Theoretical coarse-graining approach to bridge length scales in diblock copolymer liquids
A microscopic theory for coarse graining diblock copolymers into dumbbells of interacting soft colloidal particles has been developed, based on the solution of liquid-state integral equations. The Ornstein-Zernike equation is solved to provide a mesoscopic description of the diblock copolymer system at the level of block centers of mass, and at the level of polymer centers of mass. Analytical forms of the total correlation functions for block-block, block-monomer, and center-of-mass pairs are obtained for a liquid of structurally symmetric diblock copolymers as a function of temperature, density, chain length, and chain composition. The theory correctly predicts thermodynamically-driven segregation of diblocks into microdomains as a function of temperature ( parameter). The coarse-grained description contains contributions from density and concentration fluctuations, with the latter becoming dominant as temperature decreases. Numerical calculations for the block coarse-grained total correlation functions, as a function of the proximity of the system to its phase transition, are presented. Comparison with united atom molecular dynamics simulations are carried out in the athermal regime, where simulations and theory quantitatively agree with no need of adjustable parameters.
One of the challenges in understanding the properties of polymeric materials stems from the necessity of developing theoretical approaches that can describe in a comprehensive manner properties observed at many different length scales. The presence of several length scales in which relevant phenomena take place leads to the complex nature of the liquid, rendering its treatment a difficult matter complexfluids (); complexfluids1 (); complexfluids2 (). Already in the description of the structure of homopolymer melts, two length scales need to be considered, which correspond to the monomer statistical segment length, , and the overall polymer dimension, i.e. its radius of gyration, , where is the total number of monomers in the chain. For diblock copolymers, the theoretical treatment is further complicated by the presence of a new length scale, which is intramolecular in character and corresponds to the size of a block. Diblock copolymers are macromolecules in which a homopolymer chain of monomers of type is chemically bound to a second homopolymer chain of different chemical structure containing monomers of type , with . The block size is defined by its respective radius of gyration such that, for example, the spatial dimension of the block composed of -type monomers is given by .
Diblock copolymers are systems of great interest for their technological applications diblock (); balsara (). Since the two blocks are chemically different, these experience a repulsive interaction that would encourage phase separation at low temperatures where entropy cannot balance enthalpic effects. However, the chemical bond existent between the two blocks prevents a complete separation of the two phases. As a consequence, at low temperatures block copolymer liquids undergo a microphase transition from disordered systems to ordered microstructures of nanoscopic size, namely the microphase separation transition (MST). The length scale characterizing the size of the microphase is of the order of the block radius of gyration.
With the purpose of developing the technology to produce micro-ordered structures of well-controlled size and shape, an understanding is required of the processes that drive the formation of micro-ordered phases under different thermodynamic conditions of temperature and density , as well as different chain composition , monomer structure , and degree of polymerization .
Computer simulations serve as an extremely powerful tool to investigate phenomena in complex fluids simul (). However, the limited power of present-day hardware does not allow for the simultaneous study of all length scales of interest. One way to overcome this problem is through “multiscale modeling,” where a set of simulations is performed at different levels of coarse-graining of the original system and in a subsequent step, information from different length scales is combined to provide the complete physical picture simul1 (). With such an approach, however, the challenge is not only to find the appropriate computational technique for each length scale simulated, but also to know (i) the proper effective potential acting between coarse-grained units needed to carry out the simulations DAUTN (); KRMER (); Maass (); BRIELS (), and (ii) the proper procedure for combining information from different scales of modeling once simulation data is acquired.
First-principles theoretical models apt to coarse grain diblock copolymer liquids at different length scales of interest provide the potentials needed as an input in multiscale simulations, as well as the formal framework to combine information obtained from simulations of the liquid coarse grained at different length scales K2002 (); K2003 (); K2004 (); HNCOL (); MNBDY (). In a series of recent papers, we developed a coarse-graining approach that maps liquids of homopolymer chains into liquids of soft colloidal particles, providing a formal analytical “transcale” procedure YAPRL (); MLTEX (); JOPCM (). Each effective soft colloid is centered on the polymer center of mass and interacts through a Gaussian repulsive potential, which results in our formalism from the solution of Ornstein-Zernike integral equations. We later extended the same approach to describe the coarse-graining of homopolymer mixtures YAPRL (); BLNDS (). Computer simulations, performed by us, of coarse-grained polymer liquids and mixtures, where molecules interact by means of the derived effective pair potential, have been shown to reproduce quantitatively the structure and dynamics of the liquid at the center-of-mass level, while requiring considerably shorter computational time than that needed to perform united atom simulations YAPRL (); MLTEX (); JOPCM (); BLNDS ().
