Chapter 2
Theory of Coacervates and Gels
2.1 Introduction
The solubility of a colloidal dispersion in an appropriate solvent depends on the
nature of the colloid and various other factors; temperature, pH, ionic strength etc.
The nature of interactions with solvent and change in physiochemical conditions
can bring a reduction of solubility of the colloid as a result of which a larger part of
the colloid separates out in a new phase [1-5]. The original one phase-sol divides in
to colloid rich and poor phases.
According to !UPAC, Coacervation is defmed as the separation of colloidal systems
into two liquid phases. It was suggested by Oparin [6,7] that coacervates might
have played a significant role in the evolution of cells. In water, organic chemicals
do not necessarily remain uniformly dispersed, but may separate out into layers or
droplets. If the droplets, which contain a colloid, rich in organic compounds, and
are surrounded by a tight skin of water molecules then they are known as
coacervates [6-9]. It is to be distinguished from precipitation, which is observed in
the form of coagulum or flocs and occurs in colloidally unstable systems. The term
coacervation was introduced in 1929 by Bungenberg de Jong and Kruyt [1], for a
process in which aqueous colloidal solutions separate, upon alteration of the
thermodynamic condition of state, into two liquid phases, one rich in colloid, i.e.,
the coacervate, and the other containing little colloid (supernatant) [10].
Coacervation signifies the union of the colloidal particles. By colloidal particles,
one understands liquid droplets, called coacervates, primarily induced by demixing.
A wide variety of solutions can give rise to them: for example, coacervates form
spontaneously when a protein, such as gelatin, reacts with gum arabic.
Different scientists kept on defining the Gel again "and again for many times. There
are good reviews on Gel [Hermans et aI.,1949; Djabourov, 1988; 1991; Ross14
Murphy,1991; Almdal et aI., 1993] and all of them tried to answer the same century
old question-What is a Gel? Scientists and researchers working in different fields
try to bring in more and more materials into this according to their own
convenience and define it accordingly [11].
A Gel can be defmed as a three dimensional interconnected percolating
(mechanically) network structure with the continuous phase (solvent) interacting
synergistically with the network. The dispersed phase (polymer) can be present in a
disordered and non-ergodic state. Regardless, these disordered systems are often
found to be trapped far away from equilibrium, and typically relax slowly,
exhibiting complex and interesting dynamics [12]. Probably the originator of the
term gel is Thomas Grahm (1861). He proposed a different class of substances
according to their diffusive power: the colloidal substances are slowly diffusing
substances which are held in solution by feeble forces. D. Jordan Lloyd (1926)
started her survey on the gels with the famous word: gel is one which is easier to
recognize than to define. Gels must be built up from two components, one of which
is liquid and the other one is a solid at a particular concentration and temperature
considered. Gels can be broadly divided in two categories according to mechanism
of their formation: chemical gel and physical gel. A chemical gel is formed by
chemical reactions like, copolymerization, poly condensation, vulcanization leading
to formation of a branched network (cross-linked network) made up of linear
flexible chains attached by covalent bonds. They are strong gels. On the other hand
a physical gel can be formed by the formation of secondary bonds like hydrogen
bonding, vander Waal's bonding, hydrophobic interactions, etc. They are weak
gels. Gelatin gel, agarose gel etc are physical gels. Coacervates and gels are novel
and smart soft materials and constitute an area of immense interest in the recent past
due to their enormous application potentials.
15
2.2 Theory of description of Coacervates
Bungenberg de Jong (1929 - 1949)
Bungenberg de long and Kruyt (1929) gave the first theoretical explanation of
coacervate based on the stability of hydrophilic colloids. Capillary electric charge
and hydration decides the stability of these colloids. Coacervation would be the
consequence of the removal of these two stability factors. There would be shrinkage
of the solvent layer of the colloidal particles when it sets desolvation (solvent
mantle), which then would merge through their concrete solvate mantles (concrete
= after desolvation) (shown in Figure 2.1).
'~fI~IJ/il~/etfI. ,
·~MkrO·~M1IW11
·~ljq~.~"~ifM:
a
b
c
Figure 2.1: Schematic representation of the mechanism of phase separation by
coacervation. (a) Particle with a diffuse solvate mantle (dotted periphery), (b)
particle with a concrete solvate mantle, (c) fusion of the particles to a coacervate
with their concrete solvate mantle. Figure reproduced from the papers by
Bungenberg de long in 1949 [3-5].
