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Computer Study of Chromatographic Separation Process: A Monte Carlo Study of H‑Shaped and Linear Homopolymers in Good Solvent Xiu Wang,† Martin Lísal,‡,§ Karel Procházka,*,† and Zuzana Limpouchová*,† †

Department of Physical and Macromolecular Chemistry, Faculty of Science, Charles University in Prague, Hlavova 8, 128 00 Praha 2, Czech Republic ‡ Laboratory of Aerosol Chemistry and Physics, Institute of Chemical Process Fundamentals of the CAS, v. v. i., Rozvojová 135/1, 165 02 Praha-Suchdol, Czech Republic § Department of Physics, Faculty of Science, J. E. Purkinje University, Č eské Mládeže 8, 400 96 Ú stí n. Lab., Czech Republic S Supporting Information *

ABSTRACT: The partitioning of linear (L) and H-shaped polymers between bulk solvent and narrow pores with inert and attractive walls and the conformational behavior of chains in pores was studied by Monte Carlo simulations. The polymer chains were modeled as self-avoiding walks in a good solvent. The concentration profiles in the pores, partition coefficients KH and KL, and various structural characteristics were calculated as functions of pore size and interaction parameter ε, ranging from 0 to −0.26. KH is higher than KL in pores with nonattractive walls, but the difference decreases with increasing |ε|. Both partition coefficients equal for ε* ca. −0.2, and later their sequence inverts. ε* depends only slightly on chain architecture and chain length. The results are important from the experimental point of view because they show that the improperly chosen experimental conditions can deteriorate SEC analysis of branched samples.

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INTRODUCTION Size-exclusion chromatography (SEC) is a widespread method which has been used for the characterization, purification, and separation of macromolecular compounds since the middle of the 20th century.1 The basic feature of SEC is a slow transport of the solution containing a mixture of macromolecular compounds through porous media. The neat SEC mechanism does not assume enthalpic interactions of the solute with the stationary phase, and the separation of polymer compounds differing in molar mass is based on steric effects only; it is controlled by differences in the conformational entropy of polymer chains in the pores and in the bulk solvent.2 In past few decades, several chromatography techniques combining SEC with other types of separation mechanisms have been designed and successfully employed in polymer research.3−11 The development and optimization of these chromatography methods, based on temperature-induced change of the sorption strength,3,4 the interaction of polymers with hydrophobically modified stationary phase,5,6 the transient variation of the sorption strength,11 etc., opened new dimensions for the characterization and separation of polymer compounds. The general physical model of the chromatographic separation of polymer species assumes the competition of entropic effects and enthalpic interactions between the solute (polymer chains), solvent, and the stationary phase. In polymer physics, interactions that control the behavior of polymer chains in bulk solvent (in the mobile phase) are described by the Flory−Huggins (FH) parameter χ, which compares polymer−solvent interactions with the average of polymer− polymer and solvent−solvent interactions.12,13 The FH © XXXX American Chemical Society

parameter serves as a measure of the thermodynamic quality of the solvent. Good solvents (with χ < 1/2) dissolve the polymer and poor solvents (with χ > 1/2) are actually nonsolvents of long polymer chains. In the so-called ϑ-solvent (χ = 1/2), slightly unfavorable polymer−solvent interactions compensate for the effect of excluded volume of monomer units; hence, the polymer chains in ϑ-solvents behave as ideal intersecting random walks (RW). Other interactions of interest are those between the components of the mobile phase and the stationary phase. In the chromatography literature, the term “solvent strength” describes the relative strength of favorable (attractive) interactions of the solvent with the stationary phase in comparison with the solute−stationary phase interactions. Strong solvents weaken the interaction of the solute (polymer) with the stationary phase and hinder its retention, while weak solvents promote the interaction and the retention of the solute.14 As compared with the neat SEC mechanism, an accurate theoretical description of methods combining SEC with interaction chromatography (IC) is more complicated. However, the evaluation of the equilibrium partition coefficient K from pseudoexperimental computer-based data is straightforward, and the discussion of physicochemical trends can be based on the general expression8 Received: October 24, 2015 Revised: January 7, 2016

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DOI: 10.1021/acs.macromol.5b02327 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules ⎛ ΔG ⎞ ⎛ ΔH C ΔS ⎞ + K = s = exp⎜ − ⎟ = exp⎜ − ⎟ Cm kb ⎠ ⎝ k bT ⎠ ⎝ k bT

versus roughness of the surface57 have been published recently. However, the mosaic of knowledge is still incomplete, and there exists a vast room for new systematic studies. In this paper, we study and compare the conformational and retention behavior of L- and H-shaped polymers in narrow pores with attractive walls. As the behavior of L chains is basically known, we focus on H chains and use L chains as a reference system. We study the trends of the behavior and properties of chains as functions of pore size D and interaction strength ε. The aims of this study are (i) the elucidation of the role of attractive interactions between polymer beads and pore walls on different chain architectures and (ii) better understanding of the competition of confinement effects with the adsorption of polymer chains on pore walls.