In the present work, our approach is further developed to address the problem of modeling a melt of diblock copolymers as a liquid of interacting soft colloidal dumbbells. Each dumbbell represents one macromolecule composed of two effective soft colloidal particles, which in turn are sized according to the radius of gyration of each block and centered on center-of-mass coordinates of each block. Three different length scales are formally related, which correspond to coarse-graining the molecule at the monomer (the statistical segment length, ), block (the radius of gyration of block , ), and polymer (the polymer radius of gyration, ) scales. In this way, our theory represents a minimal intramolecular mesoscopic model of polymeric liquid structures.
There has been a growing interest in providing models for coarse-graining block copolymers chains.Andelman (); muller (); Liang () For example, building blocks of supermolecular structures, such as cellular membranes, could be modeled as self-assembling block copolymers chains.Pier () A recent paper proposes a model of coarse-graining for a symmetric diblock copolymer similar to ours, as the chain is modeled as two soft blobs, tethered by an entropic spring.Addison () The blobs have equal size, and the coarse-grained total distribution functions are calculated numerically from a Monte Carlo simulation of diblock copolymers described at the monomer level. Monomers occupy the sites of a simple cubic lattice, with bond along the x-, y-, or z-directions. The two blocks individually are modeled as if they where in theta solvent, while the interaction between them is self avoiding. The numerical inversion procedure to derive the coarse-grained potential is performed in the athermal regime. As the authors point out in the paper, their model is ”highly simplified”, which proves the difficulty in treating intramolecular coarse-graining. The model, coupled with a reference interaction site model (RISM) and a random-phase approximation closure, predicts the mean-field clustering of diblock copolymers in a selective solvent.Hansen ()
In this paper, we provide an analytical solution for the coarse-grained total distribution functions for a liquid of diblock copolymers represented as dumbbells of soft colloidal particles. Our model differs from the one presented in Ref.(Addison ()) in several ways. In our case, the size of the two ”blobs” varies depending on the chain composition, , degree of polymerization, , and segment length, . Moreover, repulsive interactions between segments of different chemical nature are quantified by the interaction parameter, . Concentration-fluctuation stabilization enters through the polymer reference interaction site model (PRISM) theory for the monomer-level description,PRISM (); eddavid (); eddavid1 () and deviations from mean-field theoryLEIBL () are predicted by the coarse-grained approach as well. The two blocks follow Gaussian intramolecular statistics, which is a good approximation for copolymer melts, when each block has a degree of polymerization , with , and for the region in the phase diagram from the high-temperature to the weak segregation regime ( for symmetric composition ), where the system is isotropic. Numerical mean field theory studies suggest coil stretching is not significant even below the order disorder transition until a strong segregation regime is entered, where .44a (); 44b () Analytical intermolecular total correlation functions between like and unlike coarse-grained blocks are predicted by our formalism as a function of chain composition, block size, density, temperature, as well as density- and concentration-fluctuation screening lengths.
One advantage of our approach is that it is analytical. Since a coarse-grained description is obtained by performing statistical averages over local (small-scale) degrees of freedom, it translates the energy of the system into the free energy of the renormalized fluid.Likos () In this way, the obtained total correlation functions and related effective potentials, are functions of all characteristic physical parameters defining the system under consideration, such as temperature , total site density , and degree of polymerization . For a diblock copolymer, the relevant parameters also include chain composition, , and the interaction parameter . The parameter defines the proximity of the system to its order-disorder transition. For each set of parameters, total correlation functions and free energy change. As a result, a numerical solution of the coarse-grained description obtained from microscopic simulations requires performing a number of simulations equal to the number of combinations of those parameters, partially defeating the computational gains of a coarse-grained description.
