Effect of polymer concentration and length of hydrophobic end block on the unimer-micelle transition broadness in amphiphilic ABA symmetric triblock copolymer solutions
The effects of the length of each hydrophobic end block Nst and polymer concentration φP on the transition broadness in amphiphilic ABA symmetric triblock copolymer solutions are studied using the self-consistent field lattice model. When the system is cooled, micelles are observed, i.e.,the homogen...
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| Cite this: | Effect of polymer concentration and length of hydrophobic end block on the unimer-micelle transition broadness in amphiphilic ABA symmetric triblock copolymer solutions / X.-G. Han, Y.-H. Ma, S.-L. Ouyang // Condensed Matter Physics. — 2013. — Т. 16, № 3. — С. 33601:1-8. — Бібліогр.: 22 назв. — англ. |
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Han, X.-G. Ma, Y.-H. Ouyang, S.-L. 2017-06-13T05:29:46Z 2017-06-13T05:29:46Z 2013 Effect of polymer concentration and length of hydrophobic end block on the unimer-micelle transition broadness in amphiphilic ABA symmetric triblock copolymer solutions / X.-G. Han, Y.-H. Ma, S.-L. Ouyang // Condensed Matter Physics. — 2013. — Т. 16, № 3. — С. 33601:1-8. — Бібліогр.: 22 назв. — англ. 1607-324X PACS: 61.25.Hp, 64.75.+g, 82.60.Fa DOI:10.5488/CMP.16.33601 arXiv:1309.6207 https://nasplib.isofts.kiev.ua/handle/123456789/120834 The effects of the length of each hydrophobic end block Nst and polymer concentration φP on the transition broadness in amphiphilic ABA symmetric triblock copolymer solutions are studied using the self-consistent field lattice model. When the system is cooled, micelles are observed, i.e.,the homogenous solution (unimer)-micelle transition occurs. When Nst is increased, at fixed φP, micelles occur at higher temperature, and the temperature-dependent range of micellar aggregation and half-width of specific heat peak for unimer-micelle transition increase monotonously. Compared with associative polymers, it is found that the magnitude of the transition broadness is determined by the ratio of hydrophobic to hydrophilic blocks, instead of chain length. When φP is decreased, given a large Nst, the temperature-dependent range of micellar aggregation and half-width of specific heat peak initially decease, and then remain nearly constant. It is shown that the transition broadness is concerned with the changes of the relative magnitudes of the eductions of nonstickers and solvents from micellar cores. Вплив довжини кожного гiдрофобного прикiнцевого блоку Nst i концентрацiї полiмера φ¯P на ширину переходу в симетричних ABA триблочних амфiфiльних кополiмерних розчинах дослiджується шляхом використання ґраткової моделi самоузгодженого поля. Коли система охолоджена, спостерiгаються мiцели, тобто вiдбувається перехiд однорiдний розчин (мономер)-мiцела. Якщо Nst зростає при сталому φ¯P, то мiцели виникають при високiй температурi, а температурно залежна область агрегацiї мiцел i пiвширина пiку питомої теплоємностi для переходу мономер-мiцела зростають монотонно. Порiвнюючи з асоцiативними полiмерами, знайдено, що величина ширини переходу визначається вiдношенням гiдрофобних блокiв до гiдрофiльних, а не довжиною ланцюга. Коли φ¯P зменшується при великому значеннi Nst, температурно залежна область мiцелярної агрегацiї та пiвширина пiку питомої теплоємностi спо-чатку зменшуються, а потiм залишаються майже сталими. Показано, що ширина переходу пов’язана зi змiною вiдносних величин видiлення незв’язувальної речовини i розчинникiв з мiцелярних корiв. 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 Effect of polymer concentration and length of hydrophobic end block on the unimer-micelle transition broadness in amphiphilic ABA symmetric triblock copolymer solutions Вплив концентрацiї полiмера i довжини гiдрофобного прикiнцевого блоку на ширину переходу мономер-мiцела в ABA симетричних триблочних амфiфiльних кополiмерних розчинах Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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| title |
