Partitioning of polymers between bulk and porous media: Monte Carlo study of the effect of pore size distribution

Journal of Colloid and Interface Science 567 (2020) 103–112

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Partitioning of polymers between bulk and porous media: Monte Carlo study of the effect of pore size distribution Xiu Wang, Karel Procházka ⇑, Zuzana Limpouchová ⇑ Department of Physical and Macromolecular Chemistry, Faculty of Science, Charles University, Hlavova 8, 128 00 Praha 2, Czech Republic

g r a p h i c a l a b s t r a c t

a r t i c l e

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Article history: Received 4 December 2019 Revised 27 January 2020 Accepted 28 January 2020 Available online 1 February 2020 Keywords: Phase equilibria Polymers Porous media Partition coefficient Pore size distribution Size-exclusion chromatography Interaction chromatography Monte Carlo simulation Mass balance equation

a b s t r a c t In this paper we investigated the partitioning of polymer chains between bulk solvent and porous stationary phase under conditions appropriate for the chromatography under critical conditions (LCCC) close to the critical adsorption point (CAP). We addressed the concentration effect and the thermodynamic effect of pore-size dispersity (PSD) and their impacts on chromatography, i.e., the topics which surprisingly escaped from the interest of scientists in spite that the hydrodynamic effect of PSD has been a subject of numerous studies. The phase equilibria in narrow pores (as compared with the size of polymer coil) with attractive pores are complex and the enthalpy-to-entropy interplay is very intricate. The chains are attracted into pores and the partition coefficients are larger than 1, which corresponds to the interaction chromatography (IC), but they are strongly confined and deformed. The entropy plays important role and the elution volumes of chains differing in molar mass correspond to the size-exclusion chromatography (SEC). The study thus reveals a new chromatography regime which could be easily overlooked without the awareness of its existence. The unexpected findings are important not only for chromatography, but for understanding the phase equilibria of polymers in porous systems in general. Ó 2020 Elsevier Inc. All rights reserved.

1. Introduction

⇑ Corresponding authors. E-mail addresses: [email protected] (X. Wang), karel.prochazka@natur. cuni.cz (K. Procházka), [email protected] (Z. Limpouchová). https://doi.org/10.1016/j.jcis.2020.01.119 0021-9797/Ó 2020 Elsevier Inc. All rights reserved.

Phase equilibria, particularly those between two liquids or between liquids and solids are the bases of a number of experimental methods for the characterization and separation of both low and high-molar-mass compounds. Phase equilibria in porous media and the partitioning of polymers between pores and bulk

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Abbreviations and Symbols CBMC IC LCCC MC PDF PSD SEC ITP c D D0 f K Kef L

configurational-bias Monte Carlo interaction chromatography liquid chromatography at critical conditions Monte Carlo probability density function pore-size dispersity size-exclusion chromatography ideal theoretical plate concentration pore size median of pore size pore size distribution partition coefficient effective partition coefficient length of box

play important role in polymer chemistry. An inherent feature of the overwhelming majority of porous materials is the dispersity in pore sizes. The distribution of pore sizes affects the partitioning of polymers and influences the results of separation techniques. While the confinement effects on chain conformations, the partitioning of chains between uniform pores and bulk, including the effect of adsorption [1–12] and hydrodynamic effects on chromatographic separations due to pore size dispersity (PSD), have been amply studied [13–17], the thermodynamic aspects of PSD and their direct impacts on results of chromatographic experiments have not been, to the best of our knowledge, studied at all. Size-exclusion chromatography (SEC) and interaction chromatography (IC) with a porous stationary phase are standard techniques for the characterization of polymers [18]. While SEC is based on the confinement-generated entropy effect, IC employs the adsorption enthalpy effect. IC is used less than SEC, but in last few decades its popularity among polymer scientists has increased considerably [19–23]. In most chromatography set-ups, SEC and IC mechanisms compete with each other: (i) in SEC, the confinement-exclusion dominates, (ii) both effects compensate each other in liquid chromatography at critical conditions (LCCC) and (iii) in IC, the adsorption effect prevails. The experimentalists often discern SEC and IC on the basis of the dependence of elution volumes on molar masses. We use the classification based on thermodynamic criteria. If steric constraints dominate the behavior, the chains are expelled from the pores and the partition coefficient is lower than 1. In this case, the elution volumes are between two limits given by the volume of the mobile phase and by the sum of volumes of the mobile phase and pores. Then, in accord with experimentalists, we call the process SEC and the above limits as SEC exclusion limits for large and small molecules, respectively. If the interaction energy is zero, we use the term ideal SEC. If the adsorption prevails, the polymer accumulates in pores, partition coefficients are greater than 1 and retention characteristics exceed the SEC limit for small molecules. In that case, we call the process IC. However, we will show in next parts that in narrow pores with attractive walls, steric constrains play important role and the dependence of retention characteristics on molar masses mimics the SEC elution mode, even though partition coefficients and retention volumes fulfill the criterion of IC mechanism. We have been studying the behavior of polymers and their partitioning between pores and bulk for several years [24–28]. In this paper, we substantially modified our simulation code and developed the Monte Carlo variant, which enables simultaneous

