Transcript Chimica Fisica dei Materiali Avanzati Parte I

UNIVERSITA’ DEGLI STUDI DI PADOVA
Laurea specialistica in Scienza e Ingegneria dei Materiali
Curriculum Scienza dei Materiali
Chimica Fisica dei Materiali Avanzati
Part 3 –Self-assembly
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Self-assembly
 Supramolecular self-assembly concerns the spontaneous
association of either a few or many molecular components
resulting in the generation of either discrete oligomolecular
supermolecules or extended polymolecular assemblies (layers,
films, membranes…).
 The final product is obtained entirely spontaneously when the
components are mixed together in the correct ratios under a
given set of conditions (solvent, temperature, pH, …)
 The formation of supermolecules results from the recognitiondirected spontaneous association of a well-defined and limited
number of molecular components under the intermolecular control
of the noncovalent interactions that hold them together.
J.-M. Lehn, Science 2002, 295, 2400
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Self-assembly in biology
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Self-organization
 Self-organization can be considered as ordered self-assembly.
It concerns systems presenting a spontaneous emergence of
order in either space or time or both.
Structure of the LH2 light-harvesting antenna system of Rhodopseudomonas
acidophila which contains rings of 18 (a) and 9 (b) bacteriochlorophyll molecules.
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Self-organization at the heart of life
Cy
e-
Membrane
hn
P
eBC
BP
e-
QA
A simplified view of the structure of the photosynthetic reaction center of
Rhodopseudomonas viridis.
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Self-assembly and self-organization
 Self-assembly and self-organization of a supramolecular
architecture are both multistep processes implying information
and instructed components.
 They may follow a sequence and a hierarchy of assembly steps,
and require reversibility of the connecting events, i.e., kinetic
lability and rather weak bonding (compared with covalent
bonds), in order to allow the full exploration of the energy
hypersurface of the system.
 In other words, the product formation must be completely
reversible and represent the thermodynamic minimum for the
system. In essence, all the information necessary for the
assembly to occur is coded into the constituent parts.
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Self-recognition: Instructed System Paradigm
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Fundamental thermodynamics of self-assembly
 Individual molecules in a medium possess a cohesive energy
or self energy mi
 It is the sum of its interactions with all the surrounding molecules
(including any change of the energy of the solvent caused by the
solute molecules)
 How is mi related with the pair potential w(r)?
 For an ideal gas of hard spheres of diameter s with a pair
potential wr    C r n we already obtained (Part 1, slide 3)
T ot.Energy  mi  
4C
n  3s n3
 In a condensed phase mi must also include the cavity energy,
i.e.the cost of creating the cavity in which the reference molecule
sits.
 In a closed-packed structure with 12 nearest neighbors
mi c.p.  6ws   12ws   6ws 
Cost of the
cavity
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Interactions of the
reference molecule
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Stacking of hexagonal close packed layers
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Fundamental thermodynamics … (cont’d)
 For a solute molecule (s) in a solvent medium (m) of similar
size, one might roughly estimate
i
mliq
 6wmm s  12wsm s 
 Six solvent pairs must first be separated before the solute
molecule can enter the medium and interact with 12 solvent
molecules.
 Note: the effective pair potential between two dissolved molecules
is just the change in the sum of their free energies as they
approach each other


