Self-consistent field theoretic simulations of amphiphilic triblock copolymer solutions: Polymer concentration and chain length effects
Using the self-consistent field lattice model, polymer concentration φP and chain length N (keeping the length ratio of hydrophobic to hydrophilic blocks constant) the effects on temperature-dependent behavior of micelles are studied, in amphiphilic symmetric ABA triblock copolymer solutions. When c...
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| Cite this: | Self-consistent field theoretic simulations of amphiphilic triblock copolymer solutions: Polymer concentration and chain length effects / X.-G. Han, Y.-H. Ma // Condensed Matter Physics. — 2014. — Т. 17, № 2. — С. 23604:1-10. — Бібліогр.: 32 назв. — англ. |
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Han, X.-G. Ma, Y.-H. 2019-06-14T10:38:48Z 2019-06-14T10:38:48Z 2014 Self-consistent field theoretic simulations of amphiphilic triblock copolymer solutions: Polymer concentration and chain length effects / X.-G. Han, Y.-H. Ma // Condensed Matter Physics. — 2014. — Т. 17, № 2. — С. 23604:1-10. — Бібліогр.: 32 назв. — англ. 1607-324X arXiv:1407.2382 DOI:10.5488/CMP.17.23604 PACS: 61.25.Hp, 64.75.+g, 82.60.Fa https://nasplib.isofts.kiev.ua/handle/123456789/153498 Using the self-consistent field lattice model, polymer concentration φP and chain length N (keeping the length ratio of hydrophobic to hydrophilic blocks constant) the effects on temperature-dependent behavior of micelles are studied, in amphiphilic symmetric ABA triblock copolymer solutions. When chain length is increased, at fixed φP, micelles occur at higher temperature. The variations of average volume fraction of stickers φcos and the lattice site numbers Ncols at the micellar cores with temperature are dependent on N and φP, which demonstrates that the aggregation of micelles depends on N and φP. Moreover, when φP is increased, firstly a peak appears on the curve of specific heat CV for unimer-micelle transition, and then in addition a primary peak, the secondary peak, which results from the remicellization, is observed on the curve of CV. For a long chain, in intermediate and high concentration regimes, the shape of specific heat peak markedly changes, and the peak tends to be a more broad peak. Finally, the aggregation behavior of micelles is explained by the aggregation way of amphiphilic triblock copolymer. The obtained results are helpful in understanding the micellar aggregation process. Використовуючи самоузгоджену польову ґраткову модель, вивчаються ефекти концентрацiї φ¯ P i довжини ланцюга N полiмера (при фiксованому вiдношеннi довжин гiдрофобних i гiдрофiльних блокiв) на температуро-залежну поведiнку мiцел в амфiфiльних симетричних ABA триблочних кополiмерних розчинах. Якщо довжина ланцюга зростає, при фiксованому φ¯ P , мiцели утворюються при вищiй температурi. Змiна середньої об’ємної долi стикерiв φ¯s co та числа вузлiв ґратки N ls co з температурою при мiцелярних корах залежать вiд N i φ¯ P , що вказує на залежнiсть агрегацiї мiцел вiд N i φ¯ P . Крiм того, якщо φ¯ P зростає, спочатку пiк виникає на кривiй питомої теплоємностi CV для переходу мономер-мiцела, i потiм додатково спостарiгаються на кривiй CV основний пiк, вторинний пiк, якi є результатом ремiцелiзацiї. Для довгого ланцюга при режимах промiжних i високих концентрацiй значно змiнюється форма пiку питомої теплоємностi, i пiк прямує до ширшого пiку. Накiнець, агрегацiйна поведiнка мiцел пояснюється способом агрегацiї амфiфiльного триблочного кополiмера. Отриманi результати є корисними для розумiння процесу агрегацiї мiцел. This research is financially supported by the National Nature Science Foundations of China (11147132) and the Inner Mongolia municipality (2012MS0112), and the Innovative Foundation of Inner Mongolia University of Science and Technology (2011NCL018). en Інститут фізики конденсованих систем НАН України Condensed Matter Physics Self-consistent field theoretic simulations of amphiphilic triblock copolymer solutions: Polymer concentration and chain length effects Самоузгоджене теоретико-польове моделювання амфiфiльних триблочних кополiмерних розчинiв: ефекти концентрацiї та довжини ланцюга полiмерiв Article published earlier |