The material in this paper is organized in the following manner. We start in Section II with a derivation from the Ornstein-Zernike equation of the total correlation functions of diblock copolymer liquids coarse-grained at the block level. Section III provides a theoretical description of diblock copolymer melts at the monomer level. In Section IV, we present an analytically tractable solution in reciprocal space for a structurally symmetric diblock copolymer melt. The Fourier transform of the resulting expressions leads to analytical solutions of intermolecular total block-monomer and block-block pair correlation functions in real space, treated in Section V. Our analytical approach for coarse-graining the diblock copolymer liquid at the center-of-mass level and comparisons with the corresponding presentation of the homopolymer melt are discussed in Section VI. Finally, our theoretical development in the athermal limit is compared with united atom molecular dynamics simulations of a polyethylene melt in Section VII, while temperature-dependent model calculations are presented in Section VIII. The paper concludes with a brief discussion and Appendices, where the auxiliary functions entering the exact solution of the total correlation functions, as well as the treatment of the block coarse-graining of a compositionally symmetric diblock, are presented.
Ii An integral equation approach to coarse-grain diblock copolymers at the block length scale
In this section, we derive the general expressions for the total pair correlation functions of the diblock copolymer liquid, coarse-grained at the level of the block length scale. The formalism is completely general and applies to any diblock copolymer system. The structure of a diblock copolymer liquid is characterized well by static correlation functions, which sample fluctuations of monomer units in the fluid. Given the position of monomer belonging to block comprising a chain as , monomer fluctuations at a specific wave vector are represented by . Since all chains are assumed to be equivalent, we henceforth discard the chain index . Density fluctuations for a generic monomer inside a block are defined as , while defines its complex conjugate. The two-point density correlation functions, which describe the liquid structure, are given by the partial structure factor
which includes static correlations between monomers belonging to the same chain, , i.e. the intramolecular static structure factor, and correlations between monomers belonging to different chains, , i.e. the intermolecular structure factor. Since the liquid is spatially homogeneous and isotropic, the structure factors depend only on the modulus of the wave vector, .
In our coarse-grained description, each block comprising the chain is mapped onto an effective particle, centered on the position of the block center-of-mass coordinate, . Fluctuations from the center of mass of block are defined as , and the partial structure factor becomes
which requires knowledge of both intra- and intermolecular correlations. Block structure factors are normalized by the number of blocks in the chain, which is two for a diblock copolymer molecule.
By analogy, block-monomer two-point correlation functions are given by
which for compositionally asymmetric chains, , , and consequently .
Intra- and intermolecular structure factors are related through the Ornstein-Zernike equation, which has the general matrix formula
Here, is the intermolecular direct pair correlation function matrix. In our description, the generalized Ornstein-Zernike equation includes contributions from “real” monomeric sites () and “auxiliary” sites positioned on the block center-of-mass site ().
At the block level description, matrices share similar arrangements. As an example we show the partial static structure factor, , which is defined as
where we omit the variable to simplify the notation. In an analogous way we define the intramolecular structure factor matrix, which contains the correlation between real sites, , auxiliary sites, , and the corresponding cross contributions, . Here, the block number density is , and for a diblock copolymer. Furthermore, for blocks of different type, , we have that and , while .
Consistent with its intramolecular counterpart, the matrix of total intermolecular pair correlation functions contains the correlation term for the real sites, , auxiliary sites , and cross contributions .
Here the number density of monomers is and , while the number density of blocks of type (or ) is for a diblock copolymer.
Finally, the intermolecular direct pair correlation function matrix includes the usual assumption that auxiliary sites are not directly correlated with either real or other auxiliary sites. In this theoretical framework, we obtain the correlation in fluctuations of the intermolecular block-monomer function
and the intermolecular block-block function
where , since whenever with . The general block-block relation reads
where we used the property that both and are symmetric matrices, along with the definition . Eq.(7) formally relates the total correlation function for a coarse-grained diblock copolymer, represented as a dumbbell of two soft colloidal particles, to the total correlation function and intramolecular structure factor of the monomer-level description.
and , while the block-block total correlation functions follows
Since no particular structure of the diblock copolymer has been assumed thus far, Eqs. (9) and (11) are completely general and hold for any model of diblock copolymer chains, including diblock copolymers with asymmetric chain segments, as well as any general form of an interaction potential. From the knowledge of the pair correlation functions, obtained from the Fourier transform of and , all thermodynamic properties of a polymer liquid can be formally derived hansenmacd ().