 m1i  m2i 
 Boltzmann distribution X 1  X 2 exp

k
T
B


with X1 and X2 the equilibrium concentrations of molecule X in
region (environment, phase) 1 and 2, respectively, where the selfenergy is m1i and m.2i Hence,
m1i  kBT ln X1  m2i  kBT ln X 2  m
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Chemical potential
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Self-assembly of amphiphiles
 In amphiphilic molecules, such as surfactants, lipids and certain
copolymers, one end contains a hydrophilic group while the rest of the
molecule is hydrophobic, usually a long hydrocarbon chain.
 The hydrophilic and hydrophobic interactions rely on the structure of the
water H-bonds adopted around the dissolved end groups.
Amphiphiles such as surfactants and
lipids can associate into a variety of
structures in aqueous solutions.
These can transform from one to
another by changing the solution
conditions such as the electrolyte or
lipid concentration, pH or temperature.
Most single chained surfactants form
micelles, while most double chained
surfactants form bilayers.
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Some common amphiphiles
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Vesicles for drug delivery
Figure 1: Liposomes - (left) A = aqueous soluble drug encapsulated in aqueous
compartment; (centre) B = a hydrophobic drug in the liposome bilayer; (right) C =
hydrophilic polyoxyethylene lipids incorporated into liposome
Liposomes were discovered in the mid 1960s 2 and originally studied as cell membrane models. They have
since gained recognition in the field of drug delivery. Liposomes are formed by the self-assembly of
phospholipid molecules in an aqueous environment. Shown schematically in Figure 1a, the amphiphilic
phospholipid molecules form a closed bilayer sphere in an attempt to shield their hydrophobic groups from
the aqueous environment while still maintaining contact with the aqueous phase via the hydrophilic head
group. The resulting closed sphere may encapsulate aqueous soluble drugs within the central aqueous
compartment (Figure 1, left) or lipid soluble drugs within the bilayer membrane (Figure 1, centre).
Alternatively, lipid soluble drugs may be complexed with cyclodextrins and subsequently encapsulated
within the liposome aqueous compartment.3 The encapsulation within/association of drugs with liposomes
alters drug pharmacokinetics, and this may be exploited to achieve targeted therapies. Alteration of the
liposome surface is necessary in order to optimise liposomal drug targeting.
Pharmaceutical Journal, Vol 263 No 7060 p309-318
August 28, 1999 Special Feature
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Fundamental thermodynamic equations of self-assembly
 At equilibrium, the chemical potential of all identical molecules
in different aggregates must be the same
1
 X3 
 X2 
0

m

k
T
ln

 

3
B
3
 2 
 3 
1
2
m  m10  k BT ln X 1  m 20  k BT ln
monomers
dimers
trimers
or
m  m N  m N0 
1
X 
k BT ln N 
N
 N 
 In an aggregate of aggregation number N
m N : chemical potential of a molecule
m N0 : standard chemical potential (mean interaction energy per free
molecule)
X N : concentration of molecules ( X N N
is the concentration of
aggregates)
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Fundamental thermodynamic equations … (cont’d)
 Law of mass action
 XN 
k1
NX 1 





N
kN


 Rate of association =
 Rate of dissociation =
k1 X 1N
kN X N N 
 From the equilibrium constant
X N  k
K N N  1
kN
X 1 


 N m N0  m10 
 exp

k BT


or, alternatively, from
m  m10  k BT ln X 1  m N0 
1
X 
k BT ln N 
N
 N 
one gets
The system is completely defined by
this equation and by the
conservation of solute molecules

C  X1  X 2  X 3     X N
N 1

 μ10  μ N0  
X N  N  X 1 exp

k
T
 B


N
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Note that C and XN are expressed in
mole fraction units, therefore, they
are always < 1.
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Conditions necessary for the formation of aggregates
 Aggregates form only when there is a difference in the
cohesive energies between the molecules in the
aggregate and in monomeric form
 Suppose that
m10  m 20  m30    m N0
then
X N  NX 1N
and since X 1  1 , we must have  X N N   X 1.
 Most of the molecules will be in the monomer state, the
more so the more m N0 increases with increasing N.
 The necessary condition for the formation of large
stable aggregates is that m N0  m10 for some values of N.
0
 The dependence of m N upon N determines many of the
physical properties of aggregates, such as their mean
size and the polydispersity.
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Effect of aggregate dimensionality
 The dependence of m N on N is
usually determined by the
shape of the aggregate.
0
One-dimensional case
 Consider a suspension of rodlike (linear chain) aggregates of
identical molecules
 Let k BT be the (positive)
monomer-monomer ‘bond’
energy in the aggregate
relative to monomers in
solution.
 The total interaction free-energy is Nm N0  N 1kBT
(assuming
m10  0)
 The factor (N – 1) accounts for the fact that the terminal molecules are bound on
one side only
 Hence


m N0  1 
1
k T
0
k BT  m  B
N
N
 As N increases, m N decreases asymptotically towards
energy of a molecule in an
infinite
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0
m0 , the ‘bulk’
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Effect of aggregate dimensionality (cont’d)
Two-dimensional aggregates (disks, sheets)
 The number N of molecules is proportional to the area
R 2
 The number of unbound molecules is proportional to the
1
circumference 2R , hence to N .2
 The mean free energy per molecule, considering that
is m 0  1  1 k T  m 0  k BT
N
B

1
1
1
Nm N0   N  N 2 k BT




 N 2
Three-dimensional aggregates (spheres)

