Добавил:
Sekretar
kiopkiopkiop18@yandex.ru
t.me/Prokururor I Вовсе не секретарь, но почту проверяю
Опубликованный материал нарушает ваши авторские права? Сообщите нам.
Вуз:
Предмет:
Файл:Ординатура / Хирургия / Библиотека им академика М.И. Перельмана / Книга_5907_Библиотеки_им_академика_М_И_Перельмана.pdf
4
The Fundamentals and Powerful Tool for Nano Physical Pharmaceutics
polymers resulting in an increase in the polymer concentration.
When a polymer chain with degree of polymerization N is added
to a system, it will replace N solvent molecules to maintain the
total volume unchanged, the replacement chemical potential of
the system is:
D m
rep
= D mP – NDm
S
(1.3)
where DmP
polymer chains in solution and melt. D mS
chemical potential between the solvent molecules in solution and
pure solvent. So D m
rep
can be expressed as
Dm
rep
j
1 NNln(
j
) N( −j)
= ln +− − 1 −+
c
12
(1.4)
kT
B
Dm
rep
where j is the volume fraction of the polymers. Let
∂
∂j
= 0
, we
D m
rep
/(kBT )), which
corresponds to the boundary of instability of the solution. The
expression of the boundary is:
1
N
+ = 2cN
(1.5)
j 1 − j
The instable condition of the solution is:
1
+
N
< 2cN (1.6)
j 1 − j
The dividing line between the stable and unstable regions of
the system, described by Equation 1.5, is called the spinodal
line. The relationship between the stability of the solution and
c as well as j is shown in Fig. 1.2.
The shaded part of Fig. 1.2 represents the unstable region,
and the minimum value ( cc) of the spinodal line is the critical
point. When c < cc, Equation 1.5 has no real root and the polymer
solutions are stable throughout the concentration interval. When
c > cc, it has two real roots. If j is between two roots, the polymer
5
Polymer Chains Behavior in Solutions
solution is unstable and will be separated spontaneously into
Figure 1.2 Relation between the stability of the soluon and c as well as
rep
j, when
∂Dm
= 0
. Reproduced with permission from [2].
∂j
Taking the derivative of both sides of Equation 1.5 with
c
respect to j, and let
∂
∂
j
=
0
, we get
1
(1 +N2)2 1
−
1
cc = ≈+
N
2
(1.7)
2N 2
1
−
1
2
j
c
=
1
≈ N
(1.8)
1 + N
2
With increasing the degree of polymerization N, the critical
point cc gradually approaches 1/2. It should be noted that when
0 < c < 1/2, although the polymer–solvent contact is unfavorable,
by the entropy of mixing, and the polymer can still dissolve. For
c
increase of the interaction energy, so the solubility of the polymer
became worse. Therefore, a solvent that makes c of the system
c < 0) is a good
solvent, while one that makes c > 1/2 is a bad solvent. Moreover,
with c increasing, the solvent can no longer dissolve the polymer,
so the solvent is nonsolvent.
6
The Fundamentals and Powerful Tool for Nano Physical Pharmaceutics
1.1.2 Solubility of Polymer
1.1.2.1 Solubility parameter
The solubility parameter (also known as the Hildebrand
parameter) was proposed by Hildebrand et al. [4, 5] and it is
di = (DEi/Vi)
1/2
(1.9)
where DEi is the molar energy of vaporization of substance i and V
i
is its molar volume. The unit of solubility parameter di is (MPa)
1/2
or (cal/cm3)
1/2
, and they satisfy 1 (MPa)
1/2
= 2.0455 (cal/cm3)
1/2
.
The solubility parameter is often used to express the interaction
between a polymer and a solvent. For binary systems, using simple
thermodynamics, the c parameter can be represented with the
solubility parameter:
V
2
c = S(dS − dP) + 034 .
(1.10)
RT
where dS and dP are solubility parameters of solvent and polymer,
respectively. VS is the molar volume of the solvent, and R is the
gas constant. Together with the relationship between c and
dissolvability of the solvent (described in Section 1.2), it can be
seen that polymers and solvents are miscible when their solubility
parameters are close to each other, otherwise they are not.
The so-called “like dissolves like” means that the solubility
parameters are similar.
1.1.2.2 Real polymer solutions
account the dispersion interaction, losing sight of the hydrogen
bonding and dipole-dipole interaction, it is only applicable to
the non-polar system. For polar substances, there will be large
errors in the estimation of the solubility properties. In response to
this, many researchers try to revise or improve it. In this section,
we will introduce the Hansen solubility parameter (also known
7
Polymer Chains Behavior in Solutions
Hansen thinks that the total energy of vaporization
of substance i is composed of three components: dispersion
interaction (DEid), polarity interaction (DEip) and hydrogen bonding
(DEih) [6, 7], namely
DEi = DE
id
+ DE
ip
+ DE
ih
(1.11)
Accordingly, the solubility parameter is split into dispersion
The relation between the solubility parameter and its three
components is:
2 2 2
di = dd + dp + d
h
(1.12)
For the Hansen solubility parameter, the relative energy
the “spatial distance” of the solubility parameters of solute and
solvent (Ra) to the radius of the interaction (Ro), i.e., RED = Ra/Ro.
