Effect of distribution of stickers along backbone on temperature-dependent structural properties in associative polymer solutions
Effect of distribution of stickers along the backbone on structural properties in associating polymer solutions is studied using self-consistent field lattice model. Only two inhomogeneous morphologies, i.e., microfluctuation homogenous (MFH) and micelle morphologies, are observed. If the system is...
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Інститут фізики конденсованих систем НАН України
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| Цитувати: | Effect of distribution of stickers along backbone on temperature-dependent structural properties in associative polymer solutions / X.-G. Han, X.-F. Zhang, Y.-H. Ma // Condensed Matter Physics. — 2012. — Т. 15, № 3. — С. 33602:1-10. — Бібліогр.: 35 назв. — англ. |
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irk-123456789-1201732017-06-12T03:02:36Z Effect of distribution of stickers along backbone on temperature-dependent structural properties in associative polymer solutions Han, X.-G. Zhang, X.-F. Ma, Y.-H. Effect of distribution of stickers along the backbone on structural properties in associating polymer solutions is studied using self-consistent field lattice model. Only two inhomogeneous morphologies, i.e., microfluctuation homogenous (MFH) and micelle morphologies, are observed. If the system is cooled, the solvent content within the aggregates decreases. When the spacing of stickers along the backbone is increased the temperature-dependent range of aggregation in MFH morphology and half-width of specific heat peak for homogenous solutions-MFH transition increase, and the symmetry of the peak decreases. However, with increasing spacing of stickers, the above three corresponding quantities related to micelles behave differently. It is demonstrated that the broad nature of the observed transitions can be ascribed to the structural changes which accompany the replacement of solvents in aggregates by polymer, which is consistent with the experimental conclusion. It is found that different effect of spacing of stickers on the two transitions can be interpreted in terms of intrachain and interchain associations. Вплив розподiлу центрiв зв’язування (stickers) вздовж головного ланцюга на структурнi властивостi в асоцiативних полiмерних розчинах вивчається з використанням ґраткової моделi самоузгодженого поля. Виявлено лише двi неоднорiднi морфологiї, а саме мiкрофлуктуацiйну гомогенну (МФГ) та мiцелярну морфологiї. Якщо система є охолодженою, тодi зменшується вмiст розчинника всерединi агрегатiв. Коли вiдстань мiж центрами зв’язування вздовж головного ланцюга збiльшується, тодi зростають температурно залежний дiапазон агрегацiї в морфологiї МФГ i пiвширина пiку питомої теплоємностi для переходу гомогеннi розчини-МФГ, а симетрiя пiку – зменшується. Проте, з ростом вiдстанi мiж центрами зв’язування вище згаданi три величини, пов’язанi з мiцелами, поводять себе iнакше. Показано, що рiзний характер спостережених переходiв може пояснюватися структурними змiнами, якi супроводжують замiну розчинникiв в агрегатах на полiмери, що узгоджується з результатами експерименту. Знайдено, що вплив вiдстанi мiж центрами зв’язування на цi два переходи можна також трактувати на мовi внутрiшньоланцюгових i мiжланцюгових асоцiацiй. 2012 Article Effect of distribution of stickers along backbone on temperature-dependent structural properties in associative polymer solutions / X.-G. Han, X.-F. Zhang, Y.-H. Ma // Condensed Matter Physics. — 2012. — Т. 15, № 3. — С. 33602:1-10. — Бібліогр.: 35 назв. — англ. 1607-324X PACS: 61.25.Hp, 64.75.+g, 82.60.Fa DOI:10.5488/CMP.15.3360 arXiv:1210.1726 http://dspace.nbuv.gov.ua/handle/123456789/120173 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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Effect of distribution of stickers along the backbone on structural properties in associating polymer solutions is studied using self-consistent field lattice model. Only two inhomogeneous morphologies, i.e., microfluctuation homogenous (MFH) and micelle morphologies, are observed. If the system is cooled, the solvent content within the aggregates decreases. When the spacing of stickers along the backbone is increased the temperature-dependent range of aggregation in MFH morphology and half-width of specific heat peak for homogenous solutions-MFH transition increase, and the symmetry of the peak decreases. However, with increasing spacing of stickers, the above three corresponding quantities related to micelles behave differently. It is demonstrated that the broad nature of the observed transitions can be ascribed to the structural changes which accompany the replacement of solvents in aggregates by polymer, which is consistent with the experimental conclusion. It is found that different effect of spacing of stickers on the two transitions can be interpreted in terms of intrachain and interchain associations. |