Coacervates could be regarded as a liquid, which had lost its free mobility to a
certain degree and this explanation was mainly of the simple coacervation
phenomenon, but in the case of complex coacervate when it was brought about by a
decrease of charge, Bungenberg de long needed more research. The role of
physiochemical parameters, such as ionic strength and pH, on the coacervation
16
formation, highlighted that the coacervation was a consequence of electrostatic
interactions by Bungenberg de long in his new theory of complex coacervate
(1949) based on experimental data that interaction of negative acacia gum (GA) and
positive gelatin form a complex largely [1].
Voorn-Overbeek (1957)
Overbeek and Voom developed the fIrst quantitative theory of complex
coacervation, based on the experimental results of Bungenberg de long, in which
they considered gelatin / GA coacervation as a spontaneous phenomenon [13]. They
interpreted cocervation is formed by the electrostatic interactions of the oppositely
charged molecules and entrapping solvent molecules. The presence of the solvent
contributed to the increase in the entropy of the system and allowed a number of
possible rearrangements of the molecules that is why coacervates were liquid in
nature and fully reversible. The theory was based on several assumptions: (i) the
molecules have a random chain confIguration, (ii) solvent - solute interactions are
negligible, (ii) the interactive forces are distributive in nature, with the system
behaving as though the charges are free to move, and (iv) there is no site specifIc
interaction between the molecules. The theoretical treatment of complex
coacervation was put on a quantitative basis by using the Debye-Hiickel equations
for the electrical interactions and the Flory-Huggins theory for the entropy term.
Overbeek and Voom set
Ftolal (T) = Fmixing (M)
+ Felectrostatic (e)
and then substituted the Flory-Huggins approximation for FM • Fe was calculated by
treating the polyions as the sum of single charges and approximating the total
electrical interaction free energy by the Debye-Huckel theory. Their fmal result was
(2.1)
17
where NT
= total number of lattice sites in the system, rj = Number of
occupied by particle i, crj = charge density of particle i,
fraction) of particles of type i, a
~[ 4Jre ]1/2(_1_)
2
3£ ikBTv
kBT'
v
=
<l>i
sites
= volume fraction (or site
electrical interaction constant
site volume, c = dielectric constant, e
=
=
elementary
charge.
The critical conditions for coacervation were derived from Eq. 2.1 for the
symmetrical two component case in which each polyion was of the same size (r2
r3
=
=
r, subscripts 2 and 3 refer to the polyions, 1 to the solvent) and charge density
(cr2 = cr3 = cr) and both were present in equal initial concentration (<1>2 = <1>3 = <1». For
the solvent r,
= 1, cr = O. The result,
·3
(Y
r
64[
= 9a2
1
]
(1- ¢)2 (1 + ¢) .
(2.2)
showed that coacervation would take place at ordinary temperatures in water only
when cr 3r ~ 0.4 since <1>«1. According to this theory, for a two component system
consisting of a polyion salt and water, the critical conditions for coacervation are
met when cr3r ~ 0.53, that is to say when the charge density (a) or the molar mass
(r) are sufficiently large. This model was extended to three or four-component
systems. Overbeek and Voorn explained that the suppression of coacervation by salt
excess was due to an increase of the solubility of the polyions, a decrease of the
amount of polyions in the coacervate, and a decrease of charge density through
charge screening by counterions (Figure 2.2). It was also shown that not only
polymers but also small ions were accumulated in the coacervate. The adaptations
of various theories were developed later, since it seemed that the assumption that
the Huggins interaction parameter was negligible was insufficient.
Veis-Aranyi (1960 -1970)
Veis and Aranyi developed a theory at conditions where cr3r < 0.53, i.e., when the
Voom-Overbeek theory was not applicable [14]. This theory was based on a
practical case of coacervation upon temperature reduction between two oppositely
18
charged gelatins. Veis modified the Voom-verbeek theory, including the Huggins
interaction parameter, corresponding to the solvent-solute interaction; this
parameter increases significantly on temperature reduction.
