[
Abstract
Effect of distribution of stickers along the backbone on structural properties in associating polymer solutions is studied using selfconsistent 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 temperaturedependent range of aggregation in MFH morphology and halfwidth of specific heat peak for homogenous solutionsMFH 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.
\keywordsstructural properties, selfconsistent field, associative polymer \pacs61.25.Hp, 64.75.+g, 82.60.Fa
Abstract
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 розчиниМФГ, 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. \keywordsструктурнi властивостi, самоузгоджене поле, асоцiативний полiмер
201215333602
\doinumber10.5488/CMP.15.33602
Effect of distribution of stickers along backbone]Effect of distribution of stickers along backbone on
temperaturedependent
structural properties in associative polymer solutions
X.G. Han et al.]X.G. Han^{1}^{1}1Email: , X.F. Zhang, Y.H. Ma
\address
School of Mathematics, Physics and Biology, Inmongolia
Science and Technology University,
Baotou 014010, China
\authorcopyrightX.G. Han, X.F. Zhang, Y.H. Ma, 2012
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 solutions behave reversibly when ambient conditions, such as temperature and concentrations, change. This tunable characteristic of the system produces extensive applications [1, 2, 3, 4] that have a great potential as smart materials [5, 6, 7].
Physically associating polymer is simply considered as an amphiphilic block copolymer. When dissolved in a solvent such as water, amphiphilic copolymers can selfassemble into micelles. The solvophobic blocks (attractive groups) cluster together to form a core, and the solvophilic blocks spread outward 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 solutions [8, 9, 10, 11] One appealing advantage of polymers is their versatility. Architectural parameters of associative polymers may be tuned by changing the chain length, chemical composition and distribution of attractive groups [12, 13, 14, 15, 16, 17, 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.
It is wellknown that selfconsistent field theory (SCFT), as a meanfield theory, has been applied to the study of a great deal of problems in polymeric systems [19, 20, 21, 22, 23]. Recently, SCFT is applied to the study of the properties of micelles in polymer solutions [24, 25, 26]. In previous paper [27], we focused on the thermodynamic 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 selfconsistent field lattice model. The temperatures at which the above two inhomogenous morphologies first appear, denoted by and , respectively, are dependent on the spacing of sticker along the chain. If the system is cooled from and , the solvent content within the aggregates (microfluctuation or micellar core) decreases, which is dependent on spacing of sticker and morphology. The increase in spacing of sticker has different effect on homogenous solutionsMFH and MFHmicelle 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 polymers are each composed of segments of type sticker monomer (attractive group) and segments of type nonsticky monomer, distributed over a lattice. The sticker monomers are placed at the two ends of a chain and regularly along the chain backbone, and there are nonsticky monomers between two neighboring sticker monomers. The degree of polymerization of chain is . In addition to polymer monomers, 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 . Nearest neighbor pairs of stickers have attractive interaction with , which is the only nonbonded interaction in the present system. In this simulation, however, instead of directly using the exact expression of the nearest neighbor interaction for stickers, we introduce a local concentration approximation for the nonbonded interaction similar to the references [27, 23]. The interaction energy is expressed as:
(1) 
where is the FloryHuggins interaction parameter in the solutions, which equals , is the coordination number of the lattice used, where means the summation over all the lattice sites and is the volume fraction of stickers on site , where and are the indexes of chain and monomer of a polymer, respectively. means that the th monomer belongs to sticker monomer type. We perform the SCFT calculations in the canonical ensemble, and the fieldtheoretic free energy is defined as
(2) 
where is the partition function of a solvent molecule subject to the field , which is defined as . is the partition function of a noninteracting polymer chain subject to the fields and , which act on sticker and nonsticky segments, respectively.
Equation (2) can be considered an alternative form of the selfconsistent field free energy functional for an incompressible polymer solutions [28]. When a local concentration approximation for the nonbonded interaction is introduced, the SCFT descriptions of lattice model for PAPSs presented in this work is basically equivalent to that of the ‘‘Gaussian thread model’’ chain for the similar polymer solutions [28]. The related illumination in detail refers to reference [27].
Minimizing the free energy function with and leads to the following saddle point equations:
(3) 
(4) 
where
(5) 
and
(6) 
are the average numbers of sticker and nonsticky segments at , respectively, and
is the average numbers of solvent molecules at . is expressed as
where and denote the position and orientation of the th segment of the chain, respectively. means the summation over all the possible positions and orientations of the th segment of the chain. and are the end segment distribution functions of the th segment of the chain. 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 backfolding. Although selfintersections of a chain are not permitted, the excluded volume effect is sufficiently taken into account [29].