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Self-consistent field theoretic simulations of amphiphilic triblock copolymer solutions: Polymer concentration and chain length effects |
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Self-consistent field theoretic simulations of amphiphilic triblock copolymer solutions: Polymer concentration and chain length effects Han, X.-G. Ma, Y.-H. |
| title_short |
Self-consistent field theoretic simulations of amphiphilic triblock copolymer solutions: Polymer concentration and chain length effects |
| title_full |
Self-consistent field theoretic simulations of amphiphilic triblock copolymer solutions: Polymer concentration and chain length effects |
| title_fullStr |
Self-consistent field theoretic simulations of amphiphilic triblock copolymer solutions: Polymer concentration and chain length effects |
| title_full_unstemmed |
Self-consistent field theoretic simulations of amphiphilic triblock copolymer solutions: Polymer concentration and chain length effects |
| title_sort |
self-consistent field theoretic simulations of amphiphilic triblock copolymer solutions: polymer concentration and chain length effects |
| author |
Han, X.-G. Ma, Y.-H. |
| author_facet |
Han, X.-G. Ma, Y.-H. |
| publishDate |
2014 |
| language |
English |
| container_title |
Condensed Matter Physics |
| publisher |
Інститут фізики конденсованих систем НАН України |
| format |
Article |
| title_alt |
Самоузгоджене теоретико-польове моделювання амфiфiльних триблочних кополiмерних розчинiв: ефекти концентрацiї та довжини ланцюга полiмерiв |
| description |
Using the self-consistent field lattice model, polymer concentration φP and chain length N (keeping the length ratio of hydrophobic to hydrophilic blocks constant) the effects on temperature-dependent behavior of micelles are studied, in amphiphilic symmetric ABA triblock copolymer solutions. When chain length is increased, at fixed φP, micelles occur at higher temperature. The variations of average volume fraction of stickers φcos and the lattice site numbers Ncols at the micellar cores with temperature are dependent on N and φP, which demonstrates that the aggregation of micelles depends on N and φP. Moreover, when φP is increased, firstly a peak appears on the curve of specific heat CV for unimer-micelle transition, and then in addition a primary peak, the secondary peak, which results from the remicellization, is observed on the curve of CV. For a long chain, in intermediate and high concentration regimes, the shape of specific heat peak markedly changes, and the peak tends to be a more broad peak. Finally, the aggregation behavior of micelles is explained by the aggregation way of amphiphilic triblock copolymer. The obtained results are helpful in understanding the micellar aggregation process.
Використовуючи самоузгоджену польову ґраткову модель, вивчаються ефекти концентрацiї φ¯
P i довжини ланцюга N полiмера (при фiксованому вiдношеннi довжин гiдрофобних i гiдрофiльних блокiв) на
температуро-залежну поведiнку мiцел в амфiфiльних симетричних ABA триблочних кополiмерних розчинах. Якщо довжина ланцюга зростає, при фiксованому φ¯
P , мiцели утворюються при вищiй температурi.
Змiна середньої об’ємної долi стикерiв φ¯s
co та числа вузлiв ґратки N
ls
co з температурою при мiцелярних корах залежать вiд N i φ¯
P , що вказує на залежнiсть агрегацiї мiцел вiд N i φ¯
P . Крiм того, якщо φ¯
P зростає,
спочатку пiк виникає на кривiй питомої теплоємностi CV для переходу мономер-мiцела, i потiм додатково спостарiгаються на кривiй CV основний пiк, вторинний пiк, якi є результатом ремiцелiзацiї. Для
довгого ланцюга при режимах промiжних i високих концентрацiй значно змiнюється форма пiку питомої
теплоємностi, i пiк прямує до ширшого пiку. Накiнець, агрегацiйна поведiнка мiцел пояснюється способом агрегацiї амфiфiльного триблочного кополiмера. Отриманi результати є корисними для розумiння
процесу агрегацiї мiцел.