N Rg Rs t tR V w

e

u k

r

npore NITP

length of polymer chain radius of gyration resolution time elution time volume half-width of peak interaction strength effective volume-weighted distribution coil-to-pore size ratio standard deviation total number of pores number of ITP

evaluation of the ensemble of partition coefficients of polymers with different molar masses and architectures in pores differing in pore sizes. The paper addresses some not yet fully understood observations obtained within the framework of the cooperation with the research group of T. Chang on chromatography topics [22,23]. The goal of the paper is general. We investigate the effects of PSD on polymer chains partitioning in porous media and their impacts on separation methods. 2. Computational details 2.1. Monte Carlo simulation In this study, we substantially modified our simulation variant [24] of the configurational-biased Monte Carlo (CBMC) method [29]. The algorithm employs both the exchange of chains among bulk and pores and the exchange among pores differing in sizes analogously to the exchange of molecules between two different boxes in the NpT Gibbs ensemble [30]. The stationary phase is modeled as an ensemble of 500 square pores, the sizes (diameters) of which obey the lognormal distribution,

" # 2 1 ðln D  ln D0 Þ ; f ðDÞ ¼ pffiffiffiffiffiffiffi exp  2r 2 2prD

ð1Þ

where D is the pore diameter, D0 is the median, and r is the standard deviation representing the width of the distribution [31], which we typically varied from 0 to 0.3, whereas we previously used a set of uniform pores. The total volume of pores is Lx,pore  D  D  npore, where Lx,pore stands for the x-axial length and is identical for all pores. In the simulation of chromatograms, we set the dimensions of bulk as 360  360  360 (Lx,bulk  Ly,bulk  Lz,bulk) and we adjusted Lx,pore to keep VS/VM  1, where VS and VM represent the volumes of stationary and mobile phases, respectively. Our model is illustrated in Fig. 1a and the probability density function (PDF) of the lognormal distribution is plotted in Fig. 1b. All polymers are modeled as selfavoiding chains on a cubic lattice, and the solvent molecules are implicitly included. In simulations, we varied the concentration of polymer beads from 1  105 to 2  103, but for the calculation of the effective partition coefficient, the total concentration of polymer beads was kept constant, c  7.3  104. The total number of polymer beads in these simulations is 33800, i.e., 676 chains of N = 50, 338 chains of N = 100 and 225 chains of N = 150. The interaction between two nonbonded polymer beads, epp = 0, mimics a good solvent condition, and the polymer-wall interaction, eW = 0

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2.2. Solution of the mass balance equation We modeled the chromatographic process in a column containing NITP = 103 ideal theoretical plates (ITP) filled with porous stationary phase with a broad log-normal distribution of pore sizes. After the sample injection, the polymer zone passes through the column and finally elutes. We assume that the flow is slow compared to the rates of the re-equilibration and relaxation processes. This assumption, which ensures that bulk solvent-pore equilibrium is established at any time t in any position x in the column, is an approximation, but we use it on purpose because it enables to study the neat thermodynamic impact of PSD on elution curves separately from hydrodynamic effects of PSD. Combining the bulk-pore partitioning at each plate and the hydrodynamic contribution, we numerically solved the mass balance equation describing the chromatographic process,