Effect of polymer concentration and length of hydrophobic end block on the unimer-micelle transition broadness in amphiphilic ABA symmetric triblock copolymer solutions |
| spellingShingle |
Effect of polymer concentration and length of hydrophobic end block on the unimer-micelle transition broadness in amphiphilic ABA symmetric triblock copolymer solutions Han, X.-G. Ma, Y.-H. Ouyang, S.-L. |
| title_short |
Effect of polymer concentration and length of hydrophobic end block on the unimer-micelle transition broadness in amphiphilic ABA symmetric triblock copolymer solutions |
| title_full |
Effect of polymer concentration and length of hydrophobic end block on the unimer-micelle transition broadness in amphiphilic ABA symmetric triblock copolymer solutions |
| title_fullStr |
Effect of polymer concentration and length of hydrophobic end block on the unimer-micelle transition broadness in amphiphilic ABA symmetric triblock copolymer solutions |
| title_full_unstemmed |
Effect of polymer concentration and length of hydrophobic end block on the unimer-micelle transition broadness in amphiphilic ABA symmetric triblock copolymer solutions |
| title_sort |
effect of polymer concentration and length of hydrophobic end block on the unimer-micelle transition broadness in amphiphilic aba symmetric triblock copolymer solutions |
| author |
Han, X.-G. Ma, Y.-H. Ouyang, S.-L. |
| author_facet |
Han, X.-G. Ma, Y.-H. Ouyang, S.-L. |
| publishDate |
2013 |
| language |
English |
| container_title |
Condensed Matter Physics |
| publisher |
Інститут фізики конденсованих систем НАН України |
| format |
Article |
| title_alt |
Вплив концентрацiї полiмера i довжини гiдрофобного прикiнцевого блоку на ширину переходу мономер-мiцела в ABA симетричних триблочних амфiфiльних кополiмерних розчинах |
| description |
The effects of the length of each hydrophobic end block Nst and polymer concentration φP on the transition broadness in amphiphilic ABA symmetric triblock copolymer solutions are studied using the self-consistent field lattice model. When the system is cooled, micelles are observed, i.e.,the homogenous solution (unimer)-micelle transition occurs. When Nst is increased, at fixed φP, micelles occur at higher temperature, and the temperature-dependent range of micellar aggregation and half-width of specific heat peak for unimer-micelle transition increase monotonously. Compared with associative polymers, it is found that the magnitude of the transition broadness is determined by the ratio of hydrophobic to hydrophilic blocks, instead of chain length. When φP is decreased, given a large Nst, the temperature-dependent range of micellar aggregation and half-width of specific heat peak initially decease, and then remain nearly constant. It is shown that the transition broadness is concerned with the changes of the relative magnitudes of the eductions of nonstickers and solvents from micellar cores.
Вплив довжини кожного гiдрофобного прикiнцевого блоку Nst i концентрацiї полiмера φ¯P на ширину переходу в симетричних ABA триблочних амфiфiльних кополiмерних розчинах дослiджується шляхом використання ґраткової моделi самоузгодженого поля. Коли система охолоджена, спостерiгаються мiцели, тобто вiдбувається перехiд однорiдний розчин (мономер)-мiцела. Якщо Nst зростає при сталому φ¯P, то мiцели виникають при високiй температурi, а температурно залежна область агрегацiї мiцел i пiвширина пiку питомої теплоємностi для переходу мономер-мiцела зростають монотонно. Порiвнюючи з асоцiативними полiмерами, знайдено, що величина ширини переходу визначається вiдношенням гiдрофобних блокiв до гiдрофiльних, а не довжиною ланцюга. Коли φ¯P зменшується при великому значеннi Nst, температурно залежна область мiцелярної агрегацiї та пiвширина пiку питомої теплоємностi спо-чатку зменшуються, а потiм залишаються майже сталими. Показано, що ширина переходу пов’язана зi змiною вiдносних величин видiлення незв’язувальної речовини i розчинникiв з мiцелярних корiв.
