Transition layer for the heterogeneous Allen-Cahn equation


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Learn more Check out. Citing Literature. Volume 75 , Issue 9 30 July Pages Related Information. Close Figure Viewer. Browse All Figures Return to Figure. Previous Figure Next Figure. Email or Customer ID. Forgot password? Old Password. New Password. Password Changed Successfully Your password has been changed. Returning user. Request Username Can't sign in? The gelation dynamics are expressed using universal scaling equations corresponding to the gelation mechanism, such as gelation induced by the inflow of cross-linkers Case 1 , by solvent exchange Case 2 , by the inflow of catalysts Case 3 , by the exchange of solutes having very different diffusion constants Case 4 and by nucleation at low supersaturation Case 5 see Section 2.

In Section 3 , we discuss the physical meaning of isotropic and anisotropic gelation from the standpoint of phase separation. By fitting data for gelation dynamics whose gelation mechanism is unknown to the universal scaling equation for Cases 1—5, we can identify the type of the gelation of the system. Furthermore, we can determine kinetic coefficients from the fitting parameters.

Transition layer for the heterogeneous Allen-Cahn equation

The second aim of this article is to apply the analytical method to one of the most important biomedical processes, blood coagulation. Although some properties of the blood of patients can be obtained from conventional biochemical tests and the time required for blood coagulation, we have few means of estimating kinetic properties in blood coagulation, which is part of the missing link between blood properties and disorder in blood coagulation. Analysis of the dynamics of blood coagulation by fitting the data to the universal scaling equation enables us to extract information on kinetic coefficients in the process that cannot be obtained by static measurements.

In Section 4 , we describe the application of the analytical method for gelation dynamics to two model systems of blood coagulation to determine several key kinetic coefficients relating to blood coagulation. We propose a generalized model of heterogeneous gelation and show the observed gelation dynamics of chitosan solution [ 30 ] as a typical example in Section 2.

Analysis of Heterogeneous Gelation Dynamics and Their Application to Blood Coagulation

Then we demonstrate a theoretical analysis of the dynamics based on the MB picture in Section 2. In Section 2. The gel growth behaviors from the interface of polymer solutions with various types of gelator solutions have been observed. In Figure 1 we show an illustration of one of the simplest cases, the one-dimensional growth of a gel in a polymer solution cell in contact with a gelator solution bath left-hand side of the cell. This model was proposed for the analysis of gelation dynamics of chitosan solution induced by a change of pH. Figure 2 shows a typical observed time course of the gel layer thickness X t induced at the interface of chitosan solution with NaOH solution [ 25 ].

Chitosan solution is soluble at a low pH and forms a gel at a neutral pH by hydrogen bonding.

Under the geometry shown in Figure 1 , the part of the chitosan solution where the pH changes from a low pH to a neutral pH is transformed to a gel. Here we modify the model shown in Figure 1 and generalize it by including various cases in which A and B have different roles other than those of NaOH solution and chitosan solution, respectively.

For example, the outflow of B may or may not be involved in gelation and gelation can occur with or without the consumption of A in the generalized model. Similar initial behaviors have, however, been observed for different types of gelator solutions under various geometries, whereas the late-stage behaviors were different from each other [ 28 , 29 , 30 ].

The x -axis is chosen to be perpendicular to the contact surface and is oriented in the direction from the immersing gelator solution to the polymer solution.

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The origin of the x -axis is chosen at the contact interface. X t denotes the gel layer thickness at immersion time t. In the gelation of chitosan solution, the gelator solution and polymer solution are NaOH aqueous solution and chitosan in acetic acid aqueous solution, respectively. Time course of gel thickness X observed for one-dimensional growth of chitosan gel induced in 2 wt. The inset shows the X 2 vs. The MB picture proposed by Yamamoto et al. The idea of the MB picture will be briefly explained through the gelation dynamics of chitosan solution.