When RED < 1, it indicates that the solvent is good, while RED > 1,
the solvent is bad. For more detailed introductions or applications
of the Hansen solubility parameter, please refer to other relevant
materials [8–11].
1.1.3 Self-Assembly Behavior of Polymers in Soluon
When amphiphilic polymers are dispersed in a solvent that is
selectively soluble for only one of the polymer components, the
solvophobic part of the polymer will tend to aggregate together
to minimize contact with the solvent, thereby reducing the
interfacial energy. Under these conditions, phase separation will
take place in solutions, and the polymers aggregate to form
nanoparticles. This process occurs spontaneously and is called
self-assembly in solutions. Water is a commonly used as selective
solvent in current studies. Entropy, interaction free energy and
molecule geometry of the system are combined to provide a
framework for self-assembly theory.
1.1.3.1 Dynamics of self-assembly
The free energy of a micelle composed of n amphiphilic polymers
is n
G
n
0
. Assuming that the dilute solution theory is applicative,
8
x
The Fundamentals and Powerful Tool for Nano Physical Pharmaceutics
then equilibrium thermodynamics requires that the molar fraction
of the amphiphilic polymers of the micelles above should
satisfy [12, 13]:
n
0
kT
x
n
0
G + ln
G + kT x
= const
n
=
1
ln()
1
(1.13)
n
n
where k is Boltzmann’s constant, and the subscript 1 represents an
isolated amphiphilic polymer. Or Equation 1.13 can be rewritten as
nG
1
0
−G
n
0
)/ [ ( kT ]
x = nxe
(1.14)
n 1
If
G
n
0
− G
1
0
< 0, aggregations are energetically favorable. The
transition from the dispersed state to the aggregated state is
due to the competition between entropy and enthalpy.
1.1.3.2 Free energy of self-assembly systems
The self-assembly bulk is in a thermodynamically stable state, and
the system free energy is also the lowest, correspondingly. The
morphology of self-assembly is mainly controlled by three kinds of
free energy in the system:
interfacial tension between the hydrophobic chain
segment and solvent;
Assembly
balance, such the composition
participating in
In the
experiment,
adjusting these factors on purpose.
morp
hology is essentially determined by the balance
as
self-assembly or
the
assembly
and
structure
conditions
morphology
of
the
polymer
for self-assembly,
is often
adjusted
by
1.1.3.3
Packing
micellization
Basic morphology of self-assembly systems
parameter
of amphiphilic
theory [14–16],
small molecules
proposed
based
in
solutions,
on
the
can
9
Polymer Chains Behavior in Solutions
explain the assembly morphology of polymers. Self-assembly
behavior of amphiphilic molecules in solutions is controlled by two
opposing forces, i.e., an attractive force between the hydrophobic
segments leading to aggregation and a repulsive force between
the hydrophilic segments preventing unlimited growth of the selfassembly bulk into a distinct macroscopic phase. Self-assembled
structures are stabilized in a solution thanks to the interaction
between the hydrophilic segments and the solvent. For an
amphiphilic polymer, the packing parameter P can be calculated
by the following formula:
v
P
=
(1.15)
al
0c
where v is the volume of the hydrophobic chain segment, a0 is
the equilibrium area of the hydrophilic part, and lc is the length of
the hydrophobic chain segment. As shown in Fig. 1.3, the value
of P
bulk. Spherical micelles are obtained with 0 < P < 1/3, while
cylindrical micelles with 1/3 < P < 1/2 and vesicles with ½ < P < 1.
When P = 1, one gets a planar membrane and the inverse-phase
assembly will be observed with P > 1. Note that the aggregation
number of polymers n
Figure 1.3 Relaonship between the packing parameters, the microstructure
of amphiphilic block copolymers and their self-assembly morphology in
soluon. The formula related to the table is shown in 1.10, 1.15.
10
The Fundamentals and Powerful Tool for Nano Physical Pharmaceutics
1.2 Self-Assembly of Amphiphilic Copolymers in
Solution
1.2.1 Self-Assembly of Block Copolymers
Self-assembly of molecules is a process during which molecules
factors. It is one of the most important methods to prepare
nanomaterials. Self-assembly of molecules will transform the
system from a disordered state to an ordered state, a common
phenomenon in nature such as various biomolecules selfassembling into cells, phospholipids self-assembling into cell
membranes, and surfactants into soap bubbles. Self-assembly
of polymers, an important branch of self-assembly of molecules,
utilizes the interaction between polymers, or polymers and
solvent (hydrogen bonding, hydrophilic/hydrophobic interaction,
electrostatic interaction, etc.) to assemble polymers into highly
ordered microphase structure. Compared with the self-assembly
of small molecules, aggregation products of polymers have higher
stability and better mechanical properties, and have been widely
science. The self-assembly bulk in solutions mainly includes
micelles, liposomes, microspheres and gels. In this section, the
morphologies and phase transitions of amphiphilic assembly units
(block copolymers and lipids) during the self-assembly process
are introduced.