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Article |
| author |
Han, X.-G. Zhang, X.-F. Ma, Y.-H. |
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Han, X.-G. Zhang, X.-F. Ma, Y.-H. Effect of distribution of stickers along backbone on temperature-dependent structural properties in associative polymer solutions Condensed Matter Physics |
| author_facet |
Han, X.-G. Zhang, X.-F. Ma, Y.-H. |
| author_sort |
Han, X.-G. |
| title |
Effect of distribution of stickers along backbone on temperature-dependent structural properties in associative polymer solutions |
| title_short |
Effect of distribution of stickers along backbone on temperature-dependent structural properties in associative polymer solutions |
| title_full |
Effect of distribution of stickers along backbone on temperature-dependent structural properties in associative polymer solutions |
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Effect of distribution of stickers along backbone on temperature-dependent structural properties in associative polymer solutions |
| title_full_unstemmed |
Effect of distribution of stickers along backbone on temperature-dependent structural properties in associative polymer solutions |
| title_sort |
effect of distribution of stickers along backbone on temperature-dependent structural properties in associative polymer solutions |
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Інститут фізики конденсованих систем НАН України |
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2012 |
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http://dspace.nbuv.gov.ua/handle/123456789/120173 |
| citation_txt |
Effect of distribution of stickers along backbone on temperature-dependent structural properties in associative polymer solutions / X.-G. Han, X.-F. Zhang, Y.-H. Ma // Condensed Matter Physics. — 2012. — Т. 15, № 3. — С. 33602:1-10. — Бібліогр.: 35 назв. — англ. |
| series |
Condensed Matter Physics |
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2025-07-08T17:22:08Z |
| last_indexed |
2025-07-08T17:22:08Z |
| _version_ |
1837100254140301312 |
| fulltext |
Condensed Matter Physics, 2012, Vol. 15, No 3, 33602: 1–10
DOI: 10.5488/CMP.15.33602
http://www.icmp.lviv.ua/journal
Effect of distribution of stickers along backbone on
temperature-dependent structural properties in
associative polymer solutions
X.-G. Han∗, X.-F. Zhang, Y.-H. Ma
School of Mathematics, Physics and Biology, Inmongolia Science and Technology University,
Baotou 014010, China
Received March 29, 2012, in final form June 8, 2012
Effect of distribution of stickers along the backbone on structural properties in associating polymer solutions
is studied using self-consistent field lattice model. Only two inhomogeneous morphologies, i.e., microfluctua-
tion homogenous (MFH) and micelle morphologies, are observed. If the system is cooled, the solvent content
within the aggregates decreases. When the spacing of stickers along the backbone is increased the temperature-
dependent range of aggregation in MFH morphology and half-width of specific heat peak for homogenous
solutions-MFH transition increase, and the symmetry of the peak decreases. However, with increasing spacing
of stickers, the above three corresponding quantities related to micelles behave differently. It is demonstrated
that the broad nature of the observed transitions can be ascribed to the structural changes which accompany
the replacement of solvents in aggregates by polymer, which is consistent with the experimental conclusion. It is
found that different effect of spacing of stickers on the two transitions can be interpreted in terms of intrachain
and interchain associations.
Key words: structural properties, self-consistent field, associative polymer
PACS: 61.25.Hp, 64.75.+g, 82.60.Fa
1. Introduction
Physically associating polymers are polymer chains containing a small fraction of attractive groups
along the backbones. The attractive groups tend to form physical links which can play a important role
in reversible junctions between different polymer chains. The junctions can be broken and recombined
frequently on experimental time scales. This property of the junctions makes associative polymer solu-
tions behave reversibly when ambient conditions, such as temperature and concentrations, change. This
tunable characteristic of the system produces extensive applications [1–4] that have a great potential as
smart materials [5–7].
