Numerical investigation of the growth kinetics for macromolecular microsphere composite hydrogel based on the TDGL equation Dating Wu * and Hui Zhang † School of Mathematical Sciences, Beijing Normal University Laboratory of Mathematics and Complex Systems Ministry of Education, Beijing 100875, P. R. China * [email protected]† [email protected]Received 19 June 2016 Accepted 7 October 2016 Published 28 October 2016 We present results of a detailed numerical investigation of the phase separation kinetic process of the macromolecular microsphere composite (MMC) hydrogel. Based on the Flory-Huggins-de Gennes-like reticular free energy, we use the time-dependent Ginzburg–Landau (TDGL) mesoscopic model (called MMC-TDGL model) to simulate the phase separation process. Domain growth is investigated through the pair correlation function. Then we obtain the time- dependent characteristic domain size, which re°ects the growth kinetics of the MMC hydrogel. The results indicate that the growth law based on the MMC-TDGL equation is consistent with the modi¯ed Lifshitz–Slyozov theory. Keywords: Phase transition; MMC hydrogel; MMC-TDGL equation; correlation function; growth rate. 1. Introduction Macromolecular microsphere composite (MMC) hydrogels, a kind of polymeric materials, have attracted some theoretical and experimental studies because of their well-de¯ned network structure. 1–6 The new hydrogel, as proposed in Fig. 1 predominately consists of MMS (macromolecular microsphere), chains and water molecules, which shapes its well-de¯ned structure and high mechanical strength shown in Figs. 2 and 3. But, how is it phase transition and forming these well-de¯ned micro-structure? Why the hydrogels have such high mechanical strengths? what is the structure-property relationship? How do the structural factors a®ect, such as nanoparticle size, grafting density, polymer chain length, entanglement, and so on? Theoretical studies may provide supports for optimizing the synthesis. Now there are many mathematical and physical work to help understand these questions. 4–13 † Corresponding author. Journal of Theoretical and Computational Chemistry Vol. 15, No. 8 (2016) 1650064 (19 pages) # . c World Scienti¯c Publishing Company DOI: 10.1142/S0219633616500644 1650064-1 J. Theor. Comput. Chem. Downloaded from www.worldscientific.com by WSPC on 11/07/16. For personal use only.
Transcript
Numerical investigation of the growth kinetics for macromolecular
microsphere composite hydrogel based on the TDGL equation
Dating Wu* and Hui Zhang†
School of Mathematical Sciences, Beijing Normal UniversityLaboratory of Mathematics and Complex Systems
Ministry of Education, Beijing 100875, P. R. China*[email protected]
We present results of a detailed numerical investigation of the phase separation kinetic process
of the macromolecular microsphere composite (MMC) hydrogel. Based on the Flory-Huggins-deGennes-like reticular free energy, we use the time-dependent Ginzburg–Landau (TDGL)
mesoscopic model (called MMC-TDGL model) to simulate the phase separation process.
Domain growth is investigated through the pair correlation function. Then we obtain the time-
dependent characteristic domain size, which re°ects the growth kinetics of the MMC hydrogel.The results indicate that the growth law based on the MMC-TDGL equation is consistent with
Concerning the above questions, now theoretical results of ordered micro-struc-
tures could be obtained.12 Furthermore, there was a randomly micro-structure
model.14 But up to now, there is hardly numerical results about this model. For the
mechanical strength, G. W. Wei set up some multiscale models.11 Suo et al. present
some results.15–18 The phase transition process for hydrogel was investigated19–23
Fig. 2. Photographs of a MMC hydrogel during the compression test.
Fig. 1. An MMC hydrogel microstructure quoted from Ref. 1.
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and references therein. We presented the phase transition process of the MMC
hydrogel5 based on the time-dependent Ginzburg–Landau (TDGL) mesoscopic
equation, called the MMC-TDGL equation. The key point was to construct the free
energy of the MMC hydrogel. In that work,5 we did not use the traditional Flory–
Huggins free energy for polymer, but constructed a reticular free energy of the MMC
hydrogel. Then we could simulate the phase transition process and understand that
the system shows intermittent phenomenon with increasing reaction temperature,
which is a very good explanation of chemical experiments.