Flory and Huggins have shown the free energy of mixing of a non-ionic polymer
with solvent to be given by M'M = MfM - TMM
M'M
=kBr[nlln¢1 +n2In¢2 + XI2 nl¢2]
(2.3)
nl and n2 are the numbers of molecules of solvent and solute, respectively, <\>1 and <\>2
their volume fractions. X12 is a dimensionless quantity that characterizes the
solvent-solute interaction energy per solvent molecule divide by kBT.
ij
!J.F M = kBTL nj In ¢j + kBTL Xijni¢i
(2.4)
i<}
in which Xij represents the interaction per kinetic unit i-kinetic unit} pair divided by
kBT. The modified equation obtained by Veis for the phase II, combining the FloryHuggins equation is given by
"'Y .
M'r _
¢i
ij
()3/2
-----'-_
+ L..,./LIJ n-h
-a "(j-h
N k T "-In-h
~
'f'.
.'f'.
~ l'f'l
.
r
B
I
Y;
1<)
where M'r is the free energy of mixing, and a
=
(2.5)
I
electrical interaction constant. N r =
total number of lattice sites in the system, rj = Number of sites occupied by particle
i,
(jj
= charge density of particle i, v = site volume (3.0xlO·23 cm\ ¢; = volume
fraction (or site fraction) of particles of type i. In the Veis-Aranyi theory,
coacervation is considered as a two-step process rather than a spontaneous one.
First the gelatins spontaneously aggregate by electrostatic interaction to form
neutral aggregates oflow configurational entropy, and then, these aggregates slowly
rearrange to form the coacervate phase. (Veis investigated complex Coacervation
on a system of water and symmetrical gelatins (polycation and polyanaion with
identical charge density and chain length), and he presented the following model of
phase separation,
(2.6)
19
(2.7)
where I and II represent the dilute and concentrated phases, A + and B- represent the
polycataion and polyanaion, and SAP and IRC represent the symmetrical aggregate
polymer and independent random chain, respectively. The mechanism is driven by
the gain in confimrrational entropy resulting from the fonnation of a randomly
mixed coacervate phase.
-
Polymer Concentration
Figure 2.2: Theoretical phase diagram for complex coacervation in the system
solvent / polymer PQ / univalent salt KA. The figure has been constructed for r =
1000,
(J
= 0.15. The dotted line is the spinodal, C is the critical point, eM is
positioned at the middle of the node lines and OE gives the equivalent
polyelectrolyte / salt composition. Figure reproduced from J. T. G. Overbeek and
M. J. Voom [13], Phase separation in polyelectrolyte solutions.
20
Nakajima - Sato (1972)
Nakajima and Sato applied the equation of I:!F (which was obtained by Veis for the
phase II) to both phases, and they derived the condition for phase separation. When
the polymer concentration is sufficiently high, this condition is expected to be
applicable. Nakajima and Sato studied an equivalent mixture of sulphated polyvinyl
alcohol and aminoacetalyzed polyvinyl alcohol (PVA) in microsalt aqueous
solution [15]. They adapted the Voom-Overbeek theory by including the Huggins
parameter and changing the electrostatic term. Nevertheless, they agreed with
Overbeek and Voom that the charges should be treated as uniformly distributed in
both dilute and concentrated phases. The experimental and theoretical results were
in good agreement with each other, and the study showed that for specific systems
the Overbeek-Voom model could still be used.
Tainaka (1979 - 1980)
The Tainaka theory is the most recent model developed for complex coacervation,
and is an adaptation of the Veis-Aranyi theory. The main difference from the VeisAranyi model is that the aggregates, present in both the dilute and concentrated
phase, are formed without specific ion pairing [16]. The biopolymer aggregates
present in the initial phase condense to form a coacervate. Tainaka obtained the
condition of phase separation in low concentration. He modified the Eq. 2.7 by Veis
and considered the following model.
(2.8)
In his model, each phase is considered to be the same two-component system of
water and SAP as the phase I of the Veis model. Eq. 2.8 means that coacervate is a
condensation of SAP. In this scheme, the coacervation is a condensation
phenomenon of SAP, and the random chain has been replaced by SAP in Veis's
scheme under the consideration of weak overlapping of polymer chains.
Theoretically, he applied both the phases to the virial expansion method similar to
21
TH-16i& b
5~1·~12<;;\b~~
E:~
the theory of Veis and Gates for the phase 1. Veis and Gates obtained the virial .
expansion method up to the second order, whereas the second virial coefficient A2
was obtained on the assumption that SAP is neutral. On the other hand, he
considered that A2 is directly influenced by the
cha~gys
in SAP. If SAP's overlap
each other, electrostatic energy gain is produced as a result of increase of the ion
density in the overlapped domain. Therefore he took up an interaction potential
between SAP's in order to include the electrostatic energy gain and calculate the
virial coefficients up to the fifth order. This theory led the expression of osmotic
pressure in SAP solution based on the virial expansion method. He extended his
theory to more complicated phenomenon of the counterion-containing solutions. In
his scheme, he deals with the asymmetry system of polymer charges in which the
charges of poly-cation and poly-anion are not identical and hence counterions exist.