The saddle point is calculated using the pseudodynamical evolution process [27]. The calculation is initiated from appropriately randomchosen fields and , and interrupted when the change of free energy between two successive iterations is reduced to the needed precision. The resulting configuration 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: is the FloryHuggins interaction parameter, is the degree of polymerization of chain, where equals 81 in this paper, is the spacing of stickers along the backbone and is the average volume fraction of polymers. The calculations are performed in a threedimensional simple cubic lattice with periodic boundary condition, and the effect of the lattice size is considered. The results presented below are obtained from the lattice with . Three different morphologies in PAPSs are observed, i.e., the homogenous, microfluctuation 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 . At fixed , when is increased from homogenous solutions, MFH and micelle morphologies appear in turn. and in the solutions with MFH morphology slightly fluctuate around and , respectively. The average volume fraction of stickers at the stickerrich sites (fluctuations) increases with increasing for fixed . Its maximum value is about , which is much smaller than unity for all the . 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 by Kumar et al. [27, 30]. The basic component of micelle morphology is flower micelles, which are randomly and closely distributed in the system. Each micelle has a stickerrich core, which is located at the center of a micelle, surrounded by nonsticky components of polymers. The average value of volume fraction of stickers at the micellar core is much larger than that of stickerrich 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 is changed, only MFH morphology and micelles are observed as inhomogeneous morphologies. The structural morphology of MFH morphology does not change, and the micellar shape remains spherelike. For , the value on micellar boundary () increases with decreasing . When goes down to a certain extent, micellar boundary becomes steep. The value on MFH boundary () also rises with a decrease in . MFH boundary intersects the micellar one at , which is the critical MFH concentration (). When is decreased, at fixed , the value on micellar boundary shifts to a small value, and the value on MFH boundary decreases markedly. also drops with a decrease in .
In this paper, the structural properties dependent on temperature are focused. Therefore, the quantities related to volume fractions of stickers in MFH and micelle morphologies as a function of are studied.
Figure 2 (a) shows the variations of effective average volume fractions of stickers and solvents at the stickerrich sites (microfluctuations) in MFH morphologies with different spacing of stickers , which are denoted by and , respectively, with the deviation from MFH boundary, , at , where and equal and , respectively. With the increase in , at initially rises when and then maintains a certain value, and the corresponding first decreases when 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 (). When is increased, the shapes of the curves of and are similar to the case of . However, the onset of the range independent of shifts to a larger value. The minimum of goes down with an increasing . It is demonstrated that in MFH morphology the increase in spacing of stickers augments the temperaturedependent 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 and (=), respectively, with the deviation from micellar boundary, , in the systems with different spacings of stickers at are shown in figure 2 (b). At , rises and approaches 1, and the corresponding decreases and is close to its minimum when is increased. When is increased, goes up and rapidly approaches 1, and the corresponding goes down and is quickly close to its minimum, with increasing . 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 , which is contrary to that of MFH morphology.
In order to demonstrate the property of the observed transition, the heat capacity is calculated, because the halfwidth 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 ):
(7)  
The curves for the HSMFH and MFHmicelle transitions in various at are shown in figures 3 (a) and 3 (b), respectively. For HSMFH transition, a peak appears in each curve. When is increased, the height and halfwidth of the transition peak rise, and the symmetry of the peak decreases. Meanwhile, for the MFHmicelle transition, there are some peaks in each curve. The highest of these peaks, corresponds to MFHmicelle transition. When is increased, the height of the transition peak rises, and the corresponding halfwidth does go down. The shape of the transition peak tends to be symmetric under the condition of an increasing . It is shown that an increase in causes an increase in the broadness of HSMFH transition and a decrease in the broadness of MFHmicelle transition. The effect of spacing of stickers on curve for the MFHmicelle transition is different from that for HSMFH transition.