|
| issn |
1607-324X |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/153498 |
| citation_txt |
Self-consistent field theoretic simulations of amphiphilic triblock copolymer solutions: Polymer concentration and chain length effects / X.-G. Han, Y.-H. Ma // Condensed Matter Physics. — 2014. — Т. 17, № 2. — С. 23604:1-10. — Бібліогр.: 32 назв. — англ. |
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2025-11-26T10:14:02Z |
| last_indexed |
2025-11-26T10:14:02Z |
| _version_ |
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| fulltext |
Condensed Matter Physics, 2014, Vol. 17, No 2, 23604: 1–10
DOI: 10.5488/CMP.17.23604
http://www.icmp.lviv.ua/journal
Self-consistent field theoretic simulations
of amphiphilic triblock copolymer solutions:
Polymer concentration and chain length effects
X.-G. Han1,2∗, Y.-H. Ma1,2
1 Inner Mongolia Key Laboratory for Utilization of Bayan Obo Multi-Metallic Resources:
Elected State Key Laboratory, Inmongolia Science and Technology University, Baotou 014010, China
2 School of Mathematics, Physics and Biology, Inmongolia Science and Technology University,
Baotou 014010, China
Received December 18, 2013, in final form March 28, 2014
Using the self-consistent field lattice model, polymer concentration φ̄P and chain length N (keeping the length
ratio of hydrophobic to hydrophilic blocks constant) the effects on temperature-dependent behavior of micelles
are studied, in amphiphilic symmetric ABA triblock copolymer solutions. When chain length is increased, at
fixed φ̄P , micelles occur at higher temperature. The variations of average volume fraction of stickers φ̄sco andthe lattice site numbers N lsco at the micellar cores with temperature are dependent on N and φ̄P , which demon-strates that the aggregation of micelles depends on N and φ̄P . Moreover, when φ̄P is increased, firstly a peakappears on the curve of specific heat CV for unimer-micelle transition, and then in addition a primary peak,
the secondary peak, which results from the remicellization, is observed on the curve of CV . For a long chain, inintermediate and high concentration regimes, the shape of specific heat peak markedly changes, and the peak
tends to be a more broad peak. Finally, the aggregation behavior of micelles is explained by the aggregation way
of amphiphilic triblock copolymer. The obtained results are helpful in understanding the micellar aggregation
process.
Key words:micelle, self-consistent field, amphiphilic copolymer
PACS: 61.25.Hp, 64.75.+g, 82.60.Fa
1. Introduction
Polymeric micelles constitute a unique class of nanomaterials having a typical core-shell morphol-
ogy. They are formed from amphiphilic block- or graft-copolymers in a selective solvent, where the non-
soluble parts self-assemble to form the core of the micelles and the soluble parts form the solvated shell.
The properties of micelle can be changed by the solution conditions such as concentration, temperature,
and chain architecture. Such self-assembly phenomena of amphiphilic molecules are of principal impor-
tance in many biological and industrial processes. Recently, self-assembled bolaamphiphile nanotubes
have been used as templates to produce metal-coated nanowires [1]. A detailed understanding of the
aggregation process is crucial to understand and eventually control their formation for the related appli-
cations of micelles.
The triblock copolymers, made up of poly(ethyleneoxide) (PEO) and poly(propyleneoxide) (PPO)
blocks, which are experimentally studied as amphiphilic molecules, have been the subject of intense
research over the last two decades due to their unique solution behavior [2, 3]. Furthermore, the ar-
rangement of the PPO and PEO blocks in the chain is the key factor affecting self-aggregation and phase
behavior of these copolymers, which are well documented in literature [3–8]. The temperature induced
aggregation behavior of triblock copolymers in aqueous solutions has received great attention during the
∗xghan0@163.com
© X.-G. Han, Y.-H. Ma, 2014 23604-1
http://dx.doi.org/10.5488/CMP.17.23604
http://www.icmp.lviv.ua/journal
X.-G. Han, Y.-H. Ma
recent decades due to their fundamental and practical importance [3, 5–9]. Compared with experimental
studies, however, related theoretical studies are few, especially to account for the effect of chain archi-
tecture. Han et al. [10] investigated the effects of the length of each hydrophobic end block and polymer
concentration on micellar aggregation in amphiphilic symmetric ABA triblock copolymer solutions. It is
found that the broadness of transition is affected by the length of hydrophobic end blocks (i.e., chain
length). However, in associative polymers [11], when the total length of hydrophilic blocks is decreased,
keeping the chain length constant, the broadness of transition concerned micelles increases. It is an obvi-
ous conclusion drawn that the broadness of the transition changes due to the length ratio of hydrophobic
to hydrophilic blocks. The chain length is an important parameter to understand the thermodynamics
of block copolymers in a selective solvent. In amphiphilic triblock copolymer solutions, however, the ef-
fect of chain length on micellar aggregation behavior has not been clarified so far, the length ratio of
hydrophobic to hydrophilic blocks remaining constant.