In this case, if the solution is salt-free, the number of the counterions should be
equal· to the difference· of the charges of both polymers. The influence of
counterions on the complex coacervation has been well known from the
experimental fact that the coacervation in the case of two protein species having
different isoelectric points is influenced by a pH change of the solution, that is, the
phase separation is suppressed, when the number of the counterions is increased by
the pH change. So, the effects of charge asymmetry accompanied by the existence
of counterions on the coacervation. Therefore the suffix SAP in the above Eq. 2.8 is
altered to AAP for aggregate polymer of charge asymmetry.
[A+] + [B-]-)[A+.B-]AAP
(2.9)
[A+,B-]AAP -) [A+.B-iAAP + [A+.B-]IIAA
(2.10)
In the ith volume element of AAP (Asymmetric aggregated polymer) domain, the
mixing free energy jjpi when the number of total lattice sites is N i may be
represented with the interaction paramete X ]]..' , between j and / by
22
where ¢I '¢2 and ¢3 are the volume fraction of solvent 1, of AAP segments 2, and of
counterion 3, respectively. The suffix i denotes the ith volume element, kBT is the
Boltzmann factor, e is the elementary charge. The interaction potential UA(R)
between the two aggregated AAPs at a distance R was given by Flory's method
based on Eq. 2.11 in the following form:
UA(R) = UI(R) + U 2(R) +U3(R) + U 4(R)
(2.12)
where the four potentials Uj , U2, U3 and U4 on the right hand side of Eq. 2.12
correspond to the four terms in Eq. 2.11, respectively in turn. The potential Uj is the
Flory-Krigbaum potential, and U2 is the electrostatic term, and U3 and U4 are the
newly added terms in the presence of counterions. The interaction potential Us
between SAP is expressed by Us = U3 + U2 • If U4 is neglected under the assumption
%31
= 0, the necessary calculation to be added to the previous theory is just for U3 •
This U3, which should be positive, results from the entropy increase due to the
counterion distribution. The repulsive force between AAPs therefore appears due to
this term in this system.
According to Tainaka, the driving forces for phase separation are the electrostatic
and the attractive force between the aggregates, which become stronger when the
molar mass and the charge density of the polymers increase. Charge density and
molar mass of the polymers should fall within a critical range for coacervation to
occur. If the charge density or molar mass of the polymer becomes higher than the
critical range, then a concentrated gel or a precipitate, induced by the long-range
attractive forces among the aggregates, will be formed. On the other hand, for
charge densities or molar mass below the range, short-range repulsive forces will
stabilize the dilute solution and coacervation will not occur. The Tainaka theory is
more general than all the previous theories and is applicable to both high and low
charge density systems. It provides an adequate explanation of the coacervation
process for a large number of systems.
23
,
'
Gupta and Bohidar (2005)
Gupta and Bohidar have given a systematic and comprehensive theoretical
approach that encapsulates "the understanding of kinetics of phase separation,
spinodal decomposition and syneresis with the exception from Veis model that they
included contribution from electrostatic interactions.
The kinetics of phase separation of a homogeneous polyelectrolytic solution into a
dense polymer rich coacervate and the dilute supernatant phase was discussed
through statistical thermodynamics. Physical conditions for the phase seperations
were deduced explicitly from the statistical thermodynamic treatment of the
problem that was based on the Flory-Huggins lattice model description of polymer
solutions [17]. In this lattice model, r is the number of sites occupied by the
polymer having a critical volume fraction
would
fjJ2c,
it was found that phase separation
ensue
which
«(73r/<p2C)~(64/9a2)~0.45 at 20°C for
<P2c
reduces
to
«1, where cr and a. are the
polyelectrolyte charge density and the electrostatic parameter respectively. The
separation kinetics was observed to mimic a spinodal decomposition process. For a
wide variety of experimental conditions [18,19], the onset of complexation between
complementary macroions conforms to an empirical relation given by
GV /
fj :?:
constant," where v is the charge density of complementary polyelectrolyte. Odijk
[20) has argued that for this empirical relation to be valid the interactions in system
must adhere to the following requirements: (i) Debye-Huckel approximation must
be valid, (ii) complexation is independent of polyelectrolyte chain statistics, and
(iii) excluded volume effects are not significant. For self-charge neutralization v can
be replaced by cr and it follows the equation
(7 2
/.fj ~ constant. Gupta and
Bohidar have provided a rigorous proof to the empirical condition proposed by
Dubin [21) though they deal with a single polyelectrolyte undergoing self-charge
neutralization, which is comparable to the complexation between oppositely
charged polyelectrolytes described by Dubin [21]. The system generated a simple
coacervate whereas a two-polyelectrolyte system yields a complex coacervate. It
24
has been shown that the polymer rich coacervate phase is associated with higher
internal pressure, consequently giving rise to syneresis and the rate of released of
supernatant due to syneresis was found to be independent of the initial coacervate
mass.
