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 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.


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 self-assemble 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 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][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 self-consistent field lattice model. The temperatures at which the above two inhomogenous morphologies first appear, denoted by T MFH and T m , respectively, are dependent on the spacing of sticker along the chain. If the system is cooled from T MFH and T m , 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 solutions-MFH and MFH-micelle transitions. It is found that this result can be interpreted in terms of intrachain and interchain associations.

Theory
We consider a system of incompressible PAPSs, where n P polymers are each composed of N st segments of type sticker monomer (attractive group) and N ns 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 l nonsticky monomers between two neighboring sticker monomers. The degree of polymerization of chain is N = N st + N ns . In addition to polymer monomers, n h 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 N L = n h + n P N . Nearest neighbor pairs of stickers have attractive interaction −ǫ with ǫ > 0, which is the only non-bonded 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 non-bonded interaction similar to the references [23,27]. The interaction energy is expressed as: where χ is the Flory-Huggins interaction parameter in the solutions, which equals z 2k B T ǫ, z is the coordination number of the lattice used, where r means the summation over all the lattice sites r and φ st (r ) = j s∈st δ r,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 ∈ st means 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 where Q h is the partition function of a solvent molecule subject to the field ω h (r ) = ω + (r ), which is defined as Q h = 1 n h r exp [−ω h (r )]. Q P is the partition function of a noninteracting polymer chain subject to the fields ω st (r ) = ω + (r ) − ω − (r ) and ω ns (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 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 F with ω − (r ) and ω + (r ) leads to the following saddle point equations: ω − (r ) = 2χφ st (r ), (2.3) φ st (r ) + φ ns (r ) + φ h (r ) = 1, are the average numbers of sticker and nonsticky segments at r , respectively, and is the average numbers of solvent molecules at r . Q P is expressed as where r N and α N denote the position and orientation of the N th segment of the chain, respectively. r N α N means the summation over all the possible positions and orientations of the N th segment of the chain. G α s (r, s|1) and G α s (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 backfolding. 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 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.

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 andφ st 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 obtained from the lattice with N L = 40 3 . Three different morphologies in PAPSs are observed, i.e., the homogenous, 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 and micelle morphologies appear in turn. φ P (r ) and φ st (r ) in the solutions with MFH morphology slightly fluctuate aroundφ P andφ st , respectively. The average volume fraction of stickers at the sticker-rich sites (fluctuations) r ri increases with increasing χ for fixed φ P . Its maximum value is about 2φ st , which is much smaller than unity for all theφ P . 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 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 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 l = 19, the χ value on micellar boundary (∼ 1/T m ) increases with decreasingφ P . Whenφ P goes down to a certain extent, micellar boundary becomes steep. The χ value on MFH boundary (∼ 1/T MFH ) also rises with a decrease inφ P . MFH boundary intersects the micellar one atφ CFC , which is the critical MFH concentration (φ CFC = 0.4). When l is decreased, at fixedφ P , the χ value on micellar boundary shifts to a small value, and the χ value on MFH boundary decreases markedly.φ CFC also drops with a decrease in l.
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 χ are studied. Figure 2  the sticker-rich sites (microfluctuations) in MFH morphologies with different spacing of stickers l, which are denoted byφ ri se andφ ri he , respectively, with the χ deviation from MFH boundary, χ r , atφ P = 0.8, wherē φ ri se andφ ri he equalφ ri st /φ st and (φ ri h −φ h )/φ st , respectively. With the increase in χ r ,φ ri se at l = 3 initially

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Effect of distribution of stickers along backbone rises when 0 χ r 0.2 and then maintains a certain value, and the correspondingφ ri he first decreases when 0 χ r 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 (|φ ri he | = 0.047). When l is increased, the shapes of the curves ofφ ri se (χ r ) andφ ri he (χ r ) are similar to the case of l = 3. However, the onset of the range independent of χ r shifts to a larger χ r value. The minimum ofφ ri he (χ r ) goes down with an increasing l. It is demonstrated that in MFH morphology 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 byφ r co st andφ r co hr (=φ r co h −φ h ), respectively, with the χ deviation from micellar boundary, χ r , in the systems with different spacings of stickers l atφ P = 0.8 are shown in figure 2 (b). At l = 3,φ r co st rises and approaches 1, and the correspondingφ r co hr decreases and is close to its minimum when χ r is increased. When l is increased,φ r co st goes up and rapidly approaches 1, and the correspondingφ r co hr goes down and is quickly close to its minimum, with increasing χ r . 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 χ r , which is contrary to that of MFH morphology.  In order to demonstrate the property of the observed transition, the heat capacity C V 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 k B ): The C V (χ r ) curves for the HS-MFH and MFH-micelle transitions in various l atφ P = 0.8 are shown in figures 3 (a) and 3 (b), respectively. For HS-MFH transition, a peak appears in each C V (χ r ) 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 C V (χ r ) 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