In the following sections, we present an analytical solution for the intermolecular block-monomer and block-block correlation functions for a diblock copolymer liquid. We assume a structurally and interaction symmetric diblock with variable chain composition. The molecule is modeled as a Gaussian “thread” of infinite length and vanishing thickness. This model allows for an analytical solution of the total correlation functions in real and reciprocal spaces, as a function of the thermodynamic parameters of the system.
Iii Monomer level description of diblock copolymer liquids
The coarse-graining formalism presented in Section II is simplified when structurally and interaction symmetric diblock copolymers are investigated. For these model systems, segments of different chemical nature are assumed to have equivalent statistical lengths, , while the specific chemical nature of the block enters as an effective persistence length, through the block radius of gyration. Segments of like species interact through the potentials , while unlike species repel each other through . At high temperatures, entropic effects dominate over enthalpic contributions, and block copolymer liquids resemble closely liquids of homopolymer molecules. As the temperature decreases, the effective repulsive potential increases as , leading to the phase separation transition. This phase transition is characterized by a dramatic increase of the collective concentration fluctuation static structure factor, , at a specific length scale, . At the temperature of the phase transition, only certain fluctuations become anomalously large and the liquid segregates on a molecular length scale on the order of the overall size of the molecule, . This remarkable property of copolymer liquids is due to the fact that, because of the connectivity between different blocks, even complete segregation cannot lead to macroscopic phase separation, as occurs in mixtures of two chemically different homopolymer melts. Since even at high temperatures, presents a peak due to the finite molecular size of the block copolymer chain, the peak position is largely independent of temperature.
The first theoretical approach to describe the microphase separation transition was a mean-field theory developed by Leibler LEIBL (). The theory is built on the expansion of the free-energy density of an ordered phase in powers of the order parameter, defined as the average deviation from the uniform distribution of monomers. The theory predicts a second-order phase transition for the symmetric mixture () and a first-order transition for asymmetric systems. Mean-field approaches are usually less accurate in the vicinity of the transition, where fluctuation corrections to the mean-field theory can change drastically the phase diagram. In the case of diblock copolymer melts, these corrections modify the predicted second-order phase transition for the symmetric diblock into a first-order phase transition.
A fluctuation corrected mean-field approach was later derived by Fredrickson and Helfand FRDIK () by implementing Brazovskii’s Hartree approximation of Landau-Ginzburg field theory to treat diblock copolymer systems BRAZO (). The fluctuation corrected approach recovers Leibler’s results in the limit of infinite chain length. Fredrickson-Helfand predictions have been found to agree well with scattering experiments in the whole range of temperatures encompassing disordered to weakly ordered phases across the microphase separation. Since these approaches focus on the single-chain free energy and the liquid is assumed to be incompressible, fluctuation stabilization is mainly of entropic origin.
Schweizer and coworkers developed an integral equation description of block copolymer melts that formally recovers the scaling behaviors obtained in field theories with only small differences eddavid (); eddavid1 (); MDCMP (); MDDYN (); MDDYN1 (). In this case, however, the stabilization of the disordered state close to the MST is of enthalpic origin. Moreover, since the theory does not enforce incompressibility as a starting point in the treatment, the resulting structure factor contains contributions from both density and concentration fluctuations. This is consistent with the physical picture that in block copolymer melts far from their MST, the concentration fluctuation contribution is negligible while density fluctuations can still occur.
In this work, we adopt an integral equation approach to describe the block copolymer structure at the monomer level, extending Schweizer’s theory. This liquid-state description is largely compatible with the fluctuation-corrected mean-field approach, and has the advantage of providing a theoretical framework that is formally consistent with the procedure presented in the previous section. In this way, the approach presented here builds on, and merges two well-developed theoretical fields involving (i) the extension of integral equation approaches to treat complex molecular liquids blockschweizer (); PRISM (), and (ii) procedures to coarse grain macromolecular liquids YAPRL (); MLTEX (); JOPCM (); K2002 (); HNCOL (); MNBDY ().