(1)

where Cs and Cm are the equilibrium concentrations of the solute in the stationary and mobile phases, respectively, kb is the Boltzmann constant, T is the temperature, and ΔG is the change of the Gibbs function accompanying the transfer of one molecule from the bulk mobile phase into the pore, which contains the enthalpic (ΔH) and entropic (ΔS) contributions. As already mentioned, SEC is a typical entropy-driven chromatography mode; i.e., the contribution of enthalpy is zero. In this method, the high molar mass polymers pass through the column faster than the smaller ones. The enthalpic contribution (if relevant) is reflected by the term −ΔH/ kbT in eq 1. Its net effect depends (apart from other factors) also on the size of the solute, i.e., on the number of interaction sites between the solute and the stationary phase. This means that attractive interaction promotes the retention of high molar mass polymers more than that of low molar mass polymers and competes with SEC separation mechanisms; i.e., both effects can mutually compensate. The adsorption strength, which ensures that SEC separation is completely offset by the enthalpic attraction, is named the “critical adsorption point” (CAP). For the elution of ideal chains (RW) via porous materials, this does not depend on the length of the chains, but it was shown that for self-avoiding walks (SAW), it slightly depends on it.15 If the attractive interaction between the polymer and porous stationary phase is higher than the CAP, the separation process is dominated by the adsorption on the porous column packing and approaches the standard interaction liquid chromatography regime. However, enthalpic interactions in confined volumes can generate new features of the separation process because they strongly influence short chains which enter the small pores and almost do not apply for long chains.16 Branched polymers exhibit smaller hydrodynamic volumes as compared to their linear analogues of the same molar mass, which offers the possibility to separate them from the linear ones by SEC.17,18 Various IC techniques have also been employed for the characterization of branched polymers.19 However, the chromatographic separation and characterization of polymers differing simultaneously in chain architecture and in molar mass is very difficult, and in spite of great effort by a number of research teams, the progress is slow as compared with other chromatography methods; the interpretation of experimental data is far from being satisfactory.19−21 Computer simulations (Monte Carlo and molecular dynamics) nowadays belong to standard methods of polymer research. They have been successfully used for the solution of a number of topics in almost all fields of polymer science. The number of simulation papers is very high, and it is why we included only several interesting papers published recently22−34 to demonstrate progress in this field and further we focus only on articles devoted directly to the studied topic. The behavior of various polymer chains close to planar surfaces, in narrow slits and in the pores, and the mechanism of the chromatographic separation of polymer chains have already been studied theoretically35−37 by scaling approaches,38,39 by mean-field calculations,40−43 and particularly by lattice Monte Carlo (MC) simulations.44−59 A number of individual studies concerning the effects of chain architecture44,45 and flexibility,46,47 molar mass,48 polymer concentration,49,50 solvent quality,51 temperature,48,52 polymer−wall interaction,45,48,49 and smoothness

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MODEL, SIMULATION METHOD, AND CALCULATED FUNCTIONS We simulate three types of fairly long flexible chains: linear chains (L), and regular H-chains with a fairly long backbone (H), and (ii) pseudostars with a short backbone (S) and vary their length (from 100 to 250). Schematic representation of a regular H chain and a pseudostar S chain is shown in Figure 1. Simulated systems contain only one type of chains (mixtures will be the subject of upcoming papers).

Figure 1. Architecture of H-shaped polymers: (a) regular H-shaped polymer; (b) pseudostar H-shaped polymer.

The MC simulations of dilute polymer systems were performed on a simple cubic lattice. The size and geometry of the simulation box are depicted in Figure 2. The whole box consists of 400 (x-axis) × 100 (y-axis) × 100 (z-axis) lattice sites. The pore (stationary phase) is modeled as a square channel with variable dimensions (200 × D × D) in x, y, and z directions (in lattice units). D ranges from 16 to 60. Note that we do not study very narrow pores because they do not play an important role in chromatography. The bulk solvent compartment in touch with the pore (mobile phase) has the dimensions (200 × 100 × 100). Each lattice site can be occupied either by one polymer bead or by the solvent. Fixed particles are used to model the porous stationary phase. The pore walls are impermeable both for the polymer and for the solvent, and their interaction with the polymer beads is controlled by the interaction parameter ε (0.0 to −0.26). A short-range attractive (adsorption) interaction ε exists only between the four inner surfaces of the pore walls and the polymer beads located in neighboring layers of each surface. No interaction occurs at the two flat interfaces outside the pore. Analogously to the model used by Zhu et al.,60 the polymer beads at the corner of the squared channel experience the same adsorption energy ε as the others located in regular surface sites, although the former come into contact with two different surfaces. Periodic boundary conditions are imposed in all three dimensions and B