Here we will continuously investigate the growth kinetics for the MMC hydrogel.
The kinetics of phase separation of the MMC hydrogel is distinctive because one
needs to take the collective movements of the chains between molecules into account
in the process. Thus the important objectives in the study of the hydrogel phase
transition are: (1) to investigate universality in the dynamics of phase separation in
general; (2) to discover possible unique characteristics originating from the presence
of chains in the system. So the investigation of kinetics domain growth is deeply
meaningful.
For the self-consistence of this work, we ¯rstly present the reticular free energy
and the MMC-TDGL equation.5 In order to investigate the grow kinetics of the
MMC hydrogel, we have to choose robust and high e±cient numerical scheme to
compute. We design a semi-implicit di®erence scheme, which keeps conservation of
mass and descent of the total energy. So we claim that this scheme keeps some
physical properties reasonable for physical phenomena. Moreover we can rigorously
prove them. Further we numerically verify these properties.
In MMC-TDGL model, we adopt the Flory–Huggins-de Gennes-like reticular free
energy with the Huggins parameter �. As well-known, the Huggins parameter �
denotes the enthalpic interaction, which plays a critic role. So we show the di®erent
property by numerical calculation for di®erent �, such as double-well or one-well
Fig. 3. The stress–stain curves for the NS and MMC hydrogel: full range.
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potential. These determine the number of states of the hydrogel system. During the
process of numerical calculations, the initial state consists of uniform distributions
with a small amplitude random °uctuations at an average value. Since the correla-
tion function is constructed in a statistical sense, we discrete the correlation function
on a spatial grid, and solve them many times then average the solutions. And it is
possible for the initial state to in°uence the growth of the characteristic domain size.
We will compute cases for di®erent initial data. As a conclusion, numerical investi-
gation suggests that the characteristic size RðtÞ of the domain follows a modi¯ed
Lifshitz–Sloyzov growth law,24 i.e. RðtÞ � t13 .
The paper is organized as follows: In Sec. 2, we introduce the Flory-Huggins-de
Gennes-like reticular free energy and the MMC-TDGL equation. In order to make a
quantitative analysis of the domain growth, we de¯ne the correlation function in the
last part of this section. In Sec. 3, we construct the semi-implicit di®erence scheme and
rigorously prove the scheme is descent of energy and conservation of mass. In Sec. 4,
we present our numerical experiments to illustrate the domain growth rate is con-
sistent with the modi¯ed Lifshitz-Slyozov theory. At last we draw the conclusions.
2. Dynamic Model
In this section, we will introduce the Flory-Huggins-de Gennes-like reticular free
energy and the MMC-TDGL equation to describe the phase transition of the MMC
hydrogel for the self-consistence.5 Then we de¯ne the correlation function to quantify
the characteristic domain size of the phase transition of the MMC hydrogel.
2.1. The reticular free energy
Now we denote the concentration of the polymer chains by � in the MMC hydrogel.
The Flory-Huggins-de Gennes-like reticular free energy density of the MMC
hydrogel4,5 as follows
V ð�Þ ¼ �
�ln
��
�þ �
Nln
��
�þ ð1� ��Þlnð1� ��Þ þ ��ð1� ��Þ; ð1Þ
where � is the temperature-dependent Huggins interaction parameter, which re°ects
the change of interaction energy between the polymer chains and water molecules.
The parameters �; �; �; � depend on the micro-parameters N , the degree of the po-
lymerization of the chains, and M, the lattice number of one MMS occupies through
the following relations
� ¼ cN ; c ¼ffiffiffiffiffiffiffiffiM�
p; � ¼ 1þ M
�;
� ¼ �
ffiffiffiffiffiffiM
p ffiffiffi�
p þ N
2
!2
; � ¼ �
c:
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The details can be found.5 The relation of these parameters implies that,
�þ M
��þ ð1� ��Þ ¼ 1:
M� � represents the volume fraction of the MMS, and ð1� ��Þ is the volume fraction
of the solvent molecules. The Ginzburg–Landau free energy functional is of the
form4,5,25
F ð�Þ ¼Z�
�ð�Þjr�j2 þ V ð�Þ� �dr; ð2Þ
where � is the space domain occupied by the system. It should be three dimension.