We focus on a structurally and interaction symmetric diblock copolymer. In the framework of an Ornstein-Zernike approach for this model system, repulsive intermolecular interactions between and species, at the monomer level, are defined by the direct pair correlation function containing hard-core intermolecular repulsive interactions between like species, , and a sum of repulsive hard-core and tail potentials for intermolecular interactions between monomers of unlike species, . The effective interaction parameter, , controls the amplitude of microdomain scale concentration fluctuations and increases as the system approaches its MST. Realistic diblock copolymer systems can be mapped onto this simplified model, which has been extensively investigated in the past blockschweizer (). In the theoretical coarse-graining approach presented here, the different chemical nature of the two blocks is accounted for by the difference in their radii of gyration, which is a function of the polymer local flexibility. This assumption is well justified in our approach since the monomeric structure is averaged out by the coarse-graining procedure, while local flexibility enters through the block radii of gyration.
As a starting point in our derivation, we focus on the monomeric quantities which are input to our coarse-grained description for a diblock copolymer system, Eqs. (9) and (11). As a first approximation, we assume that all monomers comprising a block are equivalent, so that each component in Eqs. (9) and (11) represents a site-averaged quantity. This is the conventional approximation adopted for treating analytically integral equations for polymeric liquids, and becomes correct when each block in the copolymer includes a number of monomers large enough to minimize chain end effects. The approximation is formally consistent with our coarse-graining theory, where physical quantities at the monomer level are averaged over the monomer distribution.
The monomer total pair correlation function , with , in reciprocal space is defined as the difference between the total static structure factor and its intramolecular contribution as
where the total pair correlation function obeys the relationship
We adopt here the thread model for the monomer-level description of the polymer chain. This model allows us to obtain analytical equations for our coarse-grained description of the diblock copolymer system. In the thread model, the chain is treated as an infinite thread of vanishing thickness, with hard-core monomer diameter approaching zero, , and a diverging segment number density in the chain, , while remains finite. The thread model yields a good description of properties at the length scale of , including the presence of a correlation hole in the monomer pair correlation function. It cannot account for the local fine structure observed in the radial pair distribution function, , which is related to the presence of solvation shells due to monomer hard-core interactions. However, since our renormalized structures are characterized by a size comparable to the block domain, it gives a good representation of the coarse-grained structure for long chains where the block size is larger than the monomer diameter.
Each of the site-averaged components of the total static structure factor contains contributions from density and concentration fluctuations, and can be expressed as a function of the density screening length asMDDYN ()
The incompressible concentration structure factor is defined in Leibler’s mean-field approach as
and diverges at the spinodal temperature, where with the spinodal temperature defined as . The function depends on the intramolecular static structure factors with through the definition of given by Eq. (10), and
To take into account the phase stabilization due to fluctuation effects, it is convenient to rewrite the incompressible structure factor in the following approximate form FREDRICKSON (),
This expression takes into account the fact that when the spinodal condition is fulfilled, the inverse concentration contribution of the structure factor does not vanish: the disordered phase is still present and eventually the system undergoes a first-order phase transition.
With the purpose of obtaining an analytical expression for the coarse-grained system, we introduce the Otha-Kawasaki approximation OHTAK () given by , with . Here, and . At the peak position, defined as , the contribution , thus recovering the spinodal condition of . By inserting the Ohta-Kawasaki approximation into Eq. (18), the incompressible concentration fluctuation collective structure factor reduces to
which after expanding into partial fractions, leads to the following tractable analytical expression for the concentration fluctuation contribution to the static structure factor MDDYN (),
characterized by two length scales and . At the peak position, , the concentration structure factor behaves as . In the small wave vector limit , it increases as , in agreement with the observed scaling behavior for homopolymer liquids. Consistently, for the large wave vector limit , it tends to zero as since , thereby recovering Leibler’s scaling. The scaling with at large and small wave vectors is also observed when homopolymer systems are investigated. This is a characteristic feature of block copolymer systems, signifying that at very large scales, as well as on very local scales, fluctuations are independent of the effective repulsion between monomers of unlike chemical nature.