    @cM @cM ¼ v þ PðcM ; K D ðcM Þ; ðV M =V S ÞÞ þ Q ðcM Þ; @t @x

Fig. 1. (a) Scheme of the model of mutually communicating boxes used in simulations of the partitioning of polymer chain between the mobile phase and the ensemble of pores differing in size. (b) Lognormal distribution of pore sizes, f(D), for D0 = 30 and for r2 = 0.1, 0.2 and 0.3. The right-hand vertical axis represents the actual number of pores used for each pore size, npore f(D), where npore is the total number of pores. Inset of Fig. 1b: The dependence of KD(c = 0.0007) vs. k for pure SEC and chain length N = 100.

and eW = 0.32, model the SEC and IC regimes, respectively. In our recent publications [25], we have shown that the critical adsorption point (CAP) depends slightly on the chain architecture and on other factors, but for linear chains in uniform pores of medium size CAP is achieved for eW = 0.3. Using eW = 0.32, we model LCCC conditions slightly above CAP. Using the blocking method [32], we estimated the statistical error of partition coefficient, K, which is lower than 1%. The effective partition coefficient, Kef, is computed from the equilibrium concentrations of the pores (stationary phase) and the bulk (mobile phase),

K ef ¼ ccMS ¼ PDmax ¼

P

Pnpore i¼1

PDmax

Pnpore i¼1

V pore;i

mbulk =V bulk

f ðDÞK D ðcM ÞV D

D¼Dmin Dmax

D¼Dmin

¼

mpore;i =

f ðDÞV D

D¼Dmin K D ðcM Þ

PDmax

¼

Pnpore kpore;i V pore;i i¼1 P ¼ npore i¼1 2

K ðcM Þf ðDÞD

D¼Dmin D Dmax

P

¼

D¼Dmin

V pore;i

ð2Þ

f ðDÞD2

uðDÞ

where m is the number of chains, KD(cM) is the concentrationdependent partition coefficient for a specific pore size, D, and the subscripts S and M represent the stationary and mobile phases, respectively. The relationship between u(D) and f(D) is

f ðDÞD2

uðDÞ ¼ PDmax

D¼Dmin f ðDÞD

2

ð3Þ

ð4Þ

where cM = cM (x, t) is the concentration of polymers in mobile phase (t and x correspond to the time and position, respectively), v represents the constant flow velocity of the eluent, P stands for the partitioning of chains between the mobile and stationary phases, and Q implies the hydrodynamic axial dispersion. The most important new feature of our model consists in the fact that the partitioning reflects the contributions of individual pores which differ in size and reflect the fact that the concentration changes across the moving zone and decreases with time. The phase equilibria are described by the concentration-dependent and pore sizedependent partition coefficients, KD(cM) and are weighted by the distribution uðDÞ defined by Eq. (3). We would like to stress that it is essential to take the concentration dependence of KD(c) into account, because experimental studies and MC simulations show that KD(cM) depends non-negligibly on c [33–38] and c significantly changes throughout the polymer zone and decreases with time. In our recent publication [24], we have shown that the variations of KD(cM) with cM appreciably affect the shape and symmetry/asymmetry of the elution curves. All details concerning the numerical solution of Eq. (4) and simulation of chromatograms were described in Ref. [24]. 3. Results and discussion 3.1. Effect of pore size distribution on the effective partition coefficient The solution of Eq. (4) assumes the knowledge of KD(cM) for a number of pores differing in D. Nevertheless, the evaluation of the effective partition coefficients, Kef, as the weighted average u (D)KD(cM) defined in Eq. (2) facilitates the discussion and understanding of the most important trends of the behavior. Fig. 2a shows the dependence of Kef on r2 for the lognormally distributed narrow pores with median D0 = 8 in the SEC mode. The partition coefficients for three chain lengths (N = 50, 100 and 150) are low and increase significantly with the increase in r2, which can be understood if we look at the curve of KD(cM = 0.0007) vs. the confinement strength, k = 2Rg,bulk/D, where Rg,bulk is the radius of gyration in bulk and D is the pore diameter, in the inset of Fig. 1b. The inset shows that KD changes negligibly in narrower pores (k > 0.8), while it changes significantly in wider pores. The presence of larger pores in the distribution thus increases Kef and improves the separation efficiency. The increase in Kef (see Fig. 2a) is pronounced for short chains with N = 50 (k = 1.06), because they feel the weakening of steric constraints in wider pores more than the long chains. Fig. 2b depicts the dependence of Kef on r2 for broad pores with D0 = 30 in the SEC mode. The Kef values are relatively high for all