|
| issn |
1607-324X |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/120834 |
| citation_txt |
Effect of polymer concentration and length of hydrophobic end block on the unimer-micelle transition broadness in amphiphilic ABA symmetric triblock copolymer solutions / X.-G. Han, Y.-H. Ma, S.-L. Ouyang // Condensed Matter Physics. — 2013. — Т. 16, № 3. — С. 33601:1-8. — Бібліогр.: 22 назв. — англ. |
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| fulltext |
Condensed Matter Physics, 2013, Vol. 16, No 3, 33601: 1–8
DOI: 10.5488/CMP.16.33601
http://www.icmp.lviv.ua/journal
Effect of polymer concentration and length of
hydrophobic end block on the unimer-micelle
transition broadness in amphiphilic ABA symmetric
triblock copolymer solutions
X.-G. Han1,2∗, Y.-H. Ma1,2, S.-L. Ouyang2
1 School of Mathematics, Physics and Biological engineering, Inmongolia Science and Technology University,
Baotou 014010, China
2 Key laboratory of Integrated Exploitation of Bayan Obo Multi-Metal Resources, Inmongolia Science and
Technology University, Baotou 014010, China
Received December 7, 2012, in final form March 28, 2013
The effects of the length of each hydrophobic end block Nst and polymer concentration φ̄P on the transition
broadness in amphiphilic ABA symmetric triblock copolymer solutions are studied using the self-consistent
field lattice model. When the system is cooled, micelles are observed, i.e.,the homogenous solution (unimer)-
micelle transition occurs. When Nst is increased, at fixed φ̄P, micelles occur at higher temperature, and the
temperature-dependent range of micellar aggregation and half-width of specific heat peak for unimer-micelle
transition increase monotonously. Compared with associative polymers, it is found that the magnitude of the
transition broadness is determined by the ratio of hydrophobic to hydrophilic blocks, instead of chain length.
When φ̄P is decreased, given a large Nst , the temperature-dependent range of micellar aggregation and half-
width of specific heat peak initially decease, and then remain nearly constant. It is shown that the transition
broadness is concerned with the changes of the relative magnitudes of the eductions of nonstickers and solvents
from micellar cores.
Key words: transition broadness, self-consistent field, amphiphilic copolymer
PACS: 61.25.Hp, 64.75.+g, 82.60.Fa
1. Introduction
Amphiphilic block copolymers are particularly versatile macromolecules because they allow for a
rich variety of different structures. Their length and the number of blocks of each species can be tuned at
will, from di- and triblock to multiblock copolymers. Their architectures can be linear, branched or star-
like, the blocks may be distributed randomly or regularly. Consequently, amphiphilic block copolymers
have a great deal of applications such as drug delivery vectors [1], nanoparticle stabilizers, nanoreser-
voirs, emulsion stabilizers, wetting agents, rheology modifiers [2, 3] or as injectable scaffold materials for
tissue engineering [4].
Amphiphilic copolymers are capable of self-assembling into micelles when temperature drops to a
critical micelle temperature. Below the critical micelle temperature there is an equilibrium region of
a certain width, where significant amounts of both free and associated copolymer molecules coexist.
Above the transition region most copolymer molecules are in micelles. It is verified theoretically [5] and
experimentally [6] that the broad nature should be ascribed to the structural changes which accompany
the replacement of micellar core solvent by polymer. However, the effect of the hydrophobic block on
the structural changes is not clarified so far, which is very important in high polymer concentration
∗E-mail: xghan0@163.com, Phone: 011+086-472-5954303, Fax: 011+086-472-5954303
© X.-G. Han, Y.-H. Ma, S.-L. Ouyang, 2013 33601-1
http://dx.doi.org/10.5488/CMP.16.33601
http://www.icmp.lviv.ua/journal
X.-G. Han, Y.-H. Ma, S.-L. Ouyang
regimes. The study of the effects of polymer concentration and length of hydrophobic end block on the
transition broadness in amphiphilic ABA symmetric triblock copolymer solutions would be quite useful
to understand the thermodynamics of block copolymers in a selective solvent.
The self-consistent field theory (SCFT) has been brought into use as a powerful tool in predicting the
morphologies of complex block copolymers [7–10]. Recently, SCFT has been applied to study the prop-
erties of micelles in polymer solutions [11–13]. In this report, a SCFT lattice model is applied [14–16].
In earlier publications, we have used the SCFT lattice model to study the phase behavior of physically
associating polymer solutions [5, 17, 18]. It is established that the temperature-dependent behavior of
aggregates is affinitive to chain architecture [5], and the effect of polymer concentration is in a way sim-
ilar to that of chain architecture [18]. Now amphiphilic ABA symmetric triblock copolymer solutions are
studied using SCFT lattice model. The focus is made on the effects of the length of hydrophobic end block
and polymer concentration on the broadness of the transition observed. It is found that the magnitude
of the transition broadness is related to the relative changes of the eductions of nonsticker and solvent
from micellar cores.
