1. Introduction
In the first-order phase transition, one phase becomes unstable or metastable, and a new stable phase appears owing to a temperature change [
1,
2,
3,
4]. The phase transition dynamics explain how the stable phase is created and grows. The late stage of the phase transition dynamics is visualized through the growth behavior of a small stable-phase domain in the unstable or metastable phase. The growth behavior of the stable-phase domain is described by the motion of the interface between the stable and unstable (or metastable) phases [
4,
5,
6,
7,
8,
9].
By adding cross-linkers to a polymer solution, a polymer network is formed, and the polymer solution is transformed into a gel [
10,
11,
12,
13,
14]. Let us pay attention to physical gelation. Physical gelation is caused by the destabilization of the sol phase and the stabilization of the gel phase of the polymer solution due to the cross-linkers. If the temperature change is replaced by the addition of a cross-linker, the gelation can be viewed as a first-order phase transition. This idea leads to the expectation that the gelation process can be analyzed by focusing on the motion of the sol-gel interface. In gelation where the cross-linkers are homogeneously mixed with the polymer solution, the sol-gel interface cannot be clearly observed. Hence, it is not possible to use analysis methods that focus on the interface motion. The heterogeneous mixing of the polymer solution and the cross-linker solution through liquid-liquid contact [
15] leads to gelation with a distinct sol-gel interface. Therefore, the dynamics of such gelation through the liquid-liquid contact between the cross-linker solution and the polymer solution can be analyzed by investigating the motion of the interface [
16,
17].
Gelation through the liquid-liquid contact between a cross-linker solution and a polymer solution has been attempted with various combinations of cross-linker and polymer solutions [
15,
16,
17,
18,
19,
20,
21,
22,
23,
24,
25,
26,
27,
28,
29,
30,
31,
32,
33,
34,
35,
36,
37,
38,
39]. First, the gelation dynamics of a curdlan solution in contact with a CaCl
2 solution [
17] and then that of DNA solutions in contact with CoCl
2 [
26] and AlCl
3 [
28] solutions were investigated. The dynamics of gelation of alginate [
27] and carboxymethylcellulose [
29] solutions by ionic cross-linking were also observed. A distinct sol-gel interface appears in these gelations, and the motion of the interface is well described by a simple theory called the moving boundary (MB) picture [
17,
35], which is based on non-equilibrium thermodynamics. In the MB picture, it is assumed that the sol state becomes unstable with the influx of the cross-linker, and the polymer solution immediately gels upon the influx of the cross-linker. The MB picture shows that gelation proceeds in a diffusion-limited process; therefore, the distance between the liquid-liquid contact surface and the sol-gel interface,
(gel thickness), is proportional to the square root of the elapsed time
from the start of liquid-liquid contact in the early stage. The gelation dynamics of simple systems, in which polymers are directly cross-linked by divalent metal ions, are well explained by the MB picture and expressed as
in the early stage.
Gelators do not necessarily always directly cross-link polymer chains, such as divalent metal ions. Chitosan solution gelates upon a change in pH caused by its contact with NaOH solution. In this case, the solution with high pH is the gelator [
40,
41,
42]. However, in the gelation of chitosan solution, not only the influx of sodium ions but also the outflow of acetic acid from the chitosan solution must be considered [
40,
41].
Blood coagulation is regarded as a gelation process caused by contact between blood and blood coagulation factors (initiators) [
38,
42,
43,
44,
45,
46,
47]. The gelation of blood plasma was analyzed from the viewpoint of the gelation induced by the liquid-liquid contact. Blood gelation is a complex phenomenon involving not only diffusion but also a cascade of enzymatic reactions. The time development of the sol-gel interface in the plasma gel growth induced by the liquid-liquid contact is the result of complex processes [
48]. In the gelation induced by the liquid-liquid contact, the complex processes are summed up in the dynamics of gel growth. For the plasma gelation in a rectangular cell, the linear gel growth behavior
in the early stage was observed. The crossover from
in the early stage to
in the late stage was also observed [
47].