Physically associating polymer is simply considered as an amphiphilic block copolymer. When dis-
solved in a solvent such as water, amphiphilic copolymers can self-assemble into micelles. The solvo-
phobic blocks (attractive groups) cluster together to form a core, and the solvophilic blocks spread out-
ward as a corona. One key effect of solvent selectivity (or equivalently, temperature) is that the micelles
dissolve into single chains at a critical micelle temperature as the solvent selectivity decreases. Several
studies have investigated the detailed structure of micelles with varying temperature in aqueous solu-
tions [8–11] One appealing advantage of polymers is their versatility. Architectural parameters of asso-
ciative polymers may be tuned by changing the chain length, chemical composition and distribution of
attractive groups [12–18]. It was suggested that the distribution of stickers along the chains can be an
important factor in controlling macroscopic properties of these systems [17, 18]. The study of the effect
of distribution of sticker along the chain in physically association polymer solutions (PAPSs) would be
useful to establish and understand the thermodynamics of block copolymers in a selective solvent.
∗E-mail: xghan0@163.com
© X.-G. Han, X.-F. Zhang, Y.-H. Ma, 2012 33602-1
http://dx.doi.org/10.5488/CMP.15.33602
http://www.icmp.lviv.ua/journal
X.-G. Han et al.
It is well-known that self-consistent field theory (SCFT), as a mean-field theory, has been applied to
the study of a great deal of problems in polymeric systems [19–23]. Recently, SCFT is applied to the study
of the properties of micelles in polymer solutions [24–26]. In previous paper [27], we focused on the ther-
modynamic properties and structure transitions in PAPSs. The microfluctuation homogenous (MFH) and
micelle morphologies were observed. The degrees of aggregation of micelle morphology is much larger
than that of MFH morphology. In this work, the effect of the distribution of stickers along the backbone
on structures in PAPSs is studied using a self-consistent field lattice model. The temperatures at which
the above two inhomogenous morphologies first appear, denoted by TMFH and Tm, respectively, are de-
pendent on the spacing of sticker along the chain. If the system is cooled from TMFH and Tm, the solvent
content within the aggregates (microfluctuation or micellar core) decreases, which is dependent on spac-
ing of sticker and morphology. The increase in spacing of sticker has different effect on homogenous
solutions-MFH and MFH-micelle transitions. It is found that this result can be interpreted in terms of
intrachain and interchain associations.
2. Theory
We consider a system of incompressible PAPSs, where nP polymers are each composed of Nst seg-
ments of type sticker monomer (attractive group) and Nns segments of type nonsticky monomer, dis-
tributed over a lattice. The sticker monomers are placed at the two ends of a chain and regularly along
the chain backbone, and there are l nonsticky monomers between two neighboring sticker monomers.
The degree of polymerization of chain is N = Nst + Nns. 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. The total number of lattice sites is NL = nh +nPN .
Nearest neighbor pairs of stickers have attractive interaction −ǫwith ǫ> 0, which is the only non-bonded
interaction in the present system. In this simulation, however, instead of directly using the exact expres-
sion of the nearest neighbor interaction for stickers, we introduce a local concentration approximation
for the non-bonded interaction similar to the references [23, 27]. The interaction energy 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 co-
ordination 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 site r , where j and s are the indexes of chain
and monomer of a polymer, respectively. s ∈ stmeans that the sth monomer belongs to sticker monomer
type. We perform the SCFT calculations in the canonical ensemble, and the field-theoretic free energy F
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
defined asQh =
1
nh
∑
r exp[−ωh(r )].QP is the partition function of a noninteracting polymer chain subject
to the fields ωst(r ) = ω+(r )−ω−(r ) and ωns(r ) = ω+(r ), which act on sticker and nonsticky segments,
respectively.
Equation (2.2) can be considered an alternative form of the self-consistent field free energy functional
for an incompressible polymer solutions [28]. When a local concentration approximation for the non-
bonded interaction is introduced, the SCFT descriptions of lattice model for PAPSs presented in this work
is basically equivalent to that of the “Gaussian threadmodel” chain for the similar polymer solutions [28].
The related illumination in detail refers to reference [27].