2. Theory
This section briefly describes the self-consistent field theory (SCFT) lattice model for nP amphiphilic
ABA symmetric triblock copolymers which is assumed to be incompressible. Each triblock molecule is
composed of 2Nst sticker segments forming two hydrophobic end A block and Nns nonsticker segments
forming the hydrophilic middle B block, distributed over a lattice; the degree of polymerization of the
chain is N (= 2Nst + Nns) and the total number of lattice sites is NL. In addition to polymer monomers,
nh solvent molecules are placed on the vacant lattice sites. Stickers, nonsticky monomers and solvent
molecules have the same size and each occupies one lattice site, and thusNL = nh+nPN . Nearest neighbor
pairs of stickers have attractive interaction −ǫ with ǫ> 0, which is the only non-bound interaction in the
present system. The approximation of the attractive interaction energy [17] is expressed as:
U
kBT
=−χ
∑
r
φ̂st(r )φ̂st(r ), (2.1)
where χ is the Flory-Huggins interaction parameter in the solutions, which equals (z/2kBT )ǫ, z is the
coordination number of the lattice used, where
∑
r means the summation over all the lattice sites r and.
φ̂st(r ) =
∑
j
∑
s∈stδr,r j ,s
is the volume fraction of stickers on the site r , where j and s are the indexes of
chain and monomer of a polymer, respectively. s ∈ st means that the sth monomer belongs to the sticker
monomer type. In this simulation, we perform the SCFT calculations in the canonical ensemble, and the
field-theoretic free energy F [17, 19] is defined as
F [ω+,ω−]
kBT
=
∑
r
{
1
4χ
ω2
−(r )−ω+(r )
}
−nP lnQP[ωst,ωns]−nh lnQh[ωh], (2.2)
where Qh is the partition function of a solvent molecule subject to the field ωh(r ) = ω+(r ), which is de-
fined asQh = (1/nh)
∑
r exp[− ωh(r )].QP is the partition function of a noninteraction polymer chain sub-
ject to the fields ωst(r )=ω+(r )−ω−(r ) and ωns(r ) =ω+(r ), which act on sticker and nonsticker segments,
respectively.
Minimizing the free energy function F withω−(r ) andω+(r ) leads to the following saddle point equa-
tions:
ω−(r )= 2χφst(r ), (2.3)
φst(r )+φns(r )+φh(r )= 1, (2.4)
where
φst(r ) =
1
zNL
nP
QP
∑
s∈st
∑
αs
Gαs (r, s|1)Gαs (r, s|N )
G(r, s)
(2.5)
and
φns(r ) =
1
zNL
nP
QP
∑
s∈ns
∑
αs
Gαs (r, s|1)Gαs (r, s|N )
G(r, s)
(2.6)
33601-2
Effect of polymer concentration and length of end block
are the average numbers of sticker and nonsticker segments at r , respectively, and
φh(r )=
1
NL
nh
Qh
exp[−ωh(r )]
is the average number of solvent molecules at r .QP is expressed as
QP =
1
zNL
∑
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
∑
αN
means the summation over all the possible positions and orientations of the N th segment of
the chain, respectively. Gαs (r, s|1) and Gαs (r, s|N ) are the end segment distribution functions of the sth
segment of the chain. G(r, s) is the free segment weighting factor.
Following the scheme of Schentiens and Leermakers [20], Gαs (r, s|1) is the end segment distribution
function of the sth segment of the chain, which 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), (2.7)
where G(r, s) is the free segment weighting factor and is expressed as
G(r, s) =
{
exp[−ωns(rs)] , s ∈ns,
exp[−ωst(rs)] , s ∈ st.
The initial condition is Gα1 (r,1|1) =G(r,1) for all the values of α1. In the above expression, the values of
λ
αs−αs−1
rs−r ′s−1
depend on the chain model used. We assume that
λ
αs−αs−1
rs−r
′
s−1
=
{
0, αs =αs−1 ,
1/(z −1), otherwise .
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 ), (2.8)
with the initial condition GαN (r, N |N ) = G(r, N ) for all the values of αN . In this work, the chain is de-
scribed as a random walk without the possibility of direct backfolding. Although self-intersections of a
chain are not permitted, the excluded volume effect is sufficiently taken into account [21].