As a mesoscopic polymer theory, the self-consistent field theory (SCFT) has its origin from the field
theoretical approach by Edwards [12] and was explicitly adopted to deal with block copolymer structures
by Helfand [13]. In recent years, Matsen and Schick proposed a powerful numerical spectral method that
could be used to deal with complex microphases [14, 15]. This method is accurate enough but requests a
prior knowledge of the symmetry of an ordered structure, which has hindered its application in predict-
ingmicrophases of complex copolymer structures. Subsequently, Drolet and Fredrickson suggested a new
combinatorial screening method [16, 17], which involves a direct implementation of SCFT in real space
in an adaptive arbitrary cell. This method proves to be very successful and can be applied to complex
copolymer melts. It has also been extended to predict the nanostructures of polymer-grafted nanopar-
ticles [18], which have potential applications in the design and synthesis of hierarchical materials. In
addition, SCFT allows us to investigate the aggregate morphology of amphiphilic block copolymers and
their blends in a dilute solution [19–22]. Recently, Matsen extends SCFT to treat diblock copolymers with
nongaussian chain of low molecular weight [23].
A lattice model is introduced to self-consistent mean-field theory to treat microphase separation for
rod-coil block copolymers [24–26]. In our previous papers [10, 11, 27, 28], we have used the SCFT lattice
model to study the phase behavior of physically associating polymer solutions. It is found that chain
architecture and polymer concentration are important factors which affect the property of temperature-
dependent aggregation behavior. Now, in amphiphilic ABA symmetric triblock copolymer solutions, we
study chain length and polymer concentration effects on aggregation behavior. It is found that although
the length ratio of hydrophobic to hydrophilic blocks remains constant, the increase in the aggregation
degree of micelles is also dependent on the chain length, and it is explained by the way of aggregation of
amphiphilic triblock copolymer.
2. Theory
This section briefly describes the self-consistent field theory (SCFT) lattice model for nP amphiphilicsymmetric ABA triblock copolymers which are assumed to be incompressible. Each block molecule con-
sists of Nns nonsticker segments forming the middle B block and Nst sticker segments forming each endA block, distributed over a lattice. At the same time, nh solvent molecules are placed on the vacant latticesites. Polymer monomers and solvent molecules have the same size and each occupies one lattice site.
The total number of lattice sites is NL = nh +nP N . The transfer matrix λ is used to describe the polymer
chain, which depends only on the chain model used. We assume that
λ
αs−αs−1
rs−r ′s−1
=
{
0, αs =αs−1 ,
1/(z −1) , otherwise . (2.1)
Here, r ′ denotes the nearest neighboring site of r . rs and αs denote the position and bond orientation ofthe s-th segment of the copolymer, respectively. α can be any of the allowed bond orientations depend-
ing on the lattice model used. z is the coordination number of the lattice. This means that the chain is
described as a random walk without a possibility of direct backfolding. Although self-intersections of a
chain are not permitted, the excluded volume effect is sufficiently taken into account [29]. Gαs (r, s|1) is
23604-2
Polymer concentration and chain length effects
the end segment distribution function of the s-th segment of the chain. Following the scheme of Schen-
tiens and Leermakers [30], it is evaluated from the following recursive relation:
Gαs (r, s|1) =G(r, s)
∑
r ′s−1
∑
αs−1
λ
αs−αs−1
rs−r ′s−1
Gαs−1 (r ′, s −1|1), (2.2)
whereG(r, s) is the free segment weighting factor and is expressed as
G(r, s) =
{
exp[−ωst(rs )] , s ∈ st,
exp[−ωns(rs )] , s ∈ ns.