The solution of Eqs. (9) and (11) relies on the definition of monomer-monomer and block-monomer intramolecular form factors. The monomer-monomer form factors are well described by the approximated Debye function,
which can be conveniently simplified into their corresponding Padé approximants.doiedw (). In analogy with the center-of-mass monomer formalism YAMAKAWA (), we approximate the block-monomer intramolecular structure factor in reciprocal space by the following Gaussian distribution function
which includes the mean-squared radius of gyration, describing the squared average distance of a monomer of type from the center of mass of an -type block,
In real space, is the generic intramolecular distribution function for any one of the segments in a -type block with respect to the center of mass of an -type block. Finally, we define the intramolecular total block-monomer structure factor as
which in the regime yields .
While Eq. (22) is a well-known expression when it applies to the distribution of monomers around the center-of-mass of an unperturbed homopolymer chain YAMAKAWA (), its extension to diblock copolymers is novel. When tested against united atom simulation data (see Fig. 1 and the discussion of Section VII), these analytical expressions are fairly accurate for both symmetric and asymmetric diblock copolymers, while their simple Gaussian form allows us to derive analytical equations for the block total correlation functions in real space.
Iv Block coarse-grained description in reciprocal space, and isothermal compressibility
In the large- regime, which is of interest in block copolymer melts due to the finite size of the microphase separation, the ratios of intramolecular structure factors follow the relations , , and . By enforcing the approximation that , which is justified by the fact that and vanishes for long polymer chains, , the total block-monomer and block-block correlation functions simplify. Including these approximations into the monomer-monomer and block-monomer structure factors reduce Eqs. (9) to the analytical general forms
Following the same procedure, the block-block total correlation functions in reciprocal space, Eqs. (11), reduce to
where the density contribution is identical to the monomer total correlation function for a homopolymer chain PRISM () , and the concentration fluctuation contribution at some thermal state point , having as a reference the athermal state , is derived from Eq. (20) as with .
The total block-monomer intermolecular pair correlation function reads
where the contribution from concentration fluctuations rigorously vanishes, as is the case for the monomer level description of a diblock copolymer melt. The total block-block intermolecular pair correlation function is given by
When compositionally asymmetric diblocks are investigated, the concentration fluctuation contribution to Eq. (28) does not vanish, but instead provides a small correction to the density fluctuation part. However, under athermal conditions or in the limit, only density fluctuations are relevant since the concentration fluctuation contribution vanishes in a manner consistent with the monomer level description. It is worth noting that in the limit of a block approaching the size of the polymeric molecule, and in the limit of blocks of identical length (see Section B of the Appendix), Eq. (28) recovers the homopolymer expression for the molecular center-of-mass total pair correlation function, with concentration fluctuations strictly vanishing.
As a test of our formalism, we present in Section VII a comparison of Eqs. (25), (26) and Eqs. (27), (28) against simulation data in the athermal regime. All equations show good agreement with simulations for both compositionally symmetric and asymmetric diblock copolymers (see also Figs. 2 and 3), thus supporting the validity of our procedure.
Finally, starting from Eqs. (27) and (28), we calculate the isothermal compressibility of the system. Since the latter is a bulk property, it has to be independent of the level of coarse-graining adopted in the description of molecules in the liquid. The isothermal compressibility of the system coarse-grained at the block-monomer level, , is obtained from the matricial definition , after taking the limit and adimensionalizing the static structure factor. Each contribution is given by the relation , which yields . In an analogous way, the compressibility of the system coarse-grained at the block-block level is calculated from the relation , and it is obtained from the matricial definition , where . Since , this result is consistent with our prior findings obtained when coarse-graining homopolymer melts at the center-of-mass level, validating the feature that bulk properties, such as , are independent of the fundamental unit (or frame of reference) chosen to represent the system.