The saddle point is calculated using the pseudo-dynamical evolution process [17]. The calculation is
initiated from appropriately random-chosen fieldsω+(r ) andω−(r ), and stopped when the change of free
energy F between two successive iterations is reduced to the needed precision. The resulting configura-
tion is taken as a saddle point one. By comparing the free energies of the saddle point configurations
obtained from different initial fields, the relative stability of the observed morphologies can be assessed.
3. Result and discussion
In our studies, the amphiphilic ABA symmetric triblock copolymers depend on three tunable molecu-
lar parameters: χ (the Flory-Huggins interaction parameter), Nns (the length of hydrophilic middle block,
in this paperNns = 9 ) andNst (the length of each hydrophobic end block). The simulation calculations are
performed in a three-dimensional simple cubic lattice with periodic boundary condition. The results pre-
sented below are obtained from the lattice with NL = 263. The focus is made on the temperature behavior
of micelle morphologies when the length of hydrophobic end block changes.
Figure 1 shows the phase diagram of the systems with different length of each hydrophobic end block
Nst. When χ is increased, micelles are observed as a inhomogenous morphology if Nst = 1 1. The χ value
on micellar boundary increases with decreasing φ̄P. When Nst is increased, at fixed φ̄P, the χ value on
33601-3
X.-G. Han, Y.-H. Ma, S.-L. Ouyang
Figure 1. The phase diagram for the systems with different lengths of each hydrophobic end block Nst .
The boundary between homogenous solutions (blow boundary) and micelle morphology (above bound-
ary) is obtained. The squares, triangles and diamonds correspond to the boundaries for Nst = 4, 2, 1,
respectively.
micellar boundary shifts to a small value. The increase in the length of hydrophobic end block is favorable
to the occurrence of micelles in the system.
The variation of the average number of stickers at the micellar cores denoted by φ̄cost with the χ devia-
tion frommicellar boundary χr is calculated. For different Nst at φ̄P = 0.8 and different φ̄P at Nst = 4, the
curves of φ̄cost (χr ) are shown in figure 2 (a) and (b), respectively. When Nst = 1, as shown in figure 2 (a),
φ̄cost rises and approaches to 1 with an increase in χr . When Nst is increased, the value of φ̄
co
st at fixed χr
decreases, and its temperature-dependent range goes up. For Nst = 4 [see figure 2 (b)], when φ̄P is de-
creased, the value of φ̄cost at fixed χr does monotonously increase only in the χr range near χr = 0. When
χr is increased to some extent, at the middle concentration regimes, φ̄
co
st at fixed χr decreases with a de-
creasing φ̄P. It is shown that the magnitude of the temperature-dependent range of micelle aggregation
does not monotonously change with φ̄P, which is different from the case of changing Nst at fixed φ̄P.
1The structural morphology of MFH morphology [17] occurs at a narrow region of ∆χ= 0.1 neighboring the micellar boundary
when φ̄P > 0.7 (not shown), which is ignored.
Figure 2. The variations of average numbers of stickers at the micellar cores with the χ deviation from
micellar boundary χr , for various lengths of each hydrophobic end block Nst at φ̄P = 0.8 and different
φ̄P at Nst = 4, are presented by figure 2 (a) and (b), respectively. In figure (a), The squares, triangles and
diamonds correspond to Nst = 4, 2, 1, respectively; In figure (b), the squares, triangles, diamonds and
hexagons denote φ̄P = 0.8, 0.4, 0.2, 0.1, respectively.
33601-4
Effect of polymer concentration and length of end block
Figure 3. The changes of specific heat capacity in different amphiphilic ABA symmetric triblock copoly-
mers with the χ deviation frommicellar boundary χr are presented by figure 3 (a) and (b) corresponding
to the systems shown by figure 2 (a) and (b), respectively.