Due to the obvious advantages of realizing the controllability of
self-assembly morphology and size, block copolymers have become
the most widely studied polymers. Microphase separation may
occur for most block copolymers because of the incompatibility
of the constituent components [17]. The self-assembly process of
block copolymers is driven by combinatorial enthalpy and
combinatorial entropy, and the covalent bonds connecting the
blocks prevent macrophase separation. The microphase separation
depends mainly on three factors [18]: (1) the volume fraction of
each block (f); (2) the total number of the segments (or degree of
polymerization) (N); (3) the Flory–Huggins interaction parameter
between the blocks (c). c parameter, which is temperaturedependent (see Equation 1.2), describes the compatibility of
Self-Assembly of Amphiphilic Copolymers in Solution
11
cN denotes the segregation power.
With temperature increasing (or cN decreasing), the rising of the
combinatorial entropy results in a higher compatibility between
blocks, and thus the copolymer may undergo a transition from
order to disorder. The critical temperature for order-to-disorder
transition (ODT) is referred to as the T
ODT
.
The phase behavior of diblock copolymers in molten state has
been investigated theoretically and experimentally [17, 19–26].
According to cN, the so-called weak segregation limit (WSL)
( cN < 10) and the strong segregation limit (SSL) ( cN » 10) have
was developed as 10 < cN < 100. For SSL and WSL, both T
ODT
and
the microdomain size (D) can be easily obtained. For the former,
states (DG
m
) can be expressed as [25]
1
2
DG = D − D = C c
N
C D
− C
(1.16)
H TS
2
+
2
m m m 1
3
D
N
where C1, C2 and C3 are positive constants independent of c, N
and D. The derivative of Equation 1.16 with respect to D gives
the value of D at the equilibrium (denoted as Deq):
2 1
3 6
D
eq
~N c
(1.17)
The temperature at which DG (Deq) = 0 is the critical
m
temperature T
ODT
. For WSL, in terms of the theory from Leibler [19],
one can get:
1
DN~ 2 c
0
(1.18)
The phase behavior of the diblock copolymer is predicted
by SCMF theory as follows. When the value of cN (>10.5) is low,
with fA increasing, transition starts from the disordered state
to body-centered cubic spheres (S phase), hexagonally packed
cylinders (C phase), bicontinuous gyroids (G phase) and lamellae
(L phase), successively. For a higher cN, the G phase is unstable
compared to C or L phase.
12
The Fundamentals and Powerful Tool for Nano Physical Pharmaceutics
Figure 1.4 Phase diagram of assembly control of amphiphilic block copolymers
in soluons.
As the amphiphilic polymers have been dispersed in a
selective solvent, only one block of the polymers is soluble in the
solvent, and the solvophobic blocks tend to aggregate together to
reduce the contact with the solvent, resulting in phase separation
between the polymers and the solvent to form nanoparticles
(polymer micelles). The process occurs spontaneously. Compared
with self-assembly in bulk, self-assembly of block copolymers in
solutions is more complex. Besides temperature, the self-assembly
of block copolymers into nanoparticles in solutions depends on
the concentration of polymers in solutions as well. Therefore, the
degree of incompatibility has been described by fcN, where
f represents the descriptive concentration of polymer in solution.
The controlled factors of assembly of amphiphilic polymers
in solutions are shown in Fig. 1.4. The variation of the selfassembled morphology of polymers in solutions with the solution
concentration (C
polymer
), the length ratio of the head to tail
(L
head/tail
) and the c parameter is displayed in Fig. 1.5. Once the
Self-Assembly of Amphiphilic Copolymers in Solution
13
critical micelle concentration (CMC) is reached, self-assembly of
the block copolymers takes place and the nanostructures in the
solution include spherical micelles, rod-like micelles, vesicles and
lamellar.
Figure 1.5 The concentraon (C
polymer
), L
head/tail
and c dependence of the
self-assembled morphology of amphiphilic copolymers in soluons.
1.2.2 Self-Assembly of Liposomes
1.2.2.1 Formation of liposomes
Besides amphiphilic block copolymers, lipids could self-assemble
to form vesicle structures in aqueous solution as well. Self-assembly
of block copolymers results from interactions of solvent-polymer
and polymer–polymer, while that of lipids is mainly based on
the hydrophobic combination between acyl chains. When lipid
molecules are in the water environment, due to the characteristics
of amphiphilicity, the hydrophilic heads will be exposed outside
and the hydrophobic tails are coated inside, forming micelles,
vesicles, regular bilayer or other structures. Lipids with large
molecules can assemble into spherical vesicles or planar bilayers
(Referring to Fig. 1.3). Liposomes are spherical vesicles composed
of one or more layers of phospholipid bilayer. Lipid assemblies