But we choose two dimension for simple computation in the following. The �ð�Þ is apositive variable coe±cient given by de Genns7:
�ð�Þ ¼ 2
36�ð1� �Þ : ð3Þ
The parameter is the Kuhn segment length of the polymer.
2.2. MMC-TDGL equations
With the free energy F ð�Þ given by Eq. (2), the system evolution with time can be
described by the following equation:
@�
@tþr � jðr; tÞ ¼ 0; ð4Þ
where the di®usion current jðr; tÞ is de¯ned as
jðr; tÞ ¼ �D � r F
�ðr; tÞ ;
among which, F�ðr;tÞ represents the local chemical potential. D is the mobility, a
phenomenological constant. Without loss of generality, we set D ¼ 1 for the
following numerical computation.
We ¯rstly calculate the variational derivative F� from (2), and substitute it
into (4). Then we obtain the MMC-TDGL equation
@�
@¼ �½V 0ð�Þ � �0ð�Þjr�j2 � 2�ð�Þ���: ð5Þ
Now we choose r ¼ ðx; yÞ 2 ½0; d�2(d is a positive constant) and � is equipped with
the periodic boundary conditions, and
V 0ð�Þ ¼ 1
�ln
��
�þ 1
Nln
��
�� �lnð1� ��Þ þ �ð1� 2��Þ þ 1
�þ 1
N; ð6Þ
�0ð�Þ ¼ 2ð2�� 1Þ36�2ð1� �Þ2 : ð7Þ
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Remark 1. There is a thermal noise term in Eq. (4) for a traditional TDGL
equation.4,5 Here the thermal noise has been omitted in the sense that the
characteristic domain size becomes much larger than the thermal correlation length
in the late time of the phase transition. Therefore, thermal noise does not play an
important role in the coarsening process, despite the thermal noise makes the phase
separations occur in the early time.23
2.3. Correlation function
Growth laws of the average domain size have been traditionally analyzed in terms of
the correlation function25
cðr; tÞ ¼ 1
L2
Xr0½�ðrþ r0; tÞ�ðr0; tÞ� < �ð�; tÞ>2�
* +; ð8Þ
here <> represents an average concentration of chains at time t, and hi denotes anaverage over the thermal noise. Supposing that the system is isotropic, then cðr; tÞdepends only on the radial distance, r ¼ jrj: A further circular average leads to the
radial pair correlation function
Cðr; tÞ ¼ 1
Nr
Xjrj¼r
cðr; tÞ; ð9Þ
where Nr is the number of lattice vectors of magnitude r. The correlation function
has two important features as follows:
(1) the self-correlation is largest, that is jCðr; tÞj � Cð0; tÞ;(2) when the distance r grows in¯nity, the radial pair correlation function nearly
disappears, i.e., limr!1;Cðr; tÞ ¼ 0:
Fig. 4. Diagram of spatial discretization.
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In the simulations, taking the square region of the system into account, the number
of lattice vectors of magnitude r, Nr equals to 1 for the diagonal line of the lattice,
and Nr equals to 2 for the nondiagonal line of the lattice, as shown in Fig. 4. The
distance at which Cðr; tÞ ¯rst crosses zero, denoted by RðtÞ, has been used as a
measure of domain size. With the random initial conditions, we solve the equation
many times for each series of initial conditions. Then we obtain the corresponding
correlation function, ¯nally we average the solutions.
Figure 5 shows the typical dynamical evolution of the radial correlation function.
The domain structure is apparent in the oscillations of the function about zero.
3. Semi-Implicit Di®erence Schemes
We will use the Euler method to approximate the time derivative of � °uctuation.