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)
33602-2
Effect of distribution of stickers along backbone
where
φst(r ) =
1
NL
1
z
nP
QP
∑
s∈st
∑
αs
Gαs (r, s|1)Gαs (r, s|N )
G(r, s)
(2.5)
and
φns(r ) =
1
NL
1
z
nP
QP
∑
s∈ns
∑
αs
Gαs (r, s|1)Gαs (r, s|N )
G(r, s)
(2.6)
are the average numbers of sticker and nonsticky segments at r , respectively, and
φh(r )=
1
N
L
nh
Q
h
exp{− ωh(r )}
is the average numbers of solvent molecules at r .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
∑
αN
means the summation over all the possible positions and orientations of the N th segment of
the chain.Gαs (r, s|1) andGαs (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. The expressions of the above three quantities refer to
Appendix. In this work, the chain is described as a random walk without the possibility of direct back-
folding. Although self-intersections of a chain are not permitted, the excluded volume effect is sufficiently
taken into account [29].
The saddle point is calculated using the pseudo-dynamical evolution process [27]. The calculation is
initiated from appropriately random-chosen fields ω+(r ) and ω−(r ), and interrupted when the change of
free energy F between two successive iterations is reduced to the needed precision. The resulting config-
uration 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 properties of associative polymer solutions depend on four tunable parameters: χ
is the Flory-Huggins interaction parameter, N is the degree of polymerization of chain, where N equals
81 in this paper, l is the spacing of stickers along the backbone and φ̄st is the average volume fraction
of polymers. The calculations are performed in a three-dimensional simple cubic lattice with periodic
boundary condition, and the effect of the lattice size is considered. The results presented below are ob-
tained from the lattice with NL = 403. Three different morphologies in PAPSs are observed, i.e., the ho-
mogenous, micro-fluctuation homogenous (MFH) and micelle morphologies. By comparing the relative
stability of the observed states, the phase diagram is constructed.
Figure 1 shows the phase diagram of the systems with different spacing of stickers l . At fixed l , when χ
is increased from homogenous solutions, MFH andmicelle morphologies appear in turn.φP(r ) and φst(r )
in the solutions with MFH morphology slightly fluctuate around φ̄P and φ̄st, respectively. The average
volume fraction of stickers at the sticker-rich sites (fluctuations) rri increases with increasing χ for fixed
φ̄P. Its maximum value is about 2φ̄st, which is much smaller than unity for all the φ̄P. There exists the
state of microfluctuations whose thermodynamics is adequately captured by SCFT. It is confirmed that the
MFH appearance is accompanied by the appearance of the heat capacity peak (shown below), which is in
reasonable agreement with the conclusion drawn byKumar et al. [27, 30]. The basic component ofmicelle
morphology is flower micelles, which are randomly and closely distributed in the system. Each micelle
has a sticker-rich core, which is located at the center of a micelle, surrounded by non-sticky components
of polymers. The average value of volume fraction of stickers at the micellar core is much larger than that
of sticker-rich sites in MFH morphology. It is shown that the degree of aggregation of stickers in micelle
morphology is much larger than that in MFH morphology.
When spacing of stickers l is changed, only MFHmorphology andmicelles are observed as inhomoge-
neous morphologies. The structural morphology of MFH morphology does not change, and the micellar
33602-3
X.-G. Han et al.
Figure 1. The phase diagram for systems with different spacing of stickers l . The boundaries of MFH and
micelle morphologies are obtained. The red open and solid squares, green open and solid triangles, blue
open and solid diamonds correspond to the boundaries of MFH and micelle morphologies for l = 3,9,19,
respectively.
shape remains spherelike. For l = 19, the χ value on micellar boundary (∼ 1/Tm) increases with decreas-
ing φ̄P. When φ̄P goes down to a certain extent, micellar boundary becomes steep. The χ value on MFH
boundary (∼ 1/TMFH) also rises with a decrease in φ̄P. MFH boundary intersects the micellar one at φ̄CFC,
which is the critical MFH concentration (φ̄CFC = 0.4). When l is decreased, at fixed φ̄P, the χ value on
micellar boundary shifts to a small value, and the χ value on MFH boundary decreases markedly. φ̄CFC
also drops with a decrease in l .