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Helmholtz energy ΔA and internal energy ΔU were evaluated (instead of ΔG and ΔH), when the data were analyzed and interpreted on the basis of eq 1. We performed 5 × 1010 CBMC steps for each simulation run after the equilibration period consisting of 5 × 109 steps. Each simulation run was divided into 20 sub-batches, and the “block average method” was used to yield the estimate of statistical errors. The basic Monte Carlo move is composed of two steps. The first step consists in “dissolving” and the second in the “regrowing” one chain. The chain to be dissolved and the free site where the new chain starts are selected uniformly at random. During the dissolving and regrowing chains, the modified Rosenbluth−Rosenbluth weights, W = Πiωi, are calculated. The weights for all segments except the first one are ωi =

∑ pj , j

Figure 2. Schematic illustration of the bulk-pore simulation box. Dark blue planes represent noninteracting impenetrable walls, dark cyan color corresponds to the interacting impenetrable (pore) walls, and light colors designate the permeable walls with periodic boundary conditions.

⎛ Ej ⎞ pj = exp⎜ − ⎟ ⎝ k bT ⎠

(2)

where the sum is taken over all free neighbor positions of the previous bead and Ej is the energy of a bead in position j. Note that subscript i is the running number of the segment, while j denotes the position around this segment. The weight ω1 for the first bead depends on its particular position and equals p1. When deleting a chain, the beads are removed one by one starting from one (randomly selected) chain end, and the corresponding Rosenbluth−Rosenbluth weights are calculated. When regrowing a new chain, the beads are generated and the growth proceeds according to probabilities pj/ωi. The reversibility for the canonic ensemble requires

impermeable solid interfaces define the geometry of the porous material. The model used considerably simplifies the geometry of real porous column packing materials, which have a complicated topology and which, besides pores (usually of a conical shape), contain a non-negligible convex (slightly curved) surface. Therefore, in our model, we did not try to minimize the noninteracting impermeable flat surface. For the evaluation of partition coefficients in ideal pores, we use the plateau region of constant concentration profiles in the bulk (far enough from the wall) and deep inside the pores (see below). The comparison of data from plateau regions with those evaluated in whole volumes of respective phases can reveal additional effects that may occur in real systems (e.g., certain deterioration of separation efficiency). We study polymer chains of different lengths, but we focus on chains composed of 125 beads. In most simulations, the density of polymer beads (ratio of occupied-to-all lattice sites) was ρ = 0.001 15. The constant average density of the beads requires increasing numbers of chains in simulations for systems with increasing pore diameter. The whole simulation box thus contained 19−25 polymer chains (either L or H chains), and the total number of polymer beads was ca. 2500. Supplementary simulations for different densities and lengths of chains have also been performed in order to generalize the conclusions. In real chromatographic experiments, the solution moves very slowly, and the partitioning of the solute between mobile and stationary phases is very fast, so that the flow does not disturb the equilibration of the system. Therefore, we simulate the quasi-equilibrium partitioning of polymer chains between pores and arrested (stopped) mobile phase. As already mentioned, the method of self-avoiding walk (SAW) was employed because the excluded volume of beads plays a very important role in confined volumes. We assume a good solvent condition (athermal system); i.e., there is no interaction between the polymer beads. The configuration bias Monte Carlo (CBMC) algorithm has been used because it is an efficient simulation method which secures the fast equilibration of systems composed of chain molecules.61 All simulations were performed in NVT ensemble, and therefore differences in the

πxpx → y = πypy → x ,

⎛ E ⎞ πx = exp⎜ − x ⎟ ⎝ k bT ⎠

(3)

where px→y is the probability of transition from the state x to y. The transition probability px→y can be expressed as a product of the probability px|y for generating the state y from x and the probability p′y that the state y was accepted. The probability px|y is given by px|y = p1 ∏ i

pi ωi

(4)

where the product is taken over all beads except the first one. From the above equation, it follows that px/py = Wx/Wy, which means that the new chain is accepted with the probability p′ = min(1, Wx/Wy) and the old one with the probability (1 − p′), in accordance with the Metropolis criterion. Concentration Profiles. The 2D concentration profiles describe the concentration of polymer beads as a function of two spatial coordinates: (i) the position (depth) inside the pore, x, and the radial distance from the central pore axis, r. Their determination is described in the Supporting Information. The r-averaged and x-averaged concentration profiles, Φ(x) and Φ(r), describe the concentrations of polymer beads in the direction of the pore axis and in the direction across the pore (i.e., distance from the pore axis), respectively. They were obtained by averaging the corresponding 2D profiles over x and r direction, respectively (see Supporting Information). Structural Characteristics of Chains in Bulk and in Pores. The shapes of chains and their orientations in the pores are described by equivalent ellipsoids of gyration that have the same position of the gravity center and the same principal components of the tensor of inertia as the corresponding chain C