The half-width of a specific heat peak may be an intrinsic measure of transition broadness. [5, 22]. In
this work, the heat capacity per site of amphiphilic ABA symmetric triblock copolymers is expressed as
(in the unit of kB):
CV =
(
∂U
∂T
)
NL ,nP
=
1
NL
χ2 ∂
∂χ
[∑
r
φ2
st(r )
]
. (3.1)
For various Nst at φ̄P = 0.8 and different concentrations at Nst = 4, the CV (χr ) curves of the unimer-
micelle transition are shown in figure 3 (a) and 3 (b), respectively. For unimer-micelle transition, a peak
appears in each CV (χr ) curve. When Nst is increased, as shown in figure 3 (a), the half-width of the
transition peak rises, and the symmetry and the height of the transition peak decrease. The broadness
of unimer-micelle transition increases with increasing the length of hydrophobic end block, which is in
reasonable agreement with that on temperature-dependent range of micellar aggregation. Whereas for
the case of changing φ̄P at Nst = 4, the half-width and height of the transition peak do not monotonously
change with φ̄P. When φ̄P is decreased, the symmetry of the peak always increases, the height of the tran-
sition peak firstly increases, and then decreases. Its half-width initially drops, and then nearly remains
constant with a decreasing φ̄P. In other words, at high concentrations the broadness of unimer-micelle
transition is affected by polymer concentration. In middle and low concentration regimes, the height of
the transition peak is affected by the change of polymer concentration. However, the transition broadness
is almost unrelated to polymer concentration. It is shown that the effect of polymer concentration on the
transition broadness is consistent with that on the temperature-dependent range of micellar aggregation.
It is obvious that the increase in the degree of aggregation at micellar cores results from the educ-
tions of nonsticky monomers and solvents. Micelles appear when temperature drops below the critical
micelle temperature. With a further decrease of temperature, solvents and nonstickers continue to be
expelled from micellar cores, and the degree of aggregation of micellar cores strengthens. Therefore,
the temperature-dependent behavior of micellar aggregation brings about the existence of the transition
broadness, rather than a transition point. The broadness of unimer-micelle transition increases with in-
creasing the length of hydrophobic end block (i.e., the length of chain), which is consistent with the effect
of decreasing the length of hydrophilic middle block between neighboring hydrophobic blocks, at a fixed
chain length, in associative polymer solutions [5]. It is demonstrated that the broadness of the transitions
concerned with micelles is determined by the ratio of hydrophobic to hydrophilic blocks, which is not
related to the length of polymer chain.
Furthermore, the relative magnitude of contributions of nonsticky monomers and solvents to aggre-
gation of micellar cores should be related to micelle structure and the relationship among micelles. At
high concentrations, when the length of hydrophobic end block Nst is increased, the micellar volume
fraction in the system rises and the micelle structure tends to be intricate. These factors result in the
difficulties of the eductions of nonsticky monomers and solvents from micellar cores. Therefore, with an
33601-5
X.-G. Han, Y.-H. Ma, S.-L. Ouyang
Figure 4. The variation of the ratios of the changes of average numbers of nonstickers and solvents to
that of stickers at the micellar cores when the χ deviation frommicellar boundary χr , for various lengths
of each hydrophobic end block Nst at φ̄P = 0.8 and different φ̄P at Nst = 4, is presented by figure 4 (a)
and (b), respectively. The change ∆φcos (χr ) equals φcos (χr )−φcos (χr − 0.1), where s denotes st, ns, so,
respectively. In figure (a), The open and solid squares, triangles and diamonds correspond to the cases
for Nst = 4,2,1, respectively; In figure (b), the open and solid squares, diamonds and hexagons denote the
cases for φ̄P = 0.8, 0.4, 0.1, respectively.
increase in Nst at fixed φ̄P, the temperature-dependent range of aggregation of micellar cores, as well
as the transition broadness, rises. Moreover, when Nst is relatively big, the relationship among micelles
is strong, which markedly hampers the eduction of nostickers, thus the contribution of solvents will be
rather important. As shown in figure 4 (a), when Nst is increased from Nst = 1, given a fixed φ̄P, the con-
tribution of nonstickers to aggregation of stickers goes down and that of solvents rises with an increasing
χr , at the neighborhood of χr > 0. It is noted that the evidently temperature-dependent range of the ra-
tios of the changes of average numbers of nonstickers and solvents to that of stickers at the micellar cores
rises with an increasing Nst. The larger is the evidently temperature-dependent range of the above ra-
tio, the bigger is the transition broadness. It is shown that the magnitude of the transition broadness is
concerned with the changes of the relative magnitudes of the eductions of nonstickers and solvents from
micellar cores.