In this paper, the structural properties dependent on temperature are focused. Therefore, the quan-
tities related to volume fractions of stickers in MFH and micelle morphologies as a function of χ are
studied. Figure 2 (a) shows the variations of effective average volume fractions of stickers and solvents at
(a) (b)
Figure 2. The variations of effective average volume fractions of stickers and solvents at the sticker-
rich sites in MFH morphologies with different spacing of stickers l , denoted by φ̄ri
se
[
= φ̄ri
st
/
φ̄st
]
and
φ̄ri
he
[
=
(
φ̄ri
h
− φ̄h
)/
φ̄st
]
, respectively, with the χ deviation from MFH boundary χr at φ̄P = 0.8 are pre-
sented in figure 2 (a); The variations of the average volume fraction of stickers and relative average vol-
ume fraction of solvents at the micellar cores, denoted by φ̄
rco
st and φ̄
rco
hr
[
= φ̄
rco
h
− φ̄h
]
, respectively, with
the χ deviation from micellar boundary χr in the systems with different spacing of stickers l at φ̄P = 0.8
are shown in figure 2 (b).
the sticker-rich sites (microfluctuations) in MFH morphologies with different spacing of stickers l , which
are denoted by φ̄ri
se and φ̄
ri
he
, respectively, with the χ deviation fromMFH boundary, χr, at φ̄P = 0.8, where
φ̄ri
se and φ̄ri
he
equal φ̄ri
st/φ̄st and (φ̄ri
h
− φ̄h)/φ̄st, respectively. With the increase in χr, φ̄
ri
se at l = 3 initially
33602-4
Effect of distribution of stickers along backbone
rises when 0 É χr É 0.2 and then maintains a certain value, and the corresponding φ̄ri
he
first decreases
when 0 Éχr É 0.2 and then remains constant. Although the increase of the degree of aggregation in MFH
morphology is accompanied by the penetration of solvents, the effective total quantity of penetration of
solvents is very small (|φ̄ri
he
| = 0.047). When l is increased, the shapes of the curves of φ̄ri
se(χr) and φ̄ri
he
(χr)
are similar to the case of l = 3. However, the onset of the range independent of χr shifts to a larger χr
value. The minimum of φ̄ri
he
(χr) goes down with an increasing l . It is demonstrated that in MFH mor-
phology the increase in spacing of stickers augments the temperature-dependent range of aggregation of
stickers and accelerates the effective penetration of solvents.
The variations of the average volume fraction of stickers and the relative average volume fraction of
solvents at the micellar cores, which are denoted by φ̄
rco
st and φ̄
rco
hr
(=φ̄
rco
h
− φ̄h), respectively, with the χ
deviation from micellar boundary, χr, in the systems with different spacings of stickers l at φ̄P = 0.8 are
shown in figure 2 (b). At l = 3, φ̄
rco
st rises and approaches 1, and the corresponding φ̄
rco
hr
decreases and
is close to its minimum when χr is increased. When l is increased, φ̄
rco
st goes up and rapidly approaches
1, and the corresponding φ̄
rco
hr
goes down and is quickly close to its minimum, with increasing χr. It is
demonstrated that in micelle morphology, the increase in spacing of stickers almost does not change the
total quantity of the expelled solvent, and decreases the effective range of aggregation dependent on χr,
which is contrary to that of MFH morphology.
(a) (b)
Figure 3. (Color online) The variations of the specific heat CV with the χ deviation from MFH or micellar
boundary χr in HS-MFH and MFH-micelle transition regions at φ̄P = 0.8 in the systems with different
spacing of stickers l are presented in figures 3 (a) and (b), respectively. The red squares, green triangles
and blue diamonds correspond to l = 3,9,19, respectively.
In order to demonstrate the property of the observed transition, the heat capacity CV is calculated,
because the half-width of a specific heat peak may be an intrinsic measure of transition broadness [31].