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Figure 3. Concentration profiles of H and L chains for different ε = 0 (black), −0.10 (blue), −0.15 (brown), −0.18 (green), −0.20 (red), −0.22 (magenta), −0.24 (cyan), and −0.26 (orange). The solid and dotted lines represent the H and L chains, respectively. The left column: the concentration profiles of polymer beads along axis x across the whole simulation box: (a) D = 16, (c) D = 30, and (e) D = 60. The right column: the radial concentration profiles of polymer beads in the plateau region of the pore: (b) D = 16 (d), D = 30, and (f) D = 60.

is 7.362 ± 0.002 and that the differences between individual Rg are the prerequisite of their separation by SEC. A detailed description of the partitioning of chains between the bulk and pores and of their behavior in pores is provided by the angularly averaged 2D concentration profiles ρ(x,r). They depict the local concentration of polymer beads as a function of two coordinates: (i) position on the x axis and (ii) distance from the pore axis. The 2D concentration profiles of H chains consisting of 125 beads in pores with size D = 15 and 30 are shown and discussed in the Supporting Information for several

defined, e.g., in ref 62. For the benefit of readers, its definition is also given in the Supporting Information.

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RESULTS AND DISCUSSION Retention Behavior of H-Shaped and Linear Polymers. Basic characterization of all studied systems can be found in the Supporting Information. Here we just recall that the mean gyration radii of chains (N = 125) are the following: Rg of S chain is 5.787 ± 0.002, the Rg of the corresponding H chain is 6.173 ± 0.001, and the Rg of the corresponding L chain D


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Figure 4. (a) Partition coefficient K as a function of pore size D for regular H (squares connected by solid lines) and L chains (circles connected by dotted lines) for different ε = 0 (black), −0.10 (blue), −0.15 (brown), −0.18 (green), −0.20 (red), −0.22 (magenta), −0.24 (cyan), and −0.26 (orange). (b) The double difference Δ(ΔA) = (ΔA)H − (ΔA)L between free energy of systems of H and L chains accompanying the transfer of one chain from the bulk into the pore as functions of D (for the same set of interaction parameters ε as in (a)). Note that the first operator Δ describes the difference between chains of different architecture, while the second symbol Δ designates the difference between the pore and the bulk.

values of the bead−wall interaction parameter, ε. The profiles for more pore sizes are available upon request. They were not included in the main paper because important trends of the behavior are obvious from simpler 1D distributions and their discussion is easier. Therefore, the angularly averaged 2D concentration profiles were converted into 1D concentration profiles. The conversion has been described in the Supporting Information. The 1D concentration profiles are shown in Figure 3. The left column shows the r-averaged concentration profiles (both in the pore and in the bulk) as functions of x. As expected, the concentration of polymer beads in the bulk is essentially constant. Only in a narrow region close to the wall does it nonnegligibly decrease due to the depletion effect of the flat noninteraction wall. The concentration of polymer beads inside the pore differs from that in the bulk and depends sensitively on the ratio of Rg to D and on the attractive interaction, ε. It depends also on the x position inside the pore (depth). In systems without adsorption interaction (ε = 0), it increases toward the pore entrance due to less important steric restrictions. In strongly interacting systems (ε = −0.26), we observe the opposite shape: the concentration increases inside the pore (in the x direction) due to the enhanced cooperative adsorption of several beads of the same chain in confined volumes. However, for sufficiently long pores, it is almost constant in the majority of the pore. The values in the flat concentration regions give the equilibrium concentrations of polymer beads in both compartments of the simulation box. The equilibrium concentration in the pore is affected by pore size D and the interaction with the pore wall, ε. For an ideal SEC (ε = 0) the simulation yields ΦH(x) ≈ 5 × 10−5 for the H chain and D = 16, whereas ΦH(x) ≈ 4 × 10−4 and 8 × 10−4 were obtained for D = 30 and 60, respectively. An analogous Ddependent effect applies for L chains. In the pore, the concentrations of both chain architectures generally increase with increasing |ε|, but non-negligible differences between the concentrations of the H and L chains can be clearly seen. Their evolution with increasing ε reflects the intricate interplay of entropic and enthalpic forces. For ideal SEC (ε = 0) and for pores with weak adsorption (|ε| < 0.20), the concentration of H chains in the pore ΦH(x) is higher than that of L chains. This is