Mohanty and Bohidar discussed the syneresIs exhibited by a heterogeneous
polyampholyte coacervate through non-equilibrium statistical thermodynamics. It
has been shown that the coacrevate phase is associated with fluctuating excess
internal pressure that gives rise to syneresis. The internal pressure inside the
coacervate follows a damped oscillatory behavior that relaxes slowly with time,
independent of amplitude [22].
Phase Separation Mechanism (Spinodal vs Nucleation and Growth)
Complex coacervation between polysachharides and proteins have emphasized on
the importance of parameters like pH, ionic strength, protein to polysachharide
ratio, total biopolymer concentration, biopolymer charge density, and flexibility on
the extend of phase separation and coacervation yield [23,24]. It is difficult to hope
to control the final structure and physicochemical properties of coacervated system
without better knowledge of phase ordering kinetics out of equilibrium conditions.
Small angle static light scattering is a powerful method to study both the different
stages of phase separation and the mechanism of demixing [25,26]. The shape of
the scattering functions and their temporal evolution, theoretically allow the
distinction between spinodal decomposition,
and nucleation and
growth
mechanism.
Scattering function exhibit a maximum Imax at a scattering wave vector q different
from zero, qmax when the phase separation occurs through spinodal decomposition
(SD). It is due to the instantaneous buildup of the periodic concentration
fluctuations in the whole sample [27]. In the early stages of phase separation, Imax
increases exponentially but, qmax
remains steady. In the intermediate and late
stages, Imax increases and, qmax is shifted toward smaller q values. The growth of
the domains or droplets induced by phase separation follows power law scaling in
25
the late stages, with qmax-Ca and [max-tp. The hyperscaling relation
expected to hold during late stage [28]. The values of a
p = 3a is
are indicative of the
mechanism of mass transport during coarsening of systems. Diffusion-controlled
coarsening by coalescence or Ostwald ripening gives a value of ~ [29].
Hydrodynamic interactions or droplets sedimentation make to increase a value
[30].
Nucleation and growth (NG) is the mechanism of demixing, a monotonic decrease
of [(q) with increasing scattering angles is experimentally observed [31,32]. Model
simulations based on Mie theory have confirmed this trend [33]. A correlation peak
can be recorded when a high volume fraction of particles appears, possibly coupled
to multiple scattering effects or at low volume fraction when a depletion layer
surrounds particles [33,34]. For NG early stages, possibly an induction period, [(q)
evolves as [(q) == K(t - r)2n, where K is the growth constant, r is the initial time
and t is the time for phase separation. In the case of spherical nuclei, the exponent
value of 3h for diffusion controlled growth and 3 for interfacial controlled growth
[35]. During late stage ofNG, coarsening of particles follows the same mechanism
of mass transport than during SD.
A definite difference between SD and NG, in the case ·is the presence of the
correlation peak during NG, is the temporal evolution of the maximum wave vector
qmax corresponding to the maximum scattered light intensity [max' During SD,
qmax is first constant then shifts towards smaller q values because the wavelength
of concentration fluctuations grows but during NG, qmax first shifted towards larger
q values because more particles appear that promote the decrease of the distance
R(R
=~) between particles [36], then shifted tow~ds smaller q values due to
qmax
particles coarsening [33].