The initial condition isGα1 (r,1|1) =G(r,1) for all the values of α1.∑r ′s−1
∑
αs−1 means the summation overall the possible positions and orientations of the (s − 1)-th segment of the chain. Another end segment
distribution functionGαs (r, s|N ) is evaluated from the following recursive relation:
Gαs (r, s|N ) =G(r, s)
∑
r ′s+1
∑
αs+1
λ
αs+1−αs
r ′s+1−rs
Gαs+1 (r ′, s +1|N ), (2.3)
with the initial conditionGαN (r, N |N ) =G(r, N ) for all the values of αN .In this simulation, the free energy in the canonical ensemble F is defined as
F [ω+,ω−]
kBT
=∑
r
{
1
4χ
ω2
−(r )−ω+(r )
}
−nP lnQP [ωst,ωns]−nh lnQh[ωh], (2.4)
where χ is the Flory-Huggins interaction parameter in the solutions, which equals z
2kBT ε, z is the coor-
dination number of the lattice used. Qh is the partition function of a solvent molecule subjected to thefield ωh(r ) = ω+(r ), which is defined as Qh = 1
nh
∑
r exp[− ωh(r )]. QP is the partition function of a non-interaction polymer chain subjected to the fields ωst(r ) =ω+(r )−ω−(r ) and ωns(r ) =ω+(r ), which act on
sticker and nonsticky segments, respectively. QP is expressed as QP = 1
NL
1
z
∑
rN
∑
αN GαN (r, N |1), where
rN and αN denote the position and orientation of the N -th segment of the chain, respectively. ∑rN
∑
αNmeans the summation over all the possible positions and orientations of the N -th segment of the chain.
Minimization of the free energy function F with ω−(r ) and ω+(r ) leads to the following saddle point
equations:
ω−(r ) = 2χφst(r ), (2.5)
φst(r )+φns(r )+φh(r ) = 1, (2.6)
where
φst(r ) = 1
NL
1
z
nP
QP
∑
s∈st
∑
αs
Gαs (r, s|1)Gαs (r, s|N )
G(r, s)
(2.7)
and
φns(r ) = 1
NL
1
z
nP
QP
∑
s∈ns
∑
αs
Gαs (r, s|1)Gαs (r, s|N )
G(r, s)
(2.8)
are the average numbers of sticker and nonsticky segments at r , respectively, andφh(r ) = (1/NL)(nh/ Qh )
×exp[− ωh(r )] is the average numbers of solvent molecules at r .
In our calculations, real space method is implemented to solve the SCFT equations in a cubic lattice
with periodic boundary conditions, which is similar to our previous paper [27]. The configuration from
SCFT equations is taken as a saddle point configuration. By comparing the free energies of the observed
states from different initial fields, a relative stability of the observed morphologies can be obtained.
3. Result and discussion
In our studies, the property of symmetric ABA triblock copolymers is characterized by three tunable
molecular parameters: χ (The Flory-Huggins interaction parameter), N (The chain length of copolymer)
and Nst/Nns (the length ratio of each hydrophobic end block to hydrophilic middle block). In this paper,
23604-3
X.-G. Han, Y.-H. Ma
Figure 1. The phase diagram for the amphiphilic symmetric ABA tribolck copolymers with different
chain length N . The boundary between homogenous solutions and micelle morphology is obtained. The
squares, triangles and diamonds correspond to the boundaries for N = 17,26,34, respectively.
when the chain length is changed, the value of Nst/Nns(' 0.23) remains constant. The aggregation behav-
ior of micelle morphologies is focusedwhen the length of copolymer is changed. Figure 1 shows the phase
diagram of the systems with different chain length N . When χ is increased, the unimer-micelle transition
occurs. At fixed N , the χ value on micellar boundary increases with decreasing φ̄P . When N is increased,
at fixed φ̄P , the χ value on micellar boundary shifts to a small value. It is noted that although the lengthratio of each hydrophobic end block to hydrophilic middle block remains constant, the increase in the
chain length of copolymer is also favorable to the occurrence of micelles in the system.
In order to demonstrate the effects of the chain length N and polymer concentration φ̄P on aggrega-tion of micelles, the variations of the average volume fraction of stickers φ̄sco and the lattice site numbers
N lsco at the micellar cores (φsco Ê 0.5) with χr (the χ deviation frommicellar boundary) in various polymerconcentrations, for N = 26 and N = 34, are presented in figure 2 (a) and figure 4 (a), respectively. For
N = 26, at φ̄P = 0.1, φ̄sco, as well as the corresponding N lsco, smoothly rises with χr and then remains con-stant. When χr Ê 0.5, N lsco does not change with χr , and the aggregation degree of micelles strengthensfrom the increase in φ̄sco. When φ̄P is increased the change of φ̄sco with χr is not monotonous. At φ̄P = 0.3
and φ̄P = 0.5, when χr is increased, φ̄sco firstly rises, and then a φ̄sco-lower region occurs in the range of
χr , and φ̄sco finally tends to be constant. The corresponding N lsco firstly rises, and then a jump occurs at theonset of the above lower region. When φ̄P = 0.8, φ̄sco always goes up with χr , going with the slight surgeof N lsco. It is seen that, at intermediate concentrations, when χr Ê 1.1, micelles dissolve and remicellize,
which is demonstrated by a decrease of the average volume fraction of stickers at micellar core with in-
creasing from χr = 1.3 to χr = 1.4 (see figure 3). This can accelerate the further aggregation of micelles.