Reproducing the isothermal compressibility of the system, after performing a coarse-graining procedure, is an important test of the latter. Due to the nature of the coarse-graining procedure, effective coarse-grained potentials derived from pair distribution functions are softer than their microscopic counterparts. In fact, while real units, such as chain monomers, cannot physically superimpose, auxiliary sites can and the potential at contact is finite. For this reason, the occurrence of small errors in the evaluation of the potential, which is often the case for numerically evaluated coarse-grained potentials, yields liquids that are too compressible. This shortcoming is eliminated in systems for which coarse-grained total correlation functions can be evaluated analytically, as it is in our case.
V Analytical block-level description in real space
v.1 Block-monomer total correlation function
For a structurally and interaction symmetric diblock copolymer, the total pair correlation functions for block-monomer and block-block terms in real space can be expressed analytically through a simple Fourier transform. The block-monomer expression separates into density and concentration fluctuation contributions as
with . The density fluctuation contribution is represented explicitly by the relations
where for compactness, we introduce the auxiliary function, , defined by Eq. (59) of the Appendix. More specifically, Eq. (59) represents the density fluctuation contribution for one block, and is identical in form to the expression derived in our previous work for the center-of-mass-monomer total correlation function in homopolymer melts, coarse-grained at the polymer center-of-mass level MLTEX ().
The concentration fluctuation contribution in real space is given by the relations
where each term is defined as the difference in the response of the concentration fluctuation contribution between some thermal state () and the reference athermal state (), as , with defined by Eq. (61) of the Appendix.
In the microscopic, small regime, the concentration fluctuation contribution in Eq. (31) reduces to
This is the regime most relevant for block copolymer liquids approaching their phase transition, since the microphase separation transition is characterized by segregation on spatial scales close in magnitude to the polymer radius of gyration. The temperature dependence enters Eq.(31) through the parameter in Eq.(32), evaluated at the reference athermal and thermal states. In this way, at high temperatures, i.e. far from the phase transition, density fluctuations are dominant over concentration fluctuations, and the total correlation function for diblock copolymer liquids recovers that of the homopolymer.
In proximity of the phase transition and for long polymeric chains, Eqs. (31) further simplify with the cross terms becoming negligible, while self terms yield the main contribution,
Here, is the average distance of a monomer of type from the center-of-mass of the block of type , as defined in Eq. (23). A numerical study of these approximated expressions shows that neglecting terms with the most “cross” character is a reasonable approximation in real space, valid under different temperature limits and even when the system is asymmetric.
A measure of the physical clustering with temperature is given by the parameter
where . Since the number of monomers of type included within a sphere of radius from the center-of-mass of block is given by
with , Eq. (34) represents a measure of the clustering of monomers around blocks of like species. The density fluctuation contribution to Eq. (34) exactly cancels, while the concentration fluctuation contribution increases with decreasing temperature (increasing ). At contact (), we obtain
In the athermal limit block-monomer clustering due to concentration fluctuations vanishes, and . At lower temperatures clustering of like species increases, with the leading factor being proportional to the ratio , which control the strength of finite-size coupling of microdomains () and local () correlations.
v.2 Block-block total correlation function
The block-block intermolecular total pair correlation functions can be solved in an analogous way to give in real space,
where the separation between density and concentration contributions is conserved. The density fluctuation contribution is given by
with , and defined by Eq. (60) of the Appendix. The distance , where is the average distance of a monomer of type from the center-of-mass of the block of type , as defined in Eq. (23). We note that Eq. (60) was previously derived by us, in the context of coarse-graining a homopolymer liquid at the center-of-mass level YAPRL (); MLTEX (); JOPCM (). This term represents the total pair correlation function for a liquid of interacting soft colloidal particles, centered on the spatial position of the polymer center of mass. This simple analytical expression reproduces well data from united atom molecular dynamics simulations of polymer melts.
The concentration fluctuation contribution is given by the general equation
where we define and the auxiliary function by Eq. (63) of the Appendix. In the small regime of interest here, the concentration fluctuation contribution simplifies, yielding for the generic contribution in Eq. (39) the following simplified expression
For long polymeric chains in general the cross statistical distances are larger than the self ones, , and Eqs. (39) simplify to