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Fig. 2. Effective partition coefficients, Kef, for a lognormal distribution of pore sizes as functions of the distribution width, r2, for chain lengths: N = 50 (magenta), 100 (ochre) and 150 (cyan). (a) SEC (the polymer-wall adsorption strength, eW = 0), D0 = 8; (b) SEC (eW = 0), D0 = 30; (c) IC (eW = 0.32), D0 = 8; (c) (d) IC (eW = 0.32), D0 = 30. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

the studied chains and the dependences of Kef on r2 slightly increase and level-off at high r2 because the KD(cM) vs. k curve (see the inset in Fig. 1b) for ca k = 0.43 (for N = 100 and D = 30) is almost linear and so it changes comparably both for D smaller and larger than D0 = 30. The presence of narrower/wider pores thus decreases/increases the separation efficiency almost equally and the two opposite contributions roughly compensate each other. The dependences of Kef on r2 for narrow pores with median D0 = 8 in the IC mode are shown in Fig. 2c. In spite of small pore sizes, the effective partition coefficients, Kef, are quite large (Kef > 1) and decrease with increasing r2 for short chains with N = 50, which is surprising at the first glance. Nevertheless, the observed trend is understandable. Our earlier studies showed that linear chains, which are flexible and deformable, are strongly attracted into pores where they adsorb on attractive walls changing their conformations [25]. With increasing pore size, the surface-to-volume ratio (proportional to 4/D for square pores) decreases and the adsorption effect weakens. The presence of wide pores thus slightly deteriorates the efficiency of the IC process for short and only little sterically constrained chains. The long chains with N = 100 and 150 feel strong steric constraints. Their penetration into the pores (promoted by enthalpy) is sterically hindered (by entropy), the SEC mechanism efficiently competes with the IC mechanism and the Kef curves grow with increasing r2, resembling the shape of the dependences for the SEC mode, but Kef shifts towards relatively high values. The relative increase is the most pronounced for the most constrained chains of N = 150. The IC partition coefficients, Kef, for wide pores with D0 = 30 (Fig. 2d) are larger than 1, but it is noteworthy that their values

do not dramatically differ from those in narrow pores for higher

r2. In wider pores, the chains are only slightly constrained. How-

ever, the surface-to-volume ratio (4/D for square pores) decreases with D and as the contribution of wide pore increases with increasing r2, the adsorption effect weakens and the Kef curve for N = 50 slightly decrease and is reminiscent of the curve for N = 50 in Fig. 2c. Nevertheless, different numbers of beads, which can be adsorbed for chains with increasing length, influence Kef, e.g., Kef (for r2 = 0.3) increases from 1.5 for N = 50 to 2.5 for N = 150. The Kef vs. r2 curves for SEC mode and for the symmetrical Gaussian distribution with D0 = 30 (calculated for comparison and shown in Supplementary Material, Fig. S1) increase and those for IC mode decrease analogously as in the case of the log-normal distribution. The changes are less pronounced because the fraction and contribution of wide pores are lower than those in the lognormal distribution. The comparison thus shows that the observed trends are general features of stationary phases with pores of nonuniform sizes. 3.2. Simulation of elution curves In the last part of the study, we demonstrate the practical impact of log-normal distributions of pore sizes on chromatographic results. We present the simulated elution curves obtained by the solution of Eq. (4) for a column with 1000 theoretical plates using the size and concentration-dependent partition coefficients KD(cM) as the input parameters. In Fig. 3, we show three frames depicting the chromatographic behavior of SEC systems with narrow pores, D0 = 8 and three cor-