At high concentrations, the behavior of micellar dissolution and remicellization is restrained. Only a few
micelles dissolve to strengthen the aggregation degree of micelles. It is shown that the way of an increase
in aggregation degree of micelles depends on polymer concentration. It is noted that when χr Ê 0.6, φ̄scoat fixed χr decreases with an increase in φ̄P for N = 26.
ForN = 34 (see figure 4 (a)), when polymer concentration (φ̄P = 0.1 and 0.3) is not high, the tendencies
of φ̄sco and N lsco to χr are similar to those of N = 26. At intermediate and high polymer concentrations,
they are different from those ofN = 26. When φ̄P = 0.5, φ̄sco andN lsco always smoothly increase with χr . At
φ̄P = 0.8, φ̄sco always smoothly increase with χr , butN lsco goes down slowly with χr . It is demonstrated thatmicelles almost do not dissolve at φ̄P = 0.5. Consequently, the micellar further aggregation is restrained
in a way near the micellar boundary. φ̄s
co at φ̄P = 0.5 is larger than that of φ̄P = 0.8 until χr Ê 1.5, which
is larger from the case of N = 26. It is demonstrated that, at intermediate and high concentrations, the
further aggregation of micelles is markedly affected by the increase in N .
The heat capacity is an important thermodynamic signature to test the occurrence of a phase tran-
sition in a system. The shape of specific heat peak may also be a characteristic of transition. [11, 31]. In
23604-4
Polymer concentration and chain length effects
-0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5
0.6
0.7
0.8
0.9
1.0
0
1000
2000
3000
4000
5000
6000
7000
ls
co
N
r
s
co
(a)
-1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
0.00
0.04
0.08
0.12
0.16
0.20
0.24
V
C
r
(b)
Figure 2. (Color online) The variations of average volume fractions of stickers φ̄st and lattice site num-bers N lsco at the micellar cores in different amphiphilic ABA tribolck copolymers with the χ deviationfrom micellar boundary χr , for various φ̄P at N = 26 is presented in figure 2 (a). The open and solid,
open and solid triangles, open and solid diamonds, and open and solid hexagons denote the φ̄st and
N lsco for φ̄P = 0.8,0.5,0.3,0.1, respectively; The changes of heat capacity for different φ̄P in figure 2 (a)
with χr are shown in figure 2 (b). The squares, triangles, diamonds and hexagons denote the case of
φ̄P = 0.8,0.5,0.3,0.1, respectively.
this work, the heat capacity per site of amphiphilic symmetric ABA triblock copolymers is expressed as
(in the unit of kB):
CV =
(
∂U
∂T
)
NL ,nP
= 1
NL
χ2 ∂
∂χ
(∑
r
φ2st(r )
)
. (3.1)
The CV (χr ) curves for the unimer-micelle transition in various φ̄P at N = 26 and N = 34 are shown in
figure 2 (b) and figure 4 (b), respectively. For unimer-micelle transition, an asymmetric specific heat peak
appears. For N = 26, when φ̄P = 0.1, there is only a peak on CV (χr ) curves. when φ̄P is increased, aprimary and a secondary peaks, are observed as shown in figure 2 (at intermediate and high concentra-
tions). When φ̄P = 0.3 and 0.5, the primary peak is higher than the corresponding secondary peak. When
φ̄P = 0.8, a primary and two secondary peaks occur, and one of them is nearly as high as the primary
peak. The primary and secondary peaks tend to be similar. The occurrence of the secondary peak is ac-
cording to the saltation of the χr curves of the average volume fraction of stickers φ̄sco and the lattice sitenumbers N lsco at the micellar cores. Ignoring the secondary peak, the specific heat peak becomes broadwith increasing φ̄P . For N = 34, when polymer concentration is low, the peak is narrow and similar to the
23604-5
X.-G. Han, Y.-H. Ma
Figure 3. (Color online) The distributions of the numbers of micellar core sites with φs
co at φ̄P = 0.3 and
N = 26. The squares, triangles, diamonds and pentacles correspond to χr = 0.7, 1.3, 1.4, 2.4, respectively.
corresponding case of N = 26. When φ̄P is increased, the peak shape changes and the peak also becomesbroad. With increasing φ̄P , the maximum of CV shifts to a big χr and the curves of CV (χr ) tend to be
not smooth, thus the broad peak seems to be a primary peak. For a long chain, the peak shape changes
markedly at intermediate and high concentrations.