understandable: (i) Rg and also the hydrodynamic radius Rh of the H chains are smaller than those of the L chains, which facilitates the preferential penetration of the H chains into pores. (ii) Furthermore, the excluded volume effect, which plays an important role at elevated local concentrations of beads, restricts the conformation variability of the H chains in the bulk more than that of the L chains. Consequently, the decrease in entropy due to the geometrical confinement in a small volume is lower for the H than for the L chains. Nevertheless, the situation reverses when the adsorption of chains on the walls is important. Flexible L chains are more deformable and can be more easily adsorbed, and ΦL(x) grows faster with increasing |ε| than ΦH(x), and at a certain adsorption strength (around ε = −0.20), which we call the “branching-compensation adsorption strength” (BCAS), the concentrations of both types of polymers in the pore become approximately the same. For a strong adsorption (|ε| > 0.20), the concentrations of L chains, ΦL(x), are always higher than those of H chains. Similar behavior with a reverting sequence of adsorbed amounts of highly branched and linear chains versus the adsorption strength has been observed in simulations of chains close to interacting walls.63 The radial, i.e., x-averaged concentration profiles, Φ(r) vs r in the plateau region of the pore (right column in Figure 3) are the results of the competition between the confinement effects and adsorption. A pronounced depletion effect of the impermeable pore wall is obvious in all studied systems. Radial concentration profiles Φ(r) of both chain architectures for D = 16 and 30 show a monotonous decrease in the whole region from the pore center to the wall because the depletion layer is thicker than D/2. The depletion layer thickness in the pore with D = 60 could be roughly estimated from Figure 3f. However, it is necessary to keep in mind that such a simple estimate is highly inaccurate because it is affected by adsorption (the effects of the depletion are counterbalanced by attractive interactions), and e.g. the depletion layer for ε = −0.22 is thinner than that for the ideal SEC pore. In wide pores (D = 60) with strongly attractive walls, the concentration of polymer beads reaches maximum at distances corresponding roughly to the radius of gyration of the studied chains and a shallow local minimum appears in the middle part E


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both KH and KL that secure “optimum separation” are very low, and they are not important in practical chromatographic separations. To summarize the part on partition coefficients and on the role of entropic and enthalpic contributions, it is suitable to plot partition coefficients KH and KL as functions of ε. The plot is shown in Figure 5, and the difference (KH − KL) vs ε is

of the pore. The nonmonotonous concentration profile is a result of the attractive interaction of polymer beads with the wall. The chains concentrate, deform, and partially orient close to the wall. Both contributions, i.e., (i) increase in concentration of beads due to interaction with the wall and (ii) concentration decrease due to decreased conformational entropy, decay with the distance from the wall, but due to chain connectivity, the resulting effect affecting the concentration of beads propagates to the distance from the wall corresponding roughly to Rg. The effect of the pore size on the partition coefficients K, which were determined from concentrations in the pore plateau (subscript p(pl)) and bulk plateau (subscript b(pl)) regions, K = cp(pl)/cb(pl), is summarized in Figure 4a. Both partition coefficients increase with the pore size, and for pure SEC (ε = 0), KH is always higher than KL (see Figure 4a, black curve). They increase with increasing |ε|, and it is obvious that for ε = −0.2 the dependencies of KH and KL vs D overlap in the whole region of studied D, which indicates that the so-called branching-compensation adsorption strength (BCAS) ε* does not almost depend on D (for more details and discussion of BCAS features see the Supporting Information). In pores with strongly interacting walls, i.e., for |ε| > 0.2, KL becomes higher than KH. The differences of free energy changes between H and L chains due to the transfer of the chain from bulk inside the pore, Δ(ΔA) = (ΔAH) − (ΔAL), are plotted in Figure 4b as a function of D. The theoretical value of this difference for very small pores, where both polymers are excluded from the pore, is zero. For pore sizes comparable with 2(Rg)H, the difference TΔ(ΔS) due to entropy changes between the H and L chains in the pores and the bulk increases with the pore size because H chains can enter in the pore easily. For a pore size larger than 2(Rg)L, both polymer chains can enter in the pores without major limitations. In the case of pure SEC, the difference stems only from the steric repulsion, and the absolute value of the difference between entropy changes decreases with increasing pore size because the importance of confinement and depletion effects also decreases (see Figure 4b, black curve). It is interesting and useful to discuss and explain the nonmonotonous dependencies of Δ(ΔA) vs D. It is clear that in very broad pores (D → ∞) both KH and KL equal 1 and Δ(ΔA) limits to zero. For very narrow pores with dimensions comparable with molecular dimensions (or with radius of beads in coarse-grained simulations), both partition coefficients limit to zero, which does not guarantee from the mathematical point of view that their ratio limits to zero. However, if the chains cannot enter the pore because their building units would not fit in the pore, both KH and KL should be zero for D < Dmin and the limit of KH/KL for D → 0 should yield 1. In pores with D slightly higher than Dmin, linear chains can squeeze more easily attaining the rodlike conformation and can enter the pores, the diameter of which prevents the entrance of branched chains. It means that KH changes in this D range faster than KL (even though the difference is only small). Simulation data indicate that for pure SEC (and weak interactions) we see the region of D where the separation of L and H chains is the most efficient and where |Δ(ΔA)| attains maximum (D ca. 17). The strongly adsorbing walls attract chains inside the pores, and K values increase. The chains become more deformed, and the maximum difference between KH and KL should shift to lower D. It is why we see only the monotonous decrease of |Δ(ΔA)| with increasing D. However,

Figure 5. Partition coefficients, KH (solid lines) and KL (dotted lines), as functions of ε for three selected pores with D = 16 (black), 30 (blue), and 60 (red). Inset: plot of corresponding differences (KH − KL) vs ε.