In this work, the heat capacity per site of PAPSs is expressed as follows (in the unit of kB):
CV =
(
∂U
∂T
)
NL ,nP
=
1
NL
χ2 ∂
∂χ
(∑
r
φ2
st(r )
)
. (3.1)
The CV(χr) curves for the HS-MFH and MFH-micelle transitions in various l at φ̄P = 0.8 are shown in
figures 3 (a) and 3 (b), respectively. For HS-MFH transition, a peak appears in eachCV(χr) curve. When l is
increased, the height and half-width of the transition peak rise, and the symmetry of the peak decreases.
Meanwhile, for the MFH-micelle transition, there are some peaks in each CV(χr) curve. The highest of
these peaks, corresponds to MFH-micelle transition. When l is increased, the height of the transition
peak rises, and the corresponding half-width does go down. The shape of the transition peak tends to be
symmetric under the condition of an increasing l . It is shown that an increase in l causes an increase in
the broadness of HS-MFH transition and a decrease in the broadness of MFH-micelle transition. The effect
33602-5
X.-G. Han et al.
of spacing of stickers on CV(χr) curve for the MFH-micelle transition is different from that for HS-MFH
transition.
The HS-MFH transition that took place in this work, which corresponds to the clustering transition ob-
served by Kumar et al. [27, 30], is affected by the change of spacing of stickers as discussed above. When
spacing of stickers is increased, both the magnitude of the temperature-dependent range of aggregation
and the effective total quantity of the expelled solvents inMFHmorphology increase. Under the same con-
dition, the broadness of the corresponding transition also increases. Meanwhile, for MFH-micelle transi-
tion, an increase of spacing of stickers results in a decrease of the magnitude of the effective range depen-
dent on temperature and on the transition broadness. Overall, for the above two transitions, the magni-
tude of the temperature-dependent range of aggregation and the transition broadness change simultane-
ously and consistently with the spacing of stickers. It is demonstrated that the broad nature of the transi-
tions observed in PAPSs is concernedwith the penetration of solvents from the aggregates, which is in rea-
sonable agreement with the experimental result observed by Goldmints et al. in the unimer-micelle tran-
sition [32]. At the same time, it is found that the symmetry of a specific heat peak is affected by the process
of penetration of solvents. When the transition broadness increases, the symmetry of a transition peak
decreases. Furthermore, from the above behaviors of penetration of a solvent and heat capacity, it is seen
that an increase of spacing of stickers has different effect on the HS-MFH and MFH-micelle transitions.
In order to interpret the different effect of spacing of stickers on the aggregation of stickers in MFH
and micelle morphologies, we evaluate the probability that a sticker of polymer chain forms intrachain
and interchain associations in the system using an approach similar to the one presented in reference [27,
33, 34]. We suppose that there are no other sticker aggregates in the MFH and micellar system except
sticker-rich site microfluctuations or the micellar cores. A sticker in a particular chain can form an in-
trachain association, as well as an interchain association. Ignoring the probabilities that more than two
stickers of a definite chain are attached to an aggregate, the conditional probability that the sticker s1 is
concerned with intrachain association, provided that the sticker s1 is at an aggregate of the two above
mentioned types whose position locates at rag, can be expressed as:
ploop(rag, s1) =
1
P (1)(rag, s1)
∑
s2∈st,s2,s1
P (2)(rag, s1;rag, s2), (3.2)
where
∑
s2∈st,s2,s1
means the summation over all the stickers of a polymer chain except the s1th one, while
P (1)(rag, s1) and P (2)(rag, s1;rag, s2), whose expressions are given in appendix, are the single-segment and
two-segment probability distribution functions of a chain, respectively. Then, 1−ploop(rag, s1) is the con-
ditional probability that the sticker s1 is linked with those that belong to other chains when the sticker s1
is at rag, and
Plk (s1) =
∑
rag
P (1)(rag, s1)
[
1−ploop(rag, s1)
]
is the probability that a sticker s1 of a chain is related to an interchain association, where
∑
rag
means
the summation over all the aggregates in the system. The summation of Plk (s1) over all the stickers in a
chain, 〈n
lk
〉 =
∑
s1,s1∈st Plk (s1), can be viewed as the average sticker number from a particular polymer
chain linked with other chains by sticker aggregates. The average fraction of interchain association of a
sticker is expressed as f̄te = 〈n
lk
〉/Nst. The average fraction of intrachain association of a sticker is defined
as f̄tr = (1/Nst)
∑
s1,s1∈st
∑
rag
ploop(rag, s1).