When temperature drops to a certain extent, micelles appear, and with a further decrease in tem-
perature the aggregation degree of micellar cores markedly strengthens. It is indicated above that the
temperature-dependent aggregation behavior of micelles depends on polymer concentration and the
chain length. The micellization of hydrophobic end blocks of triblock copolymer can be considered in
the following ways. One is that both end blocks of each individual molecule could be incorporated into
the same core, the other is that the two hydrophobic ends of triblock copolymer could be incorporated
into two adjacentmicelles. For a short chain, at low concentration, the first micellizationway is dominant.
When polymer concentration is increased, the possibility of the two hydrophobic ends of triblock copoly-
mer to be incorporated into two adjacent micelles will rise markedly. At the same time, the aggregation
degree of micelles on micellar boundary tends to decrease. Therefore, at intermediate concentrations,
the further micellization of triblock copolymers is delayed for a while due to the correlations from chain
connection among micelles. When temperature is decreased to some extent, many micelles will be dis-
solved, and then remicellize. On remicellization, the distribution width of the volume fraction of stickers
at micellar cores increases which is different from the general variation of distribution for volume frac-
tion of stickers at micellar cores with χr . At the same time, the relationship among the micelles becomesstronger, as shown in figure 5. At high concentrations, the correlations among micelles strengthens com-
pared with intermediate concentrations, only a fewmicelles dissolve and some newmicelles form, which
is demonstrated by keeping the tendency of φ̄st(χr ) to increase and the surge of Nl s (χr ) with χr . For along chain, when concentration is low, the case is similar to that of a short chain. When the polymer
concentration is increased, the chain length effect emerges. The two hydrophobic ends of triblock copoly-
mer will be almost absolutely incorporated into two adjacent micelles or small aggregates (φs
co < 0.5 in
figure 3). When φ̄P = 0.5, quite a few small aggregates form. The increase in aggregation degree of mi-
celles is caused by dissolution of small aggregates, and thus the corresponding N lsco does not surge. Athigh concentrations, micelles form easily, and the quantity of small aggregates decreases notably. Thus,
the further aggregation of micelles results from the dissolution of a fewmicelles. In summary, at interme-
diate concentrations for a short chain, the increase in aggregation degree of micelles is by its dissolution
and remicellization. When polymer concentration or chain length is increased, the way of dissolution
and remicellization is restrained, and thus the temperature-dependent aggregation behavior of micelles
is changed.
In the end of paper, the validity of self-consistent field theory for the above results should be illu-
minated. Capture can be an essential feature for the accounted effect of polymer concentration φ̄P and
23604-6
Polymer concentration and chain length effects
-0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5
0.5
0.6
0.7
0.8
0.9
1.0
0
1000
2000
3000
4000
5000
6000
7000
ls
co
N
s
co
r
(a)
-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5
0.00
0.04
0.08
0.12
0.16
0.20
V
C
r
(b)
Figure 4. (Color online) The variations of the average volume fractions of stickers φ̄st and lattice sitenumbers N lsco at the micellar cores in different amphiphilic symmetric ABA tribolck copolymers with the
χ deviation frommicellar boundary χr , for various φ̄P atN = 34 is presented in figure 4 (a). The open and
solid squares, open and solid triangles, open and solid diamonds, and open and solid hexagons denote
the φ̄st and N lsco for φ̄P = 0.8, 0.5, 0.3, 0.1, respectively; The changes of heat capacity for different φ̄P infigure 4 (a) with χr is shown in figure 4 (b). The squares, triangles, diamonds and hexagons denote the
case of φ̄P = 0.8, 0.5, 0.3, 0.1, respectively.
chain length N . The chain accounted in the work should belong to gaussian chain. Using a self-consistent
field lattice model, the phase diagram of coil-coil diblock copolymers for N = 20 in the three dimension
space [24] is consistent with the Matsen-Schick phase diagram. [14], and the results for the solution of
homopolymer length N = 30 in a two dimensional square lattice is also in reasonable agreement with
the theoretical prediction [27]. Furthermore, the effect of relative chain length is also accounted for poly-
mer blends [15]. The above effects from chain length N = 26 and N = 34 should be reasonable. SCFT is
extensively applied to the study of the phase behavior of dilute amphiphilic block copolymer solutions,
and the obtained results at φ̄P = 0.1 have been proved by experimental observations [22]. The specific
heat peak for the transition concerned with micelles is also calculated in physically associating polymer
solutions [27], and the effect of concentration on specific heat peak (not shown) is in reasonable agree-
ment with that of the related system [32]. Therefore, the concentration effect accounted for in the work
by self-consistent field theory is reasonably qualitative.