The dynamics introduced in Section 2. A The sodium ions flow into the sol part and the acetate ions flow out from the sol part to the NaOH solution through the gel layer. The neutralization caused by the flows instantly results in the cross-linking of the inner chitosan solution to produce a new gel layer. B The gel layer does not capture the inflow sodium-ions and the outflow acetate-ions by acting as a sink and the flows change so slowly that they can be considered to be in a steady state; all the gelators inflowing from the gelator solution arrive at the inner polymer solution through the gel layer to realize a steady state.

C The NaOH solution, gel and inner sol are in local thermodynamic equilibrium at the boundaries.

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Therefore, we have. In the above, k A c is the mobility of the acetate-ion. Let a new gel layer thickness d X be produced during period d t by neutralization caused by the sodium ion inflow. Assumption A gives the following relationship:. Note that C 0 is a function of the immersion time t.

Hence, we have the time development equation for the gel thickness:. Integrating the differential Equations 9 and 10 , we have the steady-state flows of sodium ions and acetate ions:. Therefore, the flux of the acetate-ion flow is given by. The acetate ion concentration in the sol part C 0 decreases with increasing immersion time t since acetate ions flow out from the sol part and the time development of the acetate ion concentration is given by.

Using Equations 15 and 16 , we have. This equation indicates square-root behavior in which the gel thickness increases proportionally to the square of the immersion time,. The square-root behavior is a characteristic feature of the diffusion-limited dynamics. The experimental results shown in Figure 2 are analyzed using Equations 21 and The results are plotted according to the equations in Figure 3.

The slope K in is proportional to the NaOH concentration in the immersion solution, inversely proportional to the acetic acid concentration in the chitosan solution and independent of the chitosan concentration, as predicted by Equation 23 [ 30 ] not shown.

Therefore, the time course of the gel thickness is fully explained by the MB picture. In the gelation of chitosan solution, there are three types of characters having different roles. The first character is chitosan molecules, which are the element polymers constituting the gel.

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The second character is sodium ions, which are the gelator and are consumed to produce a gel layer. The third character is acetate ions, which are the gelation inhibitor. Focusing on the gelator and the gelation inhibitor, we classify the gelation system into several types depending on the gelation mechanism.

The simplest case is Case 1, in which cross-linking occurs simply by the inflow of cross-linkers as gelators; the gelation inhibitors are absent and the gelators are consumed to produce a gel layer. In Figure 1 , the gelator A is involved in gelation whereas B is not involved in gelation. In Case 2, gelators flow into the element polymer solution and the gelation inhibitors flow out. Examples of Case 2 are polymers such as chitosan and collagen that undergo gelation via the formation of hydrogen bonds induced by a change in pH resulting from contact with a high pH solution.

When chitosan or collagen in acetic acid solution with a low pH comes in contact with an aqueous solution of NaOH with a high or medium pH, anisotropic gels are prepared [ 15 , 30 ]. The gelator may be a catalyst, as in Case 3, making it necessary to consider the repeated use of the catalyst. In Case 3, although the inhibitors are absent, as in Case 1, the gelators are not consumed in the gelation. An example of Case 3 is the system where gelatin aqueous solution is in contact with transglutaminase aqueous solution [ 34 ].

Note that transient viscoelastic change occurs without any reactions when both solutes in liquid phases have very different diffusion constants [ 35 ]. We define this case as Case 4.


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In Case 4, the inhibitors are absent, as in Case 1.

Transition layer for the heterogeneous Allen-Cahn equation Transition layer for the heterogeneous Allen-Cahn equation
Transition layer for the heterogeneous Allen-Cahn equation Transition layer for the heterogeneous Allen-Cahn equation
Transition layer for the heterogeneous Allen-Cahn equation Transition layer for the heterogeneous Allen-Cahn equation
Transition layer for the heterogeneous Allen-Cahn equation Transition layer for the heterogeneous Allen-Cahn equation
Transition layer for the heterogeneous Allen-Cahn equation Transition layer for the heterogeneous Allen-Cahn equation

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