depicted in the inset. This graphical representation confirms very clearly that the BCAS ε* does not almost depend on the pore diameter and that the compensation of KH and KL occurs for their values lower than 1, which is important from the experimental point of view. Chain Conformations in Pores and in Bulk. In the studied system, the chain conformations in the pores are affected by (i) spatial confinement and by (ii) the interaction of the beads with the pore walls. The ease of deformation depends on chain rigidity and conformational variability.51,64 We study intrinsically flexible chains, but the density of beads in the H (and S) chains is higher than that in the L chains of the same molar mass, and therefore the excluded volume of the beads restricts the conformational freedom of the former chains more than that of the latter chains. The analysis of the concentration profiles indicates that the interplay of confinement effects with interactions is complex: In large pores with relatively weakly adsorbing walls, the branched chains interact more strongly with the wall than linear ones, but the preferential adsorption of more deformed L chains prevails at high adsorption strengths. To elucidate the conformational behavior of chains in the pores and substantiate this observation at the molecular level, we performed a detailed analysis of chain conformations. We evaluated the characteristics of their equivalent ellipsoids of gyration and their orientation with respect to the fixed geometry of the system. The effects of (i) spatial confinement, (ii) polymer concentration, and (iii) interactions on the shape/conformation and orientation of linear chains have been studied by van Vliet et al.,65 Cifra et al.,50,55,66 Wang et al.,60,67 and several other authors.68−70 In their detailed studies of the behavior of linear chains consisting of 100 beads in narrow parallel slits, Cifra et al.71 investigated the dependences of Rg and the dependences of the parallel and perpendicular components to the slit walls, Rg∥ and Rg⊥, respectively, and identified (analogously to other F


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Figure 6. Mean-square characteristics of chain size as a function of pore size D for two values of interaction parameter ε = 0 (black) and −0.20 (red). The left figure (a) corresponds to the H chains and the right one (b) to the L chains. The individual size characteristics are designated by the following symbols: ⟨Rg2⟩ (□), ⟨ga2⟩ (●), ⟨Rg∥2⟩ (○), ⟨gb2⟩ (■), and ⟨gc2⟩ (▲).

authors40,49,65) several regimes depending on the confinement parameter Z = D/(2Rg,bulk). They found that the chains are almost unaffected at relatively large Z values. At Z ≈ 1, the dimensions of the chains become slightly reduced, and their equivalent ellipsoids are preferentially aligned parallel to the walls. The chains become truly deformed and squeezed in all three dimensions for Z < 1, and finally (at a very low Z) they become severely flattened. The authors of this study also found that the squeezing effect is amplified at elevated polymer concentrations. We study and compare the conformations of the H, S, and L chains in pores of sizes which play an important role in the chromatographic separation, i.e., in pores with non-negligibly small partition coefficients K. Therefore, we focus on dilute polymer solutions and Z ≥ 1. To describe the conformational behavior in detail, in addition to mean Rg and to its parallel Rg∥ and perpendicular Rg⊥ components, we evaluate all the components of the gyration radius, gi, their weighted averages and distributions, and the distributions of the gi/gj ratios in individual chains. The last distributions are quite important: they provide complete information on the distribution of chain shapes. Note that the integration over two variables, which is implicitly involved in the evaluation of 1D distributions, erases the relation between gi components in individual chains and provides incomplete information on the distribution of chain shapes. In our recent papers, we have shown that the averaging over some variables could generally hide or obscure some important features of the behavior.34 The size characteristics ⟨Rg2⟩, ⟨ga2⟩, ⟨Rg∥2⟩, ⟨gb2⟩, and ⟨gc2⟩ are shown in Figure 6 as functions of the pore size. In order to provide comprehensive picture, we did not include information on S chains in most following figures. The sizes of both L and H chains are almost unchanged in wide pores. However, appreciable changes occur in narrow pores; for H chains they start at D = 30 and for L chains at D = 40. The latter changes are more pronounced than the former. The longest component of gyration radius of equivalent ellipsoids, ga, is the most affected and decreases appreciably with decreasing pore size. As a consequence, the radius of gyration also decreases with decreasing pore size. The smallest component of gyration radius is unchanged, and the middle one is also almost constant. Thus, the chains become effectively more spherical in pores with Z ca. 1.