23604-7
X.-G. Han, Y.-H. Ma
a b
Figure 5. (Color online) The cross sections of the system are presented in figure 5. Figure 5 (a) and (b),
which demonstrate the changes concerned with remicelliztion, correspond to the cases of χr = 1.3 and
χr = 1.4, respectively, for φ̄P = 0.3 and N = 26.
4. Conclusion and summary
Using the self-consistent field lattice model, polymer concentration φ̄P and the chain length N (the
length ratio of hydrophobic to hydrophilic blocks remains constant), the effects on the aggregation be-
havior of micelles are studied in amphiphilic symmetric ABA triblock copolymer solutions. When N is
increased, at fixed φ̄P , micelles occur at a higher temperature. The variations of the average volume frac-tion of stickers φ̄sco and the lattice site numbers N lsco at the micellar cores with temperature depend on
N and φ̄P , which is demonstrated by the change of the specific heat peak. For a short chain, when φ̄P isincreased, firstly a peak appears on the curve of CV for the micellar appearance, and then, in additionto a primary peak, the secondary peak is observed. For a long chain, in intermediate and high concen-
tration regimes, the shape of a specific heat peak changes markedly, and it tends to a broader primary
peak, which is explained by the aggregation way of amphiphilic triblock copolymer. For a short chain, at
intermediate concentrations, the way of two hydrophobic ends of triblock copolymer to be incorporated
into two adjacent micelles is dominant. Therefore, the aggregation degree of the micelles increases by
its dissolution and remicellization. When polymer concentration or chain length is increased, the way of
dissolution and remicellization is restrained, and thus the temperature-dependent aggregation behavior
of the micelles changes.
Acknowledgements
This research isfinancially supported by the National Nature Science Foundations of China (11147132)
and the Inner Mongolia municipality (2012MS0112), and the Innovative Foundation of Inner Mongolia
University of Science and Technology (2011NCL018).
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X.-G. Han, Y.-H. Ma
Самоузгоджене теоретико-польове моделювання
амфiфiльних триблочних кополiмерних розчинiв:
ефекти концентрацiї та довжини ланцюга полiмерiв
К.-Г. Ган, Й.-Г.Ма
Унiверситет науки i технологiй Внутрiшньої Монголiї, Баоту 014010, Китай
Використовуючи самоузгоджену польову ґраткову модель, вивчаються ефекти концентрацiї φ̄P i дов-жини ланцюга N полiмера (при фiксованому вiдношеннi довжин гiдрофобних i гiдрофiльних блокiв) на
температуро-залежну поведiнку мiцел в амфiфiльних симетричних ABA триблочних кополiмерних розчи-
нах. Якщо довжина ланцюга зростає, при фiксованому φ̄P , мiцели утворюються при вищiй температурi.
Змiна середньої об’ємної долi стикерiв φ̄sco та числа вузлiв ґратки N lsco з температурою при мiцелярних ко-
рах залежать вiд N i φ̄P , що вказує на залежнiсть агрегацiї мiцел вiд N i φ̄P . Крiм того, якщо φ̄P зростає,
спочатку пiк виникає на кривiй питомої теплоємностi CV для переходу мономер-мiцела, i потiм дода-
тково спостарiгаються на кривiй CV основний пiк, вторинний пiк, якi є результатом ремiцелiзацiї. Для
довгого ланцюга при режимах промiжних i високих концентрацiй значно змiнюється форма пiку питомої
теплоємностi, i пiк прямує до ширшого пiку. Накiнець, агрегацiйна поведiнка мiцел пояснюється спосо-
бом агрегацiї амфiфiльного триблочного кополiмера. Отриманi результати є корисними для розумiння
процесу агрегацiї мiцел.
Ключовi слова: мiцела, самоузгоджене поле, амфiфiльний кополiмер
23604-10
Introduction
Theory
Result and discussion
Conclusion and summary
|