To explain the crossover phenomenon theoretically, Dobashi and Yamamoto [
38,
47] introduced the Landau free energy [
49] for plasma as a function of the degree of gelation. They considered that the state of plasma is changed by the inflowing gelator, and the change of the state makes the sol phase metastable and the gel phase stable. The change of the state by the inflowing gelator was called activation. They expressed the activation by the change of the functional form of the Landau free energy. The transition from the metastable sol phase to the stable gel phase in the activated plasma was described by the Ginzburg–Landau (GL) equation [
50,
51] based on the Landau free energy. In their proposed theory, the gelation of plasma is described as a sequential process consisting of the activation induced by the inflowing gelator and the subsequent relaxation induced by the free energy difference between the sol structure and the gel structure of the activated plasma. In the early stage of gelation, the relaxation process induced by the free energy difference is the rate-limiting process. The free-energy-limited process gives the gel growth behavior expressed as
. In the late stage, the activation induced by the gelator diffusion is the rate-limiting process. The diffusion-limited process gives the gel growth behavior
. Hence, in their theory, the crossover behavior is due to the change in the rate-limiting process. Their theory also shows that the gel growth behavior in the early stage provides information on the thermodynamic properties of the activated plasma and that in the late stage provides that on the diffusion properties of the gelator, independently.
Their theory is a general theory, including the MB picture as its special case. This means that we can expect to find the crossover behavior in gelation processes other than plasma gelation if the experimental results of gel formation induced by the liquid-liquid contact are carefully analyzed. However, the small gel thickness in the early stage makes it difficult to accurately measure the gelation dynamics.
The scaling law was first discovered in the analysis of cardran gel growth and holds without exception in diffusion-limited liquid-liquid contact gelation [
17]. The scaling law is explained as follows. Let the polymer solution be sealed in the cylindrical cell with the base radius,
and the cell be immersed in a gelator solution. In the diffusion-limited gel growth, the time development of gel thickness can be expressed by a radius-independent function in terms of the scaled gel thickness,
and the scaled elapsed time,
. The scaling law holds since there is no length scale characterizing the system other than the radius,
. In the free-energy-limited growth, the scaling by the radius does not hold since there is a characteristic length scale due to the free energy. In the gel growth dynamics, where the crossover occurs, the scaling law does not hold in the early stage, but it does in the late stage. In the present article, from the viewpoint of scaling, we analyze the crossover phenomenon of rate-limiting processes of the liquid-liquid contact induced gelation.
2. Theoretical Model
To analyze the scaling in the gelation, let us consider the polymer solution sealed in the cylindrical cell with the base radius,
and height,
, as shown in
Figure 1. The side of the cylindrical cell is made of a dialysis membrane. The cell is immersed in a gelator solution. Gelators can flow into the polymer solution in the cell through the membrane from the side of the cell. In the present article, we focus on the relationship between gelation dynamics and the characteristic length, i.e., the radius,
.
We choose the
–
plane so that a basal plane is located on it, and the center of the basal plane coincides with the origin (see
Figure 2). The unit vectors along the
- and
-directions are respectively denoted by
and
. For convenience, we choose the polar coordinate
, where
is the angle between the
-axis and the position vector
, and
is the distance from the origin (
). The unit vector along the radial direction is given by
.
The polymer solution in the cylindrical cell is gelled by the inflowing gelator from the gelator solution. Let us assume that the gelation consists of the following two processes occurring in sequence [
38]. From now on, let this idea be called “the sequential picture”.
Process I: The gelators bind to the gelation points of polymer chains in the polymer solution and “activate” the gelation points of polymer chains.
Process II: The polymer chains with activated gelation points bind together to form a gel.
For Process I, we make the following two assumptions regarding the flow of the gelator and the activation of the polymer solution by the gelator [
17].
Assumption I: The gelators flowing into the nonactivated polymer solution instantly activate the polymer chains at the inflowing point, and all of the inflowing gelators are consumed to activate the polymer solutions.
Assumption II: The activated polymer solution does not capture the inflowing gelators.
Assumption II indicates that Assumption I also requires that no nonactivated polymer chains exist in the polymer solution activated by the inflowing gelators. Assumption I ensures that the boundary between the nonactivated polymer solution and the activated polymer solution is macroscopically distinct and that the activation process of the polymer solution can be visualized by tracking the motion of the boundary. Let us call the boundary the activation front. From the symmetry of the system, it can be observed that the activation front forms a circular pattern whose center is the origin of the
–
plane. The distance of the activation front from the center at the immersion time
is denoted by
; the polymer solution in the outer region
is activated, and that in the inner region
is not activated. The distance
between the activation front and the edge of the cylindrical cell is given by
The growth of the activated region is expressed as the time development of the activation front,
.