The HSMFH transition that took place in this work, which corresponds to the clustering transition observed by Kumar et al. [30, 27], is affected by the change of spacing of stickers as discussed above. When spacing of stickers is increased, both the magnitude of the temperaturedependent range of aggregation and the effective total quantity of the expelled solvents in MFH morphology increase. Under the same condition, the broadness of the corresponding transition also increases. Meanwhile, for MFHmicelle transition, an increase of spacing of stickers results in a decrease of the magnitude of the effective range dependent on temperature and on the transition broadness. Overall, for the above two transitions, the magnitude of the temperaturedependent range of aggregation and the transition broadness change simultaneously and consistently with the spacing of stickers. It is demonstrated that the broad nature of the transitions observed in PAPSs is concerned with the penetration of solvents from the aggregates, which is in reasonable agreement with the experimental result observed by Goldmints et al. in the unimermicelle transition [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 HSMFH and MFHmicelle 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 [34, 33, 27]. We suppose that there are no other sticker aggregates in the MFH and micellar system except stickerrich site microfluctuations or the micellar cores. A sticker in a particular chain can form an intrachain 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 is concerned with intrachain association, provided that the sticker is at an aggregate of the two above mentioned types whose position locates at , can be expressed as:
(8) 
where means the summation over all the stickers of a polymer chain except the th one, while and , whose expressions are given in appendix, are the singlesegment and twosegment probability distribution functions of a chain, respectively. Then, is the conditional probability that the sticker is linked with those that belong to other chains when the sticker is at , and
is the probability that a sticker of a chain is related to an interchain association, where means the summation over all the aggregates in the system. The summation of over all the stickers in a chain, , 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 . The average fraction of intrachain association of a sticker is defined as .
Figure 4 shows the variations of average fractions of intrachain association and interchain association of a sticker, denoted by and , respectively, with the deviation from the MFH or micellar boundary, , in the system with different spacing of stickers at . In MFH morphology [figure 4 (a)], when is increased, at first rises when , then maintains a certain value, and the corresponding always remains constant when , and is smaller than . When is increased, the variation of with resembles that of . However, a certain value at which finally arrives rises, and the corresponding also goes up, with increasing. decreases markedly when is increased. is much smaller than the corresponding . It is shown that in MFH morphology intrachain association is independent of , and the range of interchain association which is concerned with 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 with the deviation from micellar boundary and the effect of on it are similar to those of MFH morphology. However, there exists an evident difference in the curve of , which is not smooth. The intrachain associations of micelles are dependent on . The behavior is distinct for small and different from MFH morphology. It is noted that at , when , the extent of rise of and changes alternately with increasing .
As discussed above, the increase of spacing of stickers has different effect on the HSMFH and MFHmicelle transitions, which can be explained in terms of intrachain and interchain associations. The average fraction of interchain association of a sticker in MFH morphology is much larger than that of the corresponding intrachain quantity , and is absolutely independent of temperature. Therefore, the temperaturedependent property of MFH morphology is determined by interchain association. When is increased, range concerned with temperature also rises. Therefore, an increase of spacing of stickers is favorable to an increase in the broadness of HSMFH transition. Meanwhile, in micelle morphology, the average fraction of intrachain association of a sticker is dependent on temperature, especially in the case of small . Therefore, the property of MFHmicelle transition is determined by both intrachain and interchain associations [figure 4 (b)]. At , both and are sensitive to temperature in the transition region. When is increased, the susceptibility of to temperature decreases markedly. Although an increase in is favorable to an increase in the susceptibility of to temperature, it may be weak compared with the corresponding decrease of . Therefore, when spacing of stickers is increased, the broadness of the corresponding MFHmicelle transition decreases, which is different from that of HSMFH transition.
4 Conclusion and summary
The effect of distribution of stickers along the backbone on the temperaturedependent property of aggregation structure in physically associating polymer solutions (PAPSs) is studied using the selfconsistent 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 HSMFH transition increases. However, the corresponding two quantities of MFHmicelle transition do decrease under the same condition, which is opposite to that of HSMFH transition. It is found that the temperature susceptibility of penetration 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 HSMFH and MFHmicelle 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.
Acknowledgements
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).
Appendix
Following the scheme of Schentiens and Leermakers [35], is the end segment distribution function of the th segment of the chain, which is evaluated from the following recursive relation:
(9) 
where is the free segment weighting factor and is expressed as
The initial condition is for all the values of . In the above expression, the values of depend on the chain model used. We assume that
This means that the chain is described as a random walk without the possibility of direct backfolding. Another end segment distribution function is evaluated from the following recursive relation:
(10) 
with the initial condition for all the values of .
Using the expressions of the end segment distribution functions, the singlesegment probability distribution function and the twosegment probability distribution function of the chain can be defined as follows:
(11) 
which is a normalized probability that the monomer of the chain is on the lattice site ;
and
give the probability that the monomers and of the chain are on the lattice sites and , respectively. It can be verified that , and .
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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