Thus we obtain the time-discrete scheme and semi-implicit di®erence scheme
�nþ1 � �n
4t¼ �½V 0ð�nÞ � �0ð�nÞjr�nj2 � 2�ð�nÞ��nþ1�: ð10Þ
With the space discrete h ¼ d=L and denote
�h�ij ¼�iþ1;j þ �i�1;j þ �i;jþ1 þ �i;j�1 � 4�i;j
h2;
jr�ijj2 ¼1
4h2½ð�iþ1;j � �i�1;jÞ2 þ ð�i;jþ1 � �i;j�1Þ2�;
Fig. 5. Time dependence of the radial correlation function. The blue dotted line corresponds to t ¼ 10;the red solid line corresponds to t ¼ 50.
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we can obtain the complete discrete scheme as follows
�nþ1ij � �n
ij
4t¼ �h½V 0ð�n
ijÞ � �0ð�nijÞjr�n
ijj2 � 2�ð�nijÞ�h�
nþ1ij �; i; j ¼ 1; 2; . . . ;L:
ð11ÞWith periodic boundary condition, we will rigorously prove the scheme (11) keeps
the total energy decreasing and mass conservative in the following.
3.1. Descent of the total energy
Now we will rigorously prove the scheme (10) is satis¯ed that the total energy is
decreasing under this assumption.
Theorem 1. Assume that �nðr; tÞ solve the time-discrete scheme (10) with periodic
condition, Let
F ð�nÞ ¼Z�
½�ð�nÞjr�nj2 þ V ð�nÞ�dr; ð12Þ
then we have
Fð�nþ1Þ � F ð�nÞ þOð4t2Þ: ð13ÞProof. De¯ne the chemical potential
� ¼ V 0ð�nÞ � �0ð�nÞjr�nj2 � 2�ð�nÞ��nþ1: ð14ÞThen Eq. (10) can be written as
�nþ1 � �n
4t¼ ��; ð15Þ
By taking the inner product of (15) with �nþ1, we have
�;�nþ1 � �n
4t
� �¼ �jjr�jj2 � 0: ð16Þ
From the de¯nition of � in (14), we obtain
�;�nþ1 � �n
4t
� �¼ V 0ð�nÞ; �
nþ1 � �n
4t
� �þ �0ð�nÞjr�nj2; �
nþ1 � �n
4t
� �
þ �2�0ð�nÞjr�nj2 � 2�ð�nÞ��nþ1;�nþ1 � �n
4t
� �:¼ A1 þB1 þ C1: ð17Þ
Applying the following Taylor expansions
V ð�nþ1Þ ¼ V ð�nÞ þ V 0ð�nÞð�nþ1 � �nÞ þ V 00ð�Þ2
ð�nþ1 � �nÞ2; ð18Þ�nþ1 ¼ �n þ �tð�; Þ 4 t; ð19Þ
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where � is between �n and �nþ1, is between tn and tnþ1. We have the following
equations for term A1, considering �t, V00 is bounded,
In virtue of the assumption that � is bounded, then �,��t also are bounded, we have
~C2 ¼ � 1
4t
Z�
�ð�nÞð�nþ1 � �nÞð��nþ1 þ��nþ1Þdr
¼ � 1
4t
Z�
�ð�nÞð�nþ1 � �nÞ � ð��nþ1 þ��n þ��tð�; �3Þ 4 tÞdr
¼ � 1
4t
Z�
�ð�nÞð�nþ1 � �nÞ � ð��nþ1 þ��nÞdrþOð4tÞ:
So we obtain
C1 ¼ ~C1 þ ~C2
¼ � 1
4t
Z�
ð�nþ1 � �nÞ½�0ð�nÞr�n � rð�nþ1 þ �nÞ þ �ð�nÞ
��ð�nþ1 þ �nÞ�drþOð4tÞ¼ � 1
4t
Z�
ð�nþ1 � �nÞ � r � ð�ð�nÞrð�nþ1 þ �nÞÞdrþOð4tÞ
¼ 1
4t
Z�
�ð�nÞrð�nþ1 þ �nÞ � rð�nþ1 � �nÞdrþOð4tÞ
¼ 1
4t
Z�
�ð�nÞðjr�nþ1j2 � jr�nj2ÞdrþOð4tÞ: ð25Þ
Thus,
�nþ1;�nþ1 � �n
4t
� �¼ A1 þB1 þ C1
¼ F ð�nþ1Þ � Fð�nÞ4t
þOð4tÞ � 0: ð26Þ
This theorem shows that the total energy is decreasing for this scheme. It is a
physical law of the system.