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responding frames depicting IC systems. In Fig. 3a and 3b, we plot the elution curves for chains with N = 50 in a column with lognormal distribution of narrow pores with median D0 = 8 in SEC (eW = 0) and in IC regimes (eW = 0.32), respectively. In this case, the chains are subject to a non-negligible confinement, (k = 1.06). In the SEC regime (Fig. 3a), the elution peaks shift to higher tR and broaden (the maxima decrease and the half-widths increase) with the increase in r2. Both effects are due to increasing fractions of wider pores with larger KD(cM), in which the steric constraints are weaker. The IC curves (Fig. 3b) are significantly lower and broader than the SEC curves, which is, in major part, a trivial consequence of longer elution times and longer period of axial dispersion. The impact of increasing r2 on IC curves is relatively modest as compared with SEC, but it is noteworthy that the r2 effect on chromatograms for the short chain is opposite to that in SEC. An increase in r2 leads to a slight narrowing of elution curves and to their shift towards lower tR. This suggests that promoted adsorption of chains on the walls together with synergic weakening of steric constraints in wider pores dominates the trend of the observed behavior. A more detailed discussion of this effect will be presented in the next part. Fig. 3c and 3d depict the elution curves for chains with N = 100 in SEC and IC columns with D0 = 8, respectively. In this system, the chains feel fairly strong confinement (k = 1.61). In the SEC regime, the effective partition coefficients, Kef, are low (see Fig. 2a) and the chains leave the column close to the exclusion limit for large molecules, tR, SEC-LM = 1000. The fact that the peaks start at somewhat

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lower retention times than tR, SEC-LM = 1000 is caused by the diffusion in the mobile phase. Similarly to the previous case, an increase in r2 causes the broadening of elution peaks and their shift to higher tR values. Both effects are again the results of increasing fractions of wider pores with larger KD(cM). The r-influenced IC chromatographic behavior of chains with N = 100 in narrow pores (see Fig. 3d) is qualitatively similar to SEC. The shifts and the sequence of elution curves are analogous to those in Fig. 3c. This suggests that even though the IC mechanism prevails over SEC and the chains are strongly attracted into pores at eW = 0.32 (Kef > 1 and tR > tR, SEC-SM = 2000 which represents the SEC limit for small molecules), they are there greatly sterically confined. The elution curves are relatively broad due to long tR and enhanced axial dispersion. In summary, the entropic penalty plays a significant role and the PSD effect in narrow pores is similar to that in SEC. Fig. 3e and 3f depict the SEC and IC elution curves for the most confined chains with N = 150 in pores with D0 = 8 (k = 2.05), respectively. In the SEC regime, the peaks are relatively high and narrow and their maxima are very close to the separation limit of large molecules. Their shapes and positions are little affected by r2 because corresponding partition coefficients Kef are very small (close to zero – see Fig. 2a) and peak shapes are mainly results of axial dispersion in the mobile phase. In IC regime, the partition coefficients Kef are important (see Fig. 2c) and the study shows that elution curves for long chains are strongly affected by r2. The sequence of curves corresponds to that in SEC because steric

Fig. 3. Chromatograms of polymer chains eluted from a column with lognormally distributed pore sizes with the median of the lognormal distribution D0 = 8 for different chain lengths: (a) and (b) for N = 50; (c) and (d) for N = 100; (e) and (f) for N = 150. On the left is SEC (eW = 0) and on the right is IC (eW =  0.32). The green, black, red and blue curves represent r2 = 0, 0.1, 0.2 and 0.3, respectively. The total number of theoretical plates is 1000 and the injection zone covers 20 plates. The normalized dispersion coefficient, k = 0.1. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

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constraints play important role. As the chains with N = 150 contain more beads which can adsorb on walls than those with N = 100, an increase in r2.and in the fraction of large pores translates in a broader span in peak positions as compared with Fig. 3d. The comparison of the impact of PSD on SEC and IC data for wide pores with median D0 = 30 is shown in Fig. 4. Fig. 4a and 4b depict the SEC and IC elution curves for chains with N = 50, respectively, i.e., under conditions of mild confinement (k = 0.28). Fig. 4c and 4d show data for N = 100 (k = 0.43) and Fig. 4e and 4f for N = 150 (k = 0.55), both corresponding still to relatively mild confinements. The SEC trends are qualitatively similar to those observed in a column with narrow pores. The curves broaden and their positions shift to higher tR with the increase in r2 for the same reasons as explained above. The only differences are: (i) higher tR values, (ii) lower peak maxima and (iii) broader widths of different curves which are the immediate results of larger KD(cM) and of consequently longer elution times. It is worth noting that the impact of PSD in the IC regime is similar to that observed in Fig. 3b. The curves shift towards lower tR and become narrower (the maxima slightly increase and the half-widths decrease) with the increase in r2. This trend can be understood taking into account the fact that when the chains are reversibly (temporarily) adsorbed and stick firmly to the attractive walls of pores, their conformations deform and local concentration of polymer beads close to the wall increases. This in turn provides more space in the central part of the pore for the entrance of more chains and the overall concentration inside the pores increases. However, in wide pores, the ratio of the volume close to walls, in