C. Sanchez, G. Mekhloufi and D. Renard (2006) studied the mechanism of phase
separation during complex coacervation of fl-Iactoglubin(BLG) and acacia
gum(GA) that is mainly driven by electrostatic interactions. During the
26
complexation and before coacervation took place, small angle light scattering
profiles displayed a monotonic decrease of I (q) as a function of wave vector q. A
correlation peak in the scattering functions was only observed when coacervates
appeared in the system. The wave vector qmax
corresponding to maximum
scattered intensity first shifted toward larger q values, indicating an increasing
number of coacervates, then shifted towards smaller q values, as a consequence of
the system coarsening. The power laws qmax-Ca and Imax-t P gave the values
of 1.9 and 9.2 respectively, values much larger than those expected for intermediate
and late stages of spinodal decomposition. From these results, they concluded that
complex coacervation between BLG and GA was a nucleation and growth type
process.
2.3 Theory of description of Gels
Throughout the history of mankind, gels have been important. For example, gelatin
is purified from glue and is used to make jello. Glue is extracted from bones, skins
or intestines of animals and used as adhesives, stickers, food additives, and so on,
for thousands of years. Polysachharides, such as agarose and carrageenan, are also
gelling agents and are used for similar purposes. Basically mankind has been using
gels since it emergence.
In the 20th century, gels went from being passive materials to functional materials.
Ion Exchange Resin was commercialized in Germany in 1930s. Theoretical studies
on polymer gels began in the 1940' s with the pioneering work of Stockmayer and
ofFlory-Rehner in 1940s [37].
Paul J. Flory model (1942)
The peculiar properties of matter in the so called gel state have inspired numerous
theories penetrating to gelation and the structure of gels. Their low mobility has
been attributed variously to the brush heap entanglement of highly elongated
particles to the formation of network structures which extend throughout the
27
yolume, immobilizing entrapped liquid [38]. The various molecular speCIes
occurring in the linear polymers formed by the condensation of bifunctional units
differ only in one parameter, the chain length, apart from possible differences in end
groups.
A
AB--BA--AB-B
)-AB-BA-«
A
B
I
B
-A.
I
A
A
B
I
B
A
l-AB-BA-A
B--BA
Figure 2.3: Trifunctional branched polymer composed of A-A and B-B
bifunctional units.
If some of the units combine with more than two other units, nonlinear structures
developed and the complete description of the molecular constitution assumes a
higher order of complexity. Systems in which molecules are joined together at
random to form network type molecules are amenable to statistical treatment [39].
Equations have been derived expressing the critical conditions for inftnite network
formation. For any given molecule such as the one shown in Figure 2.3 may be
regarded as an assemblage of chains connected together through polyfunctional, or
branch, units.The branching coefftcient a, which is deftned as the probability that a
given functional group of a branched unit leads via a chain of bifunctional units to
another branched unit. Considering Figure 2.3, a is the probability that a A group
selected at random from one of the trifunctional units is connected to a chain the far
28
end of which connects to another trifunctional units. The location of the gel point
and the course of the subsequent conversion of sol to gel are directly related to a.
The formation of chains can be represented by the equation shown in Figure 2.4.
Where i may have any value from 0 to 00. Assuming the reactivity of all A and of
all B functional groups, the probability that the fIrst A group of the chain shown on
the right has reacted is given by PA, the fraction of all A groups which have
reacted, similarly, the probability that the B group on the right of the first B-B unit
has reacted is given by PB.
A
-A+ A--<
A
+ B-B-r >-A( B-BA-A)i B-BA-<
A
Figure 2.4: Formation of chain from bifunctionals A-A, B-B and trifuctional of
A group.
Let P represent the ratio of A's belonging to branch units to the total number of A's
in the mixture. Then, the probability that a B group has reacted with a branch unit is
PBP ; that of the probability of connected to a bifunctional A-A unit is PA (1 - p).
Hence the probability that the A group of a branched unit is connected to the
sequence of units shown in the preceding formula is given by (PA[PB(1-
P)PAJ i PBP)
The expression of a,
a = L[PA PB(1- p)Ji pA PBP
29
(2.13)
Where i
= 0,1,2, .... and evaluation of this summation yields,
a
=
PAPBP
[l-PAPB(l-p)}
(2.14)
If r represent the ratio of A to B groups, then PB = rpA and substituting this in the
above equation
a=
rp2
ora =
A
[l-rp~(l-p)}
rp2
B
[l-rp~(1-p)}
(2.15)
Ordinarily rand p can be determined by the propositions of the initial ingredients
employed and either the unreacted A or B group will be determined analytically at
various stages of the reaction. Then
a
can be calculated using either of the above
equations and hence calculable from experimentally observed and controlled
quantities.