3.2. Mass conservation
In the MMC-TDGL equations with the periodic condition, the unknown function
�ðr; tÞ is d-periodic with space. Then we have
Theorem 2. Assume that f�nþ1ij : i; j ¼ 0; 1; 2; . . . ;Lg solve the di®erence scheme
(11) and satisfy �nþ1i�L;j ¼ �nþ1
ij and �nþ1i;j�L ¼ �nþ1
ij ; then we have
XL�1
i;i¼0
�nþ1ij ¼
XL�1
i;i¼0
�nij; ð27Þ
where L is the total number of lattice at one side.
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Proof. Summing up the di®erence scheme (11) from i ¼ 0; j ¼ 0 to
i ¼ L� 1; j ¼ L� 1, we obtain,
XL�1
i;j¼0
�nþ1ij � �n
ij
4t¼XL�1
i;j¼0
�h½V 0ð�nijÞ � �0ð�n
ijÞjr�nijj2� ��h
XL�1
i;j¼0
ð�h�nþ1ij Þ:
Firstly, we ¯gure out the following term,
XL�1
i;j¼0
�h�nþ1ij ¼ 1
h2
XL�1
i;j¼0
ð�nþ1iþ1;j þ �nþ1
i�1;j þ �nþ1i;jþ1 þ �nþ1
i;j�1 � 4�nþ1i;j Þ
¼ 1
h2
XL�1
j¼0
XL�1
i¼0
ð�nþ1iþ1;j þ �nþ1
i�1;j � 2�nþ1i;j Þ
þ 1
h2
XL�1
i¼0
XL�1
j¼0
ð�nþ1i;jþ1 þ �nþ1
i;j�1 � 2�nþ1i;j Þ:
Obviously, we have the following property,
XL�1
i¼0
ð�nþ1iþ1;j þ �nþ1
i�1;j � 2�nþ1i;j Þ ¼
XL�1
i¼0
½ð�nþ1iþ1;j � �nþ1
i;j Þ � ð�nþ1i;j � �nþ1
i�1;j�
¼ ð�nþ1L;j � �nþ1
L�1;jÞ � ð�nþ10;j � �nþ1
�1;jÞ ¼ 0:
Similarly, we obtain
XL�1
i;j¼0
�h�nþ1ij ¼ 0;
XL�1
i;j¼0
�h½V 0ð�nijÞ � �0ð�n
ijÞjr�nijj2� ¼ 0:
Thus, we have the following property,
XL�1
i;j¼0
�nþ1ij � �n
ij
4t¼ 0:
This theorem shows that the total mass conserves in this scheme. It is also a
physical law of the system.
4. Numerical Result
In this section, we will present numerical results of the scheme (11). We will verify
the numerical accuracy of the scheme (11), the in°uence of the Huggins parameter �
and initial segment average concentration �0. At last it is investigated the growth
rate of the characteristic domain size. Before calculating the numerical tests, we
simplify the reticular potential V ð�Þ ¯rstly.