The gelation dynamics induced by Process II are expressed as a relaxation behavior from high- to low-free energy states. To describe the thermodynamic state of the polymer solution in the cylindrical cell, the order parameter for the degree of gelation
is introduced such that the polymer solution is a gel for
and a sol for
. We assume that the free energy per unit volume of the polymer solution at a homogeneous state at
is given by
where
is a positive constant and
depends on whether the polymer solution is in the activated state or not. The parameter
takes either a large value of
or a small value of
(
);
Let the local free energy function
have only the minimum value at
in the nonactivated state and have two minima at
and
and one maximum at
in the activated state, where
(see
Figure 3). Therefore,
should be larger than
. In the activated state, the local free energy function
is required to have two minima at
and
and one maximum at
, where
, as shown in
Figure 3b. From this requirement,
should satisfy the following inequality:
When the condition (4) is satisfied, the following expressions are obtained:
where
and
For the one-dimensional system discussed previously [
38], Process II is described by the change in
from 0 to
caused by only the free energy difference
. Therefore, gelation does not proceed when
. For the two-dimensional cylindrical system discussed in the present article, the interface free energy between the sol and gel layers, as well as the free energy difference, also drives gelation. Therefore, even if
, gelation proceeds. The effect of interface free energy on gelation is a characteristic of two- and three-dimensional gelation processes induced by the liquid-liquid contact process.
In the region
, the polymer solution is in a nonactivated state. Therefore
. On the other hand, in the region
,
. These considerations mean that the function form of the polymer solution free energy
depends on the position,
, and the degree of gelation
must be considered as a function of the position
. Therefore, as the total free energy of the polymer solution per unit height, we introduce the following functional:
where
and
is a small positive number.
The gelation process given by Process II is regarded as the equilibration of the degree of gelation from the sol state
to the gel state
. The GL equation well describes such equilibration processes [
49,
50,
51]. Therefore, we adopt the following GL equation to describe the dynamics of the equilibration process:
where
is a positive constant called the kinetic coefficient.
For the symmetry of the system, the solution of Equation (10) is a function of the distance from the origin,
and the immersion time
;
. As in the one-dimensional system, a kink-type solution expressing a stationary gel growth is expected:
Therefore, the sol-gel boundary is given by
. Then, the gel thickness,
expressing the gel growth behavior is given by
4. Discussion: Crossover and Scaling
The time development of the scaled gel front,
is expressed as Equation (45) when the following inequality is satisfied:
where the time development of the scaled activation front,
is given by Equation (26). In the early stage
, Equations (28) and (45), respectively, give the initial behaviors for
and
as
and
. Therefore, the inequality (51) is satisfied since
in the early stage. Since the scaled velocity of the scaled gel front,
exceeds that of the scaled activation front,
as time elapses, the gel front could catch up with the activation front [
47]. The gel front must move with the activation front after the gel front catches up with the activation front. Hence, the gel front motion changes at which the gel front catches up with the activation front from the free-energy-limited motion derived from the GL Equation (10) to the diffusion-limited motion dominated by gelator diffusion; the crossover behavior of the gel front motion appears [
47].
Let us discuss the crossover behavior in the case of
. The crossover behavior appears when the two curves
and
on the
plane cross in the region
. Since
and
for small
, we have the inequality
near
. Therefore, the two curves cross if
near
. Hence, the condition under which the crossover occurs is
and the condition is written as
with
The “crossover time,”
at which the gel front motion changes are obtained from the following simultaneous equations with respect to the scaled crossover time
and the scaled gel thickness,
at the crossover time:
The motion change is expressed by the following change of the function form expressing the motion:
In the early part of gelation,
, the gel grows in the free-energy-limited process, and the growth behavior is expressed by the function
. In the latter part of gelation
, the gel grows in the diffusion-limited process, and the growth behavior is expressed by the function
.
The function
is independent of the radius
. Therefore, in terms of the scaled variables given by Equation (43), the gel growth curve is independent of the radius in the diffusion-limited growth time region. In contrast, in the free-energy-limited growth time region, the curve depends on the radius since the function
depends on the radius. The
–
curves for different radii are initially different curves depending on the radius but converge to a single curve in the late stage, as shown in
Figure 4. The change from the radius-dependent gel-growth curve to the radius-independent gel-growth curve characterizes the crossover behavior from the free-energy-limited growth to the diffusion-limited growth and facilitates the experimental observation of the crossover behavior.