In high concentrations, when polymer concentration is decreased, for a large Nst, the effect of the
relationship among micelles on eductions of nonstickers and solvents evidently dies down. Therefore,
the transition broadness decreases with a decreasing φ̄P. At intermediate and low concentrations, the
effect of the relationship among micelles on eductions of nonstickers and solvents is weak, especially
to nonstickers. Seen from figure 4 (b), with an increasing χr from χr = 0.1, the ratios of the changes
of average numbers of nonstickers and solvents to that of stickers at the micellar cores nearly remain
33601-6
Effect of polymer concentration and length of end block
constant, where solvents and nonstickers are expelled proportionally. When polymer concentration is
decreased to some extent, the aggregation of micellar cores is dominated by the eductions of solvents.
Due to the existence of a large quantity of solvents among micellar cores, it is difficult to expel a small
amount of solvent at the micellar core. Therefore, although the effect of the relationship among micelles
is already very weak, the transition broadness always remains constant with a decrease in φ̄P when the
contribution of nonstickers is rather important and is temperature-dependent in a larger range of χr
[figure 4 (b)].
4. Conclusion and summary
The effects of the length of each hydrophobic end block Nst and polymer concentration φ̄P on the
transition broadness in amphiphilic ABA symmetric triblock copolymer solutions are studied using the
self-consistent field lattice model. When Nst is changed, at fixed φ̄P, micelles occur at a higher temper-
ature, and the broadness of unimer-micelle transition also increases. Compared with associating poly-
mer solutions, it is found that the magnitude of the transition broadness is determined by the ratio of
hydrophobic to hydrophilic blocks rather than by the length of polymer chain. When φ̄P is decreased,
given a large Nst, the transition does not change monotonously. In high concentration regimes, the tran-
sition broadness decreases with decreasing φ̄P, and in intermediate and low concentration regimes, the
transition broadness remains constant with φ̄P. It is demonstrated that the magnitude of the transition
broadness is concerned with the changes of the relative magnitudes of the eductions of nonstickers and
solvents from micellar cores.
Acknowledgements
This research isfinancially supported by theNational 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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Вплив концентрацiї полiмера i довжини гiдрофобного
прикiнцевого блоку на ширину переходу мономер-мiцела в
ABA симетричних триблочних амфiфiльних кополiмерних
розчинах
К.-Г. Ган1,2, Й.-Г. Ма1,2, С.-Л. Оуянг2
1 Школа математики, фiзики i бiологiчної iнженерiї, унiверситет науки i технологiй внутрiшньої Монголiї,
Баоту 014010, Китай
2 Головна лабораторiя iнтегрованого використання мультиметалiчних ресурсiв Баян Обо, унiверситет
науки i технологiї внутрiшньої Монголiї, Баоту 014010, Китай
Вплив довжини кожного гiдрофобного прикiнцевого блоку Nst i концентрацiї полiмера φ̄P на ширину
переходу в симетричних ABA триблочних амфiфiльних кополiмерних розчинах дослiджується шляхом
використання ґраткової моделi самоузгодженого поля. Коли система охолоджена, спостерiгаються мiце-
ли, тобто вiдбувається перехiд однорiдний розчин (мономер)-мiцела. Якщо Nst зростає при сталому φ̄P,
то мiцели виникають при високiй температурi, а температурно залежна область агрегацiї мiцел i пiв-
ширина пiку питомої теплоємностi для переходу мономер-мiцела зростають монотонно. Порiвнюючи з
асоцiативними полiмерами, знайдено, що величина ширини переходу визначається вiдношенням гiдро-
фобних блокiв до гiдрофiльних, а не довжиною ланцюга. Коли φ̄P зменшується при великому значеннi
Nst , температурно залежна область мiцелярної агрегацiї та пiвширина пiку питомої теплоємностi спо-
чатку зменшуються, а потiм залишаються майже сталими. Показано, що ширина переходу пов’язана зi
змiною вiдносних величин видiлення незв’язувальної речовини i розчинникiв з мiцелярних корiв.
Ключовi слова: ширина переходу, самоузгоджене поле, амфiфiльний кополiмер
33601-8
http://dx.doi.org/10.1063/1.454931
http://dx.doi.org/10.5488/CMP.4.2.209
http://dx.doi.org/10.1063/1.2356863
Introduction
Theory
Result and discussion
Conclusion and summary
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