which the chains are affected, to the inner volume decreases with D and therefore KD(cM) also decreases with D [25]. This explains the shifts in maxima positions towards lower tR. A close inspection of curve shapes in Figs. 3 and 4 reveals that the unsymmetrical shapes of the SEC and IC elution profiles change with the increase in r2 (see the blue SEC and IC curves for D0 = 30 and N = 150 in Fig. 4e and 4f). Both profiles are unsymmetrical. The tail of the SEC curve is slightly steeper than its front, but the steepness of the IC curve tail is milder than that of the front. This difference is caused by different slopes of KD vs. c dependences in pores with different sizes for SEC and IC [24]. As all details concerning symmetry changes are not clearly visible in small Figs. 3 and 4, we performed the analysis of peak shapes proposed by Armstrong and coworkers [39] and present the results in the Supplementary Material to confirm the above conclusions on the symmetry of chromatograms, Figs. S7–S10. While the SEC behavior discussed in the Supplementary Material is well-known and our results basically confirm the observations of other authors, the IC behavior in narrow pores with D0 = 8 (see Figs. 3, S7 and S8) is non-trivial and slightly surprising. Even though the values of partition coefficients are high, i.e., both KD(cM) > 1 and Kef > 1, and unambiguously show that the clear winner of the entropy (steric constraints)-to-enthalpy (adsorption of chains on walls) competition is enthalpy, steric constraints play very important role and entropy (even though overruled by enthalpy) still controls important features and dictates some rules of the behavior. The most obvious feature controlled by entropy is the dependence of tR on N for r2 = 0 (green curves in Fig. 3), which

Fig. 4. Chromatograms of polymer chains eluted from a column with lognormally distributed pore sizes with the median of the lognormal distribution D0 = 30 for different chain lengths: (a) and (b) for N = 50; (c) and (d) for N = 100; (e) and (f) for N = 150. On the left is SEC (eW = 0) and on the right is IC (eW = 0.32). The black, red and blue curves represent r2 = 0.1, 0.2 and 0.3, respectively. The total number of theoretical plates is 1000 and the injection zone covers 20 plates. The normalized dispersion coefficient, k = 0.1. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

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corresponds to SEC. One other similarity with SEC is the entropycontrolled dependence of the peak position on r2 for chains of N = 100 and 150. As already shown, these chains are strongly sterically constrained and their positions shift towards higher retention times and their width increases with the increase in r2. The chains with N = 50 are less constrained and their r-affected behavior is similar to that observed in wide pores (see the next part). The IC behavior in wide pore systems with D0 = 30 is also interesting and at the first glance slightly surprising, but the explanation of differences between IC and SEC trends is easy and straightforward. First, the longer are the chains, the more beads can be adsorbed on pore walls. As the chains are flexible and only little sterically constrained in wide pores, they easily deform to maximize polymer-wall interactions and adsorb on pore walls. The effective adsorption strength and tR increase with molar mass of the polymer as it is common in most other chromatographic methods, except SEC. The dependence of Kef(r2 = 0) on N (values in Fig. 2d) actually obeys the Martin rule [40]. Second, as already mentioned, the ratio of the sub-surface area, where the chains are affected by the adsorption, to the inner volume of the pore, where the chains are effectively unaffected, decreases with D. As we have shown in [25], the IC dependence of KD(cM) on D (for constant N) reflects an intricate entropy-to-enthalpy interplay. Its shape is non-monotonic and passes maximum in the region of medium D. In wide pores, it decreases with D, which means that the efficiency of IC process and the contribution of wider pores

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in systems with r2 > 0 are less important than those of narrower pores. This is why the position of elution curves shift to lower retention times and the curves become narrower. 3.3. Resolution of chains differing in lengths The most important characteristic for experimentalists is the resolution