Figure 4 shows the variations of average fractions of intrachain association and interchain association
of a sticker, denoted by f̄tr and f̄te, respectively, with the χ deviation from the MFH or micellar boundary,
χr, in the system with different spacing of stickers l at φ̄P = 0.8. In MFH morphology [figure 4 (a)], when
χr is increased, f̄te at l = 3 first rises when 0 É χr É 0.2, then maintains a certain value, and the corre-
sponding f̄tr always remains constant when χr > 0, and f̄tr is smaller than f̄te. When l is increased, the
variation of f̄te with χr resembles that of l = 3. However, a certain value at which f̄te finally arrives rises,
and the corresponding χr also goes up, with χr increasing. f̄tr decreases markedly when l is increased.
f̄tr is much smaller than the corresponding f̄te. It is shown that in MFH morphology intrachain associa-
tion is independent of χr, and the range of interchain association which is concerned with χr rises when
spacing of stickers is increased. In micelle morphology [figure 4 (b)], the variation of the average fraction
of interchain association of a sticker f̄te with the deviation from micellar boundary χr and the effect of l
33602-6
Effect of distribution of stickers along backbone
(a) (b)
Figure 4. The average fractions of intrachain and interchain associations of a sticker, denoted by f̄tr and
f̄te, respectively, as functions of the χ deviation from MFH or micellar boundary χr are presented in
figures 4 (a) and (b) in MFH and micelle morphologies in the systems with different spacing of stickers l
at φ̄P = 0.8. The red open and solid squares, green open and solid triangles, blue open and solid diamonds
correspond to by f̄tr and f̄te of l = 3,9,19, respectively.
on it are similar to those of MFHmorphology. However, there exists an evident difference in the curve of
f̄te(χr), which is not smooth. The intrachain associations of micelles f̄tr are dependent on χr. The behav-
ior is distinct for small l and different from MFH morphology. It is noted that at l = 3, when 0 < χr É 0.4,
the extent of rise of f̄tr and f̄te changes alternately with increasing χr.
As discussed above, the increase of spacing of stickers l has different effect on the HS-MFH and MFH-
micelle transitions, which can be explained in terms of intrachain and interchain associations. The aver-
age fraction of interchain association of a sticker in MFH morphology f̄ ri
te is much larger than that of the
corresponding intrachain quantity f̄ ri
tr , and f̄ ri
tr is absolutely independent of temperature. Therefore, the
temperature-dependent property of MFHmorphology is determined by interchain association. When l is
increased, f̄ ri
te range concerned with temperature also rises. Therefore, an increase of spacing of stickers
is favorable to an increase in the broadness of HS-MFH transition. Meanwhile, in micelle morphology, the
average fraction of intrachain association of a sticker f̄ ro
tr is dependent on temperature, especially in the
case of small l . Therefore, the property ofMFH-micelle transition is determined byboth intrachain and in-
terchain associations [figure 4 (b)]. At l = 3, both f̄ ro
tr and f̄ ro
te are sensitive to temperature in the transition
region. When l is increased, the susceptibility of f̄ ro
tr to temperature decreases markedly. Although an in-
crease in l is favorable to an increase in the susceptibility of f̄ ro
te to temperature, it may beweak compared
with the corresponding decrease of f̄ ro
tr . Therefore, when spacing of stickers is increased, the broadness of
the corresponding MFH-micelle transition decreases, which is different from that of HS-MFH transition.
4. Conclusion and summary
The effect of distribution of stickers along the backbone on the temperature-dependent property of ag-
gregation structure in physically associating polymer solutions (PAPSs) is studied using the self-consistent
field lattice model. When spacing of stickers is increased, the temperature susceptibility of penetration of
solvents from aggregates in MFH morphology and the broadness of HS-MFH transition increases. How-
ever, the corresponding two quantities of MFH-micelle transition do decrease under the same condition,
which is opposite to that of HS-MFH transition. It is found that the temperature susceptibility of pene-
tration of solvents from the two above morphologies and the broadness of the two transitions change
simultaneously and consistently. It is demonstrated that the broadness of the transitions observed in
PAPSs is concerned with the penetration of solvent from aggregates. Furthermore, the different effect of
spacing of stickers on HS-MFH andMFH-micelle transitions is due to different contributions of intrachain
and interchain associations to MFH and micelle morphologies. This work can be extended to the study
of the effects of polymer concentration and chain architecture on the transition properties related to the
penetration of a solvent.