The chains permeating narrow pores are not only deformed but also partially oriented. Information on chain orientation can be obtained in different ways. In our study, we simply compare the mean-square distance of the polymer beads from the mass center in the x direction (i.e., along the pore axis), Rg∥2, with the square of the longest component of the gyration radius, ga2, and with the arithmetic mean of the shortest and middle components of the gyration radius, (gc2 + gb2)/2. The comparison is based on the following considerations. If the chain is oriented across the pore, Rg∥2 should approach (gc2 + ga2)/2, but if it is aligned along the x-axis, it should approach the value ga2. The simultaneous plotting of all these quantities for different D and ε (see Figure 6) thus reveals the changes in orientations of chains in individual systems. The facts that the average value ⟨Rg∥2⟩ increases with a decreasing pore size and simultaneously the average value ⟨(gc2 + gb2)/2⟩ remains almost constant and ⟨ga2⟩ decreases confirm the preferential orientation of both L and H chains parallel to the pore axis in pores with Z ca. 1. The decrease of ⟨Rg2⟩ and ⟨ga2⟩ curves and the increase of ⟨Rg∥2⟩ values in the region of small D are less pronounced for stronger attraction to the pore wall because the conformations of chains are, on average, appreciably deformed and oriented also in broad pores. In Figure 7, the distribution functions F(ga), F(gb), and F(gc) of all three components of the gyration radius, ga, gb, and gc of both L (dotted lines) and H chains (full lines) in a narrow pore with D = 16 are depicted for several values of the interaction parameter ε = 0, −0.15, −0.20, and −0.26. In inset (a), the comparison of distribution functions F(ga), F(gb), and F(gc) of H chains in the bulk and in the pore is shown for two values ε = 0 and −0.20, and inset (b) offers the same comparison as the previous one for the L chains. The curves shown in the insets confirm the above-mentioned observation that the conformations of chains in fairly narrow pores (i.e., for Z ca. 1) are the most influenced by the geometrical confinement effect in the direction of the longest gyration component, ga. However, they indicate that it concerns only a fraction (even though an important fraction) of the chains−presumably those that are close to the pore wall. Simultaneously, this component is also the most strongly influenced by the wall attraction. The effect of attraction between polymer beads and the wall is evident on curves in the main part of the figure. Increasing attraction almost does not influence the short principal G


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for chains in the bulk, and its maximum is shifted to smaller values, which reflects the fact that the shape of the chains is ellipsoidal in the bulk and becomes more spherical in pores with Z ca. 1. A slight broadening of the distributions of the sij ratios in pores with respect to corresponding distributions in the bulk is due to the fact that the chains located very close to the wall (and directly interacting with it) are affected and deformed more than those in the middle part of the pore. This conclusion has already been deduced (indirectly) from radial concentration profiles and from previous plots of structural characteristics. The analysis of the curves in Figure 8 thus confirms the features observed in previous parts. This fact cannot be considered as independent proof of already described results because all curves have been evaluated from the same simulation data. Nevertheless, the information on chain shapes elucidates the mechanism of the chromatographic separation of the H and L chains in detail at the molecular level. The general conclusions from the detailed conformational analysis of chain conformations in pores aimed at the deformation and orientation of both types of chains in pores with adsorbing walls can be summarized as follows: Even though the spatial confinement and interaction of chains with the wall appreciably affect the partitioning of both L and H chains in a relatively broad range of confinement parameters Z, the deformation and partial orientation of chains becomes important only for Z approaching 1 (and for smaller Z, which we did not study in detail because they are not important from the chromatography point of view). The study shows that the L chains are effectively more deformable than the H chains because the density of beads in the L chain domains is lower and the excluded volume plays a less important role. The higher conformational variability of the L chains explains their more important adsorption on strongly adsorbing walls as compared to the H chains. Practical Aspects of the Study. The performed simulation study shows that the attractive interaction of polymers with porous stationary phase strongly affects the chromatographic separation of linear and branched chains. An improper choice of experimental conditions in combined SEC and IC experiments can suppress the effect of branching on the partitioning of chains between the pores and bulk mobile phase

Figure 7. Distribution functions of components of the gyration radius, ga, gb and gc in a pore with D = 16. The solid lines represent the H chains, and the dotted lines represent the L chains in the pore. Different colors represent different values of interaction parameter ε = 0 (black), −0.15 (blue), −0.20 (red), and −0.26 (green). In the insets, solid lines represent the distributions in the pore and the dotted lines the distributions in the bulk. Inset (a) stands for H chains and inset (b) for L chains.

components of equivalent ellipsoids. However, it appreciably affects the long component and shifts its entire distribution to larger distances. The comparison of the behaviors of the H and L chains and their detailed analysis indicate that the more conformationally variable L chains are more expanded and more ellipsoidal than the H chains not only in the bulk but also in medium narrow pores (Z ca. 1) with attractive walls due to the interaction with them. More details can be found in the Supporting Information. To get more detailed information on chain shapes in pore and in bulk, we evaluated the distributions of the ratios of the components of the gyration tensor in individual chains. Figure 8 depicts the distributions G(sca) and G(sba) of ratios sca = gc/ga and sba = gb/ga in the bulk and in a narrow pore with D = 16 for two values of attraction strengths: ε = 0 for pure SEC and ε = −0.20. The distributions are monomodal and fairly broad. The distribution (particularly for sca) is narrower

Figure 8. Distributions G(sca) and G(sba) of ratios sca = gc/ga and sba = gb/ga. The left figure (a) stands for the H chains and the right one (b) for the L chains; the pore size is D = 16. The different colors represent different interaction parameters ε = 0 (black) and −0.20 (red). The solid lines in the main figures represent the chains in the pore and the dotted lines in the bulk. In the insets, the solid and dotted lines represent the H and L chains, respectively. H