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of spacing of stickers on C V (χ r ) 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 observed 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 in MFH morphology increase. Under the same condition, the broadness of the corresponding transition also increases. Meanwhile, for MFH-micelle 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 temperature-dependent 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 unimer-micelle 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 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 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 s 1 is concerned with intrachain association, provided that the sticker s 1 is at an aggregate of the two above mentioned types whose position locates at r ag , can be expressed as: p loop (r ag , s 1 ) = 1 P (1) (r ag , s 1 ) s 2 ∈st,s 2 s 1 P (2) (r ag , s 1 ; r ag , s 2 ), where s 2 ∈st,s 2 s 1 means the summation over all the stickers of a polymer chain except the s 1 th one, while P (1) (r ag , s 1 ) and P (2) (r ag , s 1 ; r ag , s 2 ), whose expressions are given in appendix, are the single-segment and two-segment probability distribution functions of a chain, respectively. Then, 1 − p loop (r ag , s 1 ) is the conditional probability that the sticker s 1 is linked with those that belong to other chains when the sticker s 1 is at r ag , and P l k (s 1 ) = r ag P (1) (r ag , s 1 ) 1 − p loop (r ag , s 1 ) is the probability that a sticker s 1 of a chain is related to an interchain association, where r ag means the summation over all the aggregates in the system. The summation of P l k (s 1 ) over all the stickers in a chain, 〈n l k 〉 = s 1 ,s 1 ∈st P l k (s 1 ), 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 asf t e = 〈n l k 〉/N st . The average fraction of intrachain association of a sticker is defined asf tr = (1/N st ) s 1 ,s 1 ∈st r ag p loop (r ag , s 1 ). Figure 4 shows the variations of average fractions of intrachain association and interchain association of a sticker, denoted byf tr andf te , respectively, with the χ deviation from the MFH or micellar boundary, χ r , in the system with different spacing of stickers l atφ P = 0.8. In MFH morphology [figure 4 (a)], when χ r is increased,f te at l = 3 first rises when 0 χ r 0.2, then maintains a certain value, and the correspondingf tr always remains constant when χ r > 0, andf tr is smaller thanf te . When l is increased, the variation off te with χ r resembles that of l = 3. However, a certain value at whichf te finally arrives rises, and the corresponding χ r also goes up, with χ r increasing.f tr decreases markedly when l is increased. f tr is much smaller than the correspondingf te . It is shown that in MFH morphology intrachain association is independent of χ r , and the range of interchain association which is concerned with χ r rises when spacing of stickers is increased. In micelle morphology  on it are similar to those of MFH morphology. However, there exists an evident difference in the curve of f te (χ r ), which is not smooth. The intrachain associations of micellesf tr are dependent on χ r . The behavior is distinct for small l and different from MFH morphology. It is noted that at l = 3, when 0 < χ r 0.4, the extent of rise off tr andf te changes alternately with increasing χ r . As discussed above, the increase of spacing of stickers l has different effect on the HS-MFH 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 morphologyf ri te is much larger than that of the corresponding intrachain quantityf ri tr , andf ri tr is absolutely independent of temperature. Therefore, the temperature-dependent property of MFH morphology is determined by interchain association. When l is increased,f 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 stickerf ro tr is dependent on temperature, especially in the case of small l. Therefore, the property of MFH-micelle transition is determined by both intrachain and interchain associations [ figure 4 (b)]. At l = 3, bothf ro tr andf ro te are sensitive to temperature in the transition region. When l is increased, the susceptibility off ro tr to temperature decreases markedly. Although an increase in l is favorable to an increase in the susceptibility off ro te to temperature, it may be weak compared with the corresponding decrease off 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.

Conclusion and summary
The effect of distribution of stickers along the backbone on the temperature-dependent property of aggregation 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. However, 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 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 HS-MFH and MFH-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.