When , the quantity is regarded as a characteristic length attributable to the free energy difference between the sol and gel phases. In the free-energy-limited growth, there are two characteristic lengths and . Therefore, the growth behavior cannot be scaled by the radius . In the diffusion-limited growth, however, the radius is the only characteristic length scale. Hence, the gel growth behavior scaled by the radius is described by the radius-independent function .
When the free-energy-limited growth is slow, the diffusion-limited growth process does not appear. In this case, the activation front reaches the center of the cylindrical cell before the gel front catches up with the activated front, and the gel growth proceeds only through the free-energy-limited process. The condition under which the diffusion-limited growth does not appear is
. The condition is satisfied not only when the free-energy-limited growth is slow but also when
is small. Therefore, for cells with small radii, the entire gel growth process is free energy-limited (the orange solid curve in
Figure 4).
The properties of the activated polymer solution can be investigated in terms of the radius dependence of the gel growth rate. Equation (38) shows that in the early stage, the rate of increase in gel thickness is independent of elapsed time and is a function of the radius
as follows.
By measuring the rate of increase in gel thickness in cells with different radii, we obtain the two parameters, and , characterizing the activated polymer solution. Hence, by the scaling analysis, all the parameters , and that determine the progress of gelation are obtained. This means that the gelation progression can be controlled.
Next, the case of is discussed. In this case, gelation does not proceed spontaneously, even if the polymer solution is activated by the influx of the gelator. The gel film of macroscopically negligible thickness on the dialysis membrane, which is necessary for the initial condition of the gelation dynamics, does not form spontaneously. The gel film must be formed on the dialysis membrane in advance.
In this case, the characteristic length attributable to the free energy difference disappears, and the cell radius
is the only characteristic length in the free-energy-limited growth. The free-energy-limited growth behavior is expressed by
. The function
has no parameters characterizing the system at all, not just the radius
. The coefficient
is the only parameter characterizing the free-energy-limited growth. The crossover condition is independent of the radius
and is expressed as
The scaled crossover time
and the scaled crossover thickness
are also independent of the radius
. An example of an
–
curve is shown in
Figure 5. The gel growth curve is invariant to the scale transformation Equation (43). Therefore, we cannot find any crossover from the
-dependence of the gel growth curve.
Even if is not exactly zero but is a sufficiently small positive value, i.e., when , the function can be regarded as the function . Hence, when is sufficiently large if the crossover appears, the gel growth behavior is practically scale-transformation-invariant. It would take time for a thin gel film necessary for the initial condition of the gel dynamics to form on the dialysis membrane. Hence, a lag time would be observed before gel growth begins.
When , is negative. For a negative , the time development equation in the early stage Equation (56) still holds. However, for the equation to be meaningful as the equation for gel growth, the radius should be smaller than , and a gel layer must be formed previously on the dialysis membrane as the initial condition. In this case, the gel phase is metastable, not stable. Gelation is driven by interface free energy, and the free energy difference between the sol and gel phases rather inhibits gel growth. The crossover condition is given by Equation (52) with Equation (53) for a negative .
Finally, let us consider the case where the entire gel growth process is diffusion-limited from the viewpoint of the sequential picture. The first idea is that the observation of the free-energy-limited growth in the early stage is missed because it appears only for a very short time. In fact, it is difficult to accurately measure the gel thickness in the early stages of gelation. It would be difficult to establish that the gel growth is free-energy-limited on the basis of only the data measured during a short period of time in the early stages of gelation. However, the scaling-based analysis proposed in the present article may enable the finding of the short-time free-energy-limited gel growth process. Even in the case of cross-linking by multivalent metal ions, the crossover phenomenon may be observed.
One of the other possible scenarios is when there is no maximum in the free energy of the activated polymer solution and the sol state is unstable, and the gel state is the only stable state. In this case, the equilibration process expressing Process II is given by
with the initial condition
. In the above,
is a positive time constant and
From Equation (58), the gel front is obtained as
Since the motion of the gel front follows that of the activation front except for a delay of only a short relaxation time
, the gel proceeds in the diffusion-limited process; in this scenario, the crossover phenomenon does not appear.