Special cases of particular interest. ,
= 1 and
When there are no A-A units, p
a
p2
'
= -1!r = rp~.
(2.16)
When A and B groups are present in equivalent quantities, r = 1, PA = PB = P and
a
2
= [1-p2(1-p)}
P P
(2.17)
And in a system consisting of bifunctional A-A units and f-functional units
R - At, where A may condense with A,
a=
PP
[l-p(l-p)}
(2.18)
, The critical values of a at which the formation of the infinite network is possible. If
a < ~2 ,the network cannot possibly continue indefinetly. Eventually termination ' of
chains must outweigh continuation of the network through branching and hence all
molecular structures must be finite in size. If a
30
> ~ , branching
of successive
chains may continue the structure indefinitely. Then unlimited structures are
possible and hence a
= !2
represents the critical condition for incipient of infinite
formation' of infinite networks. However, beyond a
=~
by no means all of the
material will be combined into infinite molecules, Infinite networks will form as
1
long as -
2
< a < 1 [40].
Stauffer-de Gennes (1976)
The percolation model deals with the effects of varying the number of
interconnections in a random system [11]. In the bond percolation simulation, all
the sites situated at the intersections are occupied by molecules and are capable of
forming bonds with the neighbor molecules at random. The fraction of the reacted
bonds increased above certain threshold, an infmite cluster will form which extends
thought the lattice and connects all edges of the lattice. At the thermodynamic limit,
the size of the cluster is very large compared to the size of the· bond. The
macromolecules dissolved in the solution establish increasing number of links in
course of the chemical reaction. Macromolecules are linked within clusters. It
appears that at a certain stage of the process a dramatic change to the connectivity
will occur, called as percolation threshold or gel point. Pierre-Gilles de Gennes
(1979) suggested the percolation transition is equivalent to a second order phase
transition or critical phase transition, where temperature plays a major role as the
amount of the reacted bond formation. The fraction of the reacted bonds, at infmite
clusters be pF, called gel fraction and is the order parameter of the transition. p F
follows a power law with an exponent
P> 0
pF -(P - PdP; P
> Pc
(above gel point)
(below gel point)
Below the gel point, i.e., P
< Pc, clusters with fmite molecular weight (Mw) will
formed and it will start diverging as one approaches Pc. As(Pc - P)-70, Mwwill
show a power law with an exponent y > 0
Mw-(P - Pd- Y ; P
31
< Pc
Equilibrimll
modulus
Viscosity
o
gel point
Degree of crosslinking -----;:..
Figure 2.5: Divergence of viscosity is a signature of the incipient gel phase' and
once the gel is realized, the system develops an equilibrium modulus Ge .
In the case of average radius of the clusters with an exponent {) > 0
Rav-(P - pc)-tJ ; P
The critical exponents
fl,
< Pc
y and {) are connected to each other and depend on the
dimension of the space. There are some important features which characterize the
sol-gel transition and' are common to a large number of gelling systems. From
Landau theory, exponents
fl, y and {) are -3 ,
1
4
3
2
1
1
3
2
2
and - for 3D space and - ,1 and
for 2D space respectively [ 11 a].
The Newtonian viscosity 1J below gel point and the relaxed shear modulus G above
the gel point. As expected the Newtonian viscosity will have finite value below the
gel point and start diverging as one approaches (P - Pc)---70 and obey a power law
with an exponent k > 0
32
\
And for shear modulus, exponent t > 0
G-(P - pc)-t ; P > Pc
But all these theoretical predictions are for cross-linked gels (chemical gels).
Almost all the physical gels (secondary bonding) are thermoreversible gels. As the
temperature plays a major role for bonding, all these laws can apply in the case of
thermoreversible gels.
Non-ergodicity in Gels: Pusey-Van Megen(1989)
Molecules are able only to make limited Brownian excursions about fixed average
positions in the case of non-ergodic media, such as glasses or gels. For such media,
the time average function is different from the ensemble average function.
They considered a medium that contains discrete particles (scatters) for which the
instantaneous field amplitude of light scattered by N particles,
I
N
E(k, t) =
bjexp [ik o1j(t)]
j=l
Where, bj is the field amplitude,
k is
the wave vector and i} (t) is the position
vector of the j'h particle.
