starting data set (isotropic
indirect couplings and Chemical Shifts, Dij)
Analyse the spectrum
definition of spin system
Fit experimental spectrum by calculated one to
extraxt Dij
Analise the problem in terms of
observations/parameters
Searching for a starting geometry
Fit the experimental data ( Dijs )
Searching for a potential curve
Search or calculate Force Field
Fit experimental Dij by calculated ones
optimising parameters
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ARCANA
SPECTRUM
Dij
S , G
RESULTS
ANACON
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A rigid molecule VinylBromide
N of parameters
Analysis of the problem
Order parameters 3 (simmetry Cs )
geometrical parameters 5 (atoms) x 2 (xi, zi)
4 (CC fixed) 6
TOTAL 9
N of observables
3
With only DHH couplings
Br

H
3
DC1 H couplings

C
C
13C
13C
3
DC2 H couplings
H
H
TOTAL 9
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Searching for the Jij
An amount of VinylBromide (gas) was dissolved in
ZLI1132, CCN55 and in a mixture (4555) of
ZLI1132 and EBBA (Magic Mixture, MM), by gurgling
it in an NMR tube and keeping all at liquid
nitrogen temperature
Prepare the sample
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The spectra
ZLI1132
MM
CCN55
13C1
13C2
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What have we found?
The force field of molecule was calculated by
Gaussian 98W package at three levels of
approximation, PM3, B3LYP/6-31Gand MP2/6-31G
but experimental frequencies were used1.
r12 1.3320 from 2 r62 1.0870 from
preliminary calculation
1 W. A. Herrebout, B.J. Van der Veken and J. R.
Durig, J. Mol. Struct., (1995), 332, 231-240
2 D. Coffey, J. B. Smith and L. Radom, J. Chem.
Phys., (1993), 98, 5, 3952-3959
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A flexible molecule Styrene
Analysis of the problem
N of parameters
Order parameters 3 (if planar, Cs) or 5 (if
nonplanar C1)
geometrical parameters 16 (atoms) x 3 (xi, yi,
zi) constrains ?
N of observables
C
13C
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With only DHH couplings

C
13C
6
DC1 H couplings

6
DC2 H couplings
TOTAL 30
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13C-a-Styrene dissolved in a namatic phase (I52)
Styrene dissolved in CClD3
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An amount of 13C-1-styrene and 13C-2-styrene
(10 wt ) was dissolved in ZLI1132 and in I35.
The samples were bought from Aldrich.
Prepare the sample
The spectra
13C-a-styrene in ZLI1132
13C-b-styrene in ZLI1132
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In order to reach a full conformational analysis
of the whole molecule
A maximum is found with the ene out of ring plane
of about 18 and with a standard deviations of 8
.
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The End (Many Thanks)
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Programs
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