Article

Macromolecules

KH = KL. The study for two significantly different types of H chains (which we call “regular” H chains and “pseudostar”, S) shows that this value of the interaction parameter depends only slightly on chain architecture and on the chain length (total number of beads). This finding is important from the practical point of view. On the one hand, it shows that an improper choice of experimental conditions, which would promote undesirable interactions of polymers with pore walls, could deteriorate the efficiency of chromatographic analysis. On the other hand, it suggests a possibility to exploit the compensation effect in properly combined SEC and IC chromatographic experiments because the comparison of results of pure SEC with those obtained under BCAS conditions could provide information on the branching and on its impact on chromatography results. Another interesting observation concerns the conformational behavior of chains in pores with the confinement parameter Z = D/(2Rg,bulk) ca. 1. The obtained data indicate that in these medium narrow pores (i.e., under the conditions of mild deformations) the chains are slightly compressed, but they are mostly affected (shortened) in the direction of the longest axis of their equivalent ellipsoid; i.e., they become effectively more spherical than the chains in the bulk.

and deteriorate the efficiency of chromatographic separation. Simulation data show that the best separation is achieved in the case of pure SEC, but the SEC separation mechanism is in real chromatographic setups usually influenced by enthalpic interactions of components of the studied system with stationary phase. These interactions depend on the properties of the porous stationary phase and of the solvent used (on its thermodynamic quality and chromatographic strength). An important message of our study for experimentalists is the following: experimental conditions of the SEC analysis have to be carefully chosen and optimized, and all parasite adsorption effects have to be minimized as much as possible (this is particularly important if strongly polar or water-soluble polymers and copolymers are studied). Nevertheless, the possibility to suppress the separation of differently branched polymer chains without affecting the separation with respect to molar masses could find certain applications. The comparison of results of pure SEC with those of its proper combination with IC could yield indirect information on the degree of branching and on its role in chromatographic separation processes. Even though the suggestion to suppress intentionally the resolution of an analytical method is not very appealing idea for experimentalists, the compensation effect may come in handy in special applications. More details on the dependences of the “branchingcompensation adsorption strength” (BCAS) interaction parameter ε* on pore size D, length of the chain, chain architecture, etc., can be found in the Supporting Information. Here, we just summarize that the dependences of ε* on all aforementioned factors are weak (in the region of studied parameters) which means that for Z ≥ 1 the BCAS ε* ca. −0.2 seems to play a role of an almost constant “universal parameter” (which is convenient from the point of view of potential applications).

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.macromol.5b02327. General description of the studied system, definition of characteristics used for the description of the behavior of studied systems (i.e., concentration profiles and structural characteristics of chains), simulation data not included in the main text (i.e., radius of gyration and scaling behavior and detailed analysis of the retention behavior), details on the “branching-compensation adsorption strength” regime, and supplementary data on the conformation behavior of chains in pores (PDF)

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SUMMARY AND CONCLUSIONS We performed an extensive series of Monte Carlo simulations aimed at the partitioning of flexible chains of different architectures between bulk solvent and narrow pores with attractive walls. We focused on two types of H-shaped chains (with fairly long and short backbones) and compared their chromatographic and conformational behaviors with those of linear chains. The study shows that the behavior of both types of chains is the result of an intricate interplay of entropic and enthalpic forces. It reflects the conformational variability of chains differing in chain architecture and ease of their deformation due to external effects. The analysis of chain conformations elucidates the differences in the behavior of H and L chains at the molecular level and helps to explain the aspects of their chromatographic separation. The simulations revealed interesting differences between the behavior of H and L chains in relatively narrow pores with attractive walls: In a pure SEC regime (i.e., for interaction parameter ε = 0), the partition coefficient KH of branched polymers is higher because their hydrodynamic volume is lower than that of L chains. However, the conformational variability of L chains is higher than that of H chains because the density of the beads in the polymer coil is lower and the excluded volume of beads plays a less important role. Therefore, the deformation of L chains due to attractive interaction with the wall is easier, KL increases faster with |ε| than KH, and at a certain critical ε* value both partition coefficients equal; i.e.,

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected] (Z.L.). *E-mail: [email protected] (K.P.). Notes

The authors declare no competing financial interest. The authors appreciate useful comments by P. Cifra.

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ACKNOWLEDGMENTS This work was funded by Grant 328514 of the Grant Agency of the Charles University and by Czech Science Foundation Grants P106-12-0143 and P106-13-02938S. Access to the computing and storage facilities of Metacentrum (LM2010005) and CERIT-SC computing (Operational Program Research and Development for Innovations, Reg. No. CZ. 1.05/3.2.00/ 08.0144) is greatly appreciated. Xiu Wang acknowledges support from the Ministry of Education, Youth and Sports of the Czech Republic (Grant LK21302).

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