Non-ergodicity allows,
i}
= Rj + Llj(t) where < Rj >T=< i} >T and < Llj(t) >T= 0
Again intensity function
I(k, t)
= IE(k, t)12 = E(k, t)E*(k, t)
(2.19)
They considered the total scattered field consists of fluctuating and time
independent term where the time average of the total scattered field equals to the
time independent term.
33
Doing ensemble average,
II
N
< I(k, t) >E=
N
i=l j=l
<
bibj exp
[ik . (ri - i})~?E
., ,<
For identical particles, bi, bj = b
I
N
< I(k, t) >E=
<bibjexp
i,j=l
[ik. (ri(O) - i}(0»)] >E= Nl?F(k, 0)
Where , l]2 = ~"b~and
NL... }
N
F(k, 0)
=I
<bibjexp
[ik. (ri(O) - i}(0»)] >E
i,j=l
Again the normalized ensemble average intensity correlation function (ICF),
2 (k
BE
r)
I
= <1(k,O)I(k;r»E
(2.20)
<l(k»~
One can model the non-ergodicity of a medium by satisfying all non-ergodic
conditions so that the total scattered field can be expressed as the sum of the
fluctuating component EF(k, r) and a time independent component Ee(k) which is
given by
E(k, r)
= EF(k, r) + Ee(k)
(2.21)
The ergodicity or non-ergodicity can be well demonstrated by allowing ensemble
average of E(k, r) to be zero. Now the time average of the Eq.(2.21) can be taken
as the usual heterodyne situation in which Gaussian field and constant field are
mixed. The expression will contain homodyne as well as heterodyne contributions,
to the total intensity correlation function. If this idea is connected to the usual
definition of normalized ensemble average ICF, one can obtain,
gfCk, r)
= 1 + y2[f2(k, r) -
f2(k, 00)]
+ 2Y(1- Y)[f(k, r) 34
f(k, (0)]
(2.22)
where Y =
</(k»E
</(k»T
; f(k, r)
.
and f(k, 00) are dynamIc structure factor at any delay
time r and 't~oo. The dynamic structure factor f(k, r) and f(k, oo)can be expressed
. terms 0 f mean square .
. fluctuatlons,
.
2 I'
.
by (fl2
m
mtensIty
(fl'
t IS gIven
2
(k»T
= </<I (k)
>T
2
-
1•
By solving the above quadratic equation, the dynamic structure factor f(k, r) can
write in terms of experimentally measured gfCk, r) and the value found for Y.
f(k, r)
= (Yy- 1) + [g¥(k,T)-yU/U))l/
2
and f(k,
:"I
DO)
(Y-I)
1/2
= -Y+ [l-U/U))
Y
. The reason
is (fl can be estimated experimentally and it represents the reduction of initial
amplitude ofICF compared to the ideal situation. For fully ergodic mediumpl = 1,
and Y=l which leads to f(k, 00)
= 0 and gfCk, r) = 1 + If(k, r)1 2
which is the
Siegert relation. Again by short term expansion of f(k, r), f(k, r) :::; (1 - Dk 2 r
. . . ) and (gfCk, r) - 1)
= (fl(1 -
2DAk 2 r + ....)
where D
= D(q),
+.
DA =
D; are diffusion and apparent diffusion coefficient.
~
.
Panyukov-Rabin (1996)
In 1996, Panyukov and Rabin proposed a comprehensive theory on the statistical
physics of gels [41]. They showed that frozen inhomogeneities in polymer gels can
be treated as the thermal fluctuations in replica space [42] and the seemingly
interactive calculations of frozen disorder can be performed using the usual
methods of equilibrium statistical mechanics. In reality, cross-links playa major
role in the unique properties of gels, such as shape sustainability, swelling
capability, solvent absorbency, elasticity, molecular recognition, etc. By focusing
cross-linking and inhomogeneities, it can be carried out a series of works on,
decomposition of the concentration fluctuations in gels to the dynamic fluctuations
and the static inhomogeneities, the effect off charges on the microphase separation
in polymer gels, and the development of the time resolved dynamic light scattering
method which allows determination of gelation threshold.
Concentration
fluctuations in polymer gels are composed of dynamic thermal fluctuations,
originated from Brownian motion of the solvent and/or the solute and frozen
35
inhomogeneities. Particularly, the latter is a characteristic of gels and is due to
cross-linking. The frozen inhomogeneities manifest unique properties in polymer
gels [37].
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38