V ð�Þ ¼ �
�ln
��
�þ �
Nln
��
�þ ð1� ��Þlnð1� ��Þ þ ��ð1� ��Þ
¼ 1
�ln
�
�þ 1
Nln
�
�
� ��þ 1
�þ 1
N
� ��ln�þ ð1� ��Þlnð1� ��Þ þ ��ð1� ��Þ:
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Noticing that the MMC-TDGL Eq. (5) is a fourth-order equation, so ignoring the
linear term ð1� ln �� þ 1
N ln �� �, we obtain,
V1ð�Þ ¼1
�þ 1
N
� ��ln�þ ð1� ��Þlnð1� ��Þ þ ��ð1� ��Þ;
which is simple and plays the same role in the equation. As proposed in Fig. 6, V1ð�Þhas two minima corresponding to the stable phase. We use the special potential V1ð�Þto substitute V ð�Þ in the whole calculation. We will apply the developed semi-
implicit di®erence scheme (11) to investigate the phase transition process of
the MMC hydrogel. The Eq. (5) is solved on a two-dimensional region � ¼½0; 50� � ½0; 50� with periodic boundary conditions. We set the structural parameters
M ¼ 0:16, N ¼ 4:34. The lattice size is 256� 256, and the time step is taken as
4t ¼ 0:001. The initial condition �0 ¼< �ðr; 0Þ > consists of uniformly distributed
small amplitude °uctuations at an average value in each step.
4.1. Numerical accuracy of the semi-implicit scheme
In order to test the numerical accuracy, we take the numerical solution obtained by
using 4t ¼ 10�5 as the \exact" solution at t ¼ 10. The L1�norm of the error is
shown in Table 1. It is clearly observed that the semi-implicit scheme (11) gives ¯rst-
order accuracy in time.
4.2. In°uence of the Huggins parameter
According to the above analysis and Fig. 6, the reticular potentials have di®erent
height of the well for the di®erent �. We know that the two distinct minima of V1ð�Þ
Fig. 6. The evolution of potential V1ð�Þ with �.
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correspond to the two equilibrium states, where a phase transition is possible. So here
we choose � ¼ 2:0, 2:37 and 2:8, �0 ¼ 0:6. We present three pictures to show the time
evolution of the system in t ¼ 5; 100; 200 for each �. The red area, corresponding to
the larger value of �, means there are enough chains in the system in Figs. 7–9. Here
we can see the phase transition process. Moreover, we also show that the total energy
is decreasing for each case from each sub-¯gure (d) in Figs. 7–9.
Table 1. t ¼ 10, L1-error with di®erent time step.
the growth exponent is close to 1=3. Thus the numerical results are matched to the
modi¯ed Lifshitz–Slyozov theory.24
5. Conclusion
The MMC hydrogel is a kind of polymeric material, which possesses well-de¯ned
network structure and high mechanical strength. So it will have good applications in
new materials designing. Here we hope to provide theoretical studies for optimizing
the synthesis. We introduce Flory-Huggins-de Gennes-like reticular free energy
density for this well-de¯ned network structure MMC hydrogel. Then it can be
obtained the mesoscopic simulation equation, the MMC-TDGL model, which is a
microscale method to simulate the structural evolution of phase-separation in the
MMC hydrogel. This model possesses the physical properties: conversion of mass and
descent of the total energy. This make us design a numerical scheme keeping these
Fig. 12. Domain growth rate � with 128� 128. Up: time evolution of domain growth RðtÞ; Down: time
evolution of domain growth lnRðtÞ.
Fig. 13. Domain growth rate � with 256� 256. Up: time evolution of domain growth RðtÞ; Down: time
evolution of domain growth lnRðtÞ.
Numerical investigation of the growth kinetics
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two properties. Now we do it. Thus we hope a reasonable physical representation can
be obtained. Based on the structural evolution of phase-separation, we analyze the
correlation function, which can label the growth of the average domain size. Nu-
merical results show that the growth law based on the MMC-TDGL equation is
consistent with the modi¯ed Lifshitz–Slyozov theory. This implies that the MMC
hydrogel also grows by the rate t1=3.
Acknowledgment
H. Zhang is partially supported by NSFC/RGC Joint Research Scheme
No. 11261160486, NSFC grant No. 11471046, 11571045 and the Ministry of Edu-
cation Program for New Century Excellent Talents Project NCET-12-0053.
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