N
2
N is proportional to 4 3 R 3 , while the number of unbounded surface
molecules is proportional to the area 4R 2 and hence to N 3.
2
 By arguments similar to the previous ones, the mean free energy per
molecule is
m N0  m0 
k BT
N
2
3
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Effect of aggregate dimensionality (cont’d)
m N0
 Summarizing,
 For the simplest shaped structures
– rods, sheets and spheres – the
interaction free energy per
molecule can be expressed as
m N0  kBT
3-D
2-D
m N0  m0 
1-D
 k BT
Np
m 0
  : positive constant dependent on
the strength of the intermolecular
interaction
0
1
N
1
2
 p: exponent dependent on the
shape or dimensionality of the
aggregates
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The critical micelle concentration
 At what concentration will aggregates form?
N
 k BT
0
0


  one gets
μ

μ
1
N
 Using m  m 
into
X N  N  X 1 exp

Np
k
T
 B


N

N
 
1  
X N  N  X 1 exp 1  p    N X 1e
  N  

0
N
0



for small X 1 , i.e., when X 1  e  , we have
X 1  X 2  X 3  .
Most of the molecules will be isolated monomers, or X 1  C
X 1 approaches exp m10  mN0  kBT  or e , it can increase
no further (otherwise, X N could even exceed unity, which is not
 When
possible by definition).
 The saturation of
X1
occurs at the critical micelle
concentration (CMC) denoted
 X1 crit
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Critical Micelle Concentration



 At  X1 crit  CMC  exp  m10  mN0 kBT or  X1   CMC  e for all
crit
p, further addition of solute molecules results in the formation of
more aggregates with the monomer concentration unchanged at
CMC.
Most single chain surfactants
containing 1216 C atoms per
chain have their CMC in the
range 102105 M while the
corresponding double-chained
surfactants have much lower
CMC values due to their
greater hydrophobicity.
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Phase separation vs. micellization
 For simple disc-like and spherical aggregates

1


exp  N 2 


1

for discs  p  
2


1


3

exp  N 



for spheres  p 


 N

 N
X N  N X 1e
X N  N X 1e
 Above the CMC where
2

3
X 1e  1 the above two eqns. become
1
1




3  , respectively.
and
2

X N  N exp   N 
X N  N exp   N 




 For positive values of  , usually > 1, there will be very few
aggregates of any appreciable size.
 There is a phase separation, strictly to an aggregate of
infinite size at the CMC (e.g., formation of droplets)
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Phase separation vs. micellization (cont’d)
 Polydispersity comes about for rod-like (p = 1)aggregates,
such as cylindrical micelles , fibrous structures, microtubules,
etc.
N
For p  1 X N  N X 1e e 


the second exponentia l term is a constant
rather tha n a rapidly decreasing function of N .
 Since above the CMC X 1e  1 ,
XN  N
for small N
The concentration of molecules grows in proportion to the
aggregate size and there is no phase separation.


 For very large N the term X 1e N begins to dominate,
eventually bringing XN to zero as N approaches infinity.
 The distribution is highly polydisperse.
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Properties of polydispersity
 The density distribution of
molecules in aggregates of N
molecules peaks at
Nmax  M  Ce
i.e., the mean aggregation
number is concentration
dependent
 With an expectation value of N
N   NX N
 X N   NX N
C
 1  4Ce
 2 Ce  2 M
s2 
i.e., the distribution is skewed
towards large N.
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 log N
 log C
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More complex amphiphilic structures
 The value of the exponent p in
m N0  m0 
 k BT
is constant only for aggregates consisting of very
simple geometrical shapes (spheres, disks, rods).
Np
 Complex amphiphilic molecules can assemble into more complex
shapes (bilayers, vesicles, liposomes; cf. slide 10).
 This is determined by the type of anisotropic binding forces acting
between different parts of the amphiphilic molecules.
 Different degrees of molecular flexibility also affect the ability of the
molecules to adopt different type of aggregation.
 In all such cases:
m N0
leading to monodisperse
aggregates
N
M
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Parameters determining packing types
Two opposing forces
 The packing shape depends on the balance between hydrophobic
attraction at the hydrocarbon/water interface and the hydrophilic,
ionic or steric repulsion of the head-groups
 This balance determines an optimal surface area a0.
 The chain volume v and chain length lc set limits on how the fluid
chains can pack together.
 Different structures may be consistent with these constraints.
Eventually, entropy favors
theCFMA.
structure
with the smallest aggregation
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Packing shapes of
amphiphilic molecules
 For dimensionless packing
parameter:   v a0lc

 1 3
1
, spherical micelles
3    1 2 , non spherical
micelles


1 2    1 , vescicles or bilayers
  1 , ‘inverted’ structures
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