33602-7
X.-G. Han et al.
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).
Appendix
Following the scheme of Schentiens and Leermakers [35], 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), (4.1)
where G(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 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.
This means that the chain is described as a random walk without the possibility of direct backfolding.
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 ), (4.2)
with the initial condition GαN (r, N |N ) =G(r, N ) for all the values of αN .
Using the expressions of the end segment distribution functions, the single-segment probability dis-
tribution function P (1)(r, s) and the two-segment probability distribution function P (2)(r1, s1;r2, s2) of the
chain can be defined as follows:
P (1)(r, s) =
1
zNLQP
∑
r ′s
∑
αs
Gαs (r ′, s|1)Gαs (r ′, s|N )
G(r ′, s)
δr ′s ,r , (4.3)
which is a normalized probability that the monomer s of the chain is on the lattice site r ;
P (2)(r1, s1;r2, s2) =
1
zNLQP
∑
r ′s1
∑
αs1
∑
r ′s2
∑
αs2
Gαs1 (r ′, s1|1)δr ′s1
,r1
(4.4)
×G (r ′, s1;r ′, s2)Gαs2 (r ′, s2|N )δr ′s2
,r2
and
G (r, s1;r, s2) =
∑
rs1+1
∑
αs1+1
. . .
∑
rs2−1
∑
αs2−1
{
s2−1∏
s=s1+1
λ
αs−αs−1
rs−rs−1
G(r, s)
}
λ
αs2−αs2−1
rs2−rs2−1
(for s2 > s1)
give the probability that themonomers s1 and s2 of the chain are on the lattice sites r1 and r2, respectively.
It can be verified that
∑
r P (1)(r, s) = 1, and
∑
r2
P (2)(r1, s1;r2, s2) = P (1)(r1, s1).
33602-8
Effect of distribution of stickers along backbone
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Вплив розподiлу центрiв зв’язування вздовж головного
ланцюга на температурно залежнi структурнi властивостi в
асоцiативних полiмерних розчинах
К.-Г. Ган, К.-Ф. Жанг, Й.-Г. Ма
Школа математики, фiзики i бiологiї, унiверситет науки i технологiй Внутрiшньої Монголiї, Баоту 014010,
Китай
Вплив розподiлу центрiв зв’язування (stickers) вздовж головного ланцюга на структурнi властивостi в асо-
цiативних полiмерних розчинах вивчається з використанням ґраткової моделi самоузгодженого поля.
Виявлено лише двi неоднорiднi морфологiї, а саме мiкрофлуктуацiйну гомогенну (МФГ) та мiцелярну мор-
фологiї. Якщо система є охолодженою, тодi зменшується вмiст розчинника всерединi агрегатiв. Коли вiд-
стань мiж центрами зв’язування вздовж головного ланцюга збiльшується, тодi зростають температурно
залежний дiапазон агрегацiї в морфологiї МФГ i пiвширина пiку питомої теплоємностi для переходу го-
могеннi розчини-МФГ, а симетрiя пiку – зменшується. Проте, з ростом вiдстанi мiж центрами зв’язування
вище згаданi три величини, пов’язанi з мiцелами, поводять себе iнакше. Показано, що рiзний характер
спостережених переходiв може пояснюватися структурними змiнами, якi супроводжують замiну розчин-
никiв в агрегатах на полiмери, що узгоджується з результатами експерименту. Знайдено, що вплив вiдста-
нi мiж центрами зв’язування на цi два переходи можна також трактувати на мовi внутрiшньоланцюгових
i мiжланцюгових асоцiацiй.
Ключовi слова: структурнi властивостi, самоузгоджене поле, асоцiативний полiмер
33602-10
Introduction
Theory
Result and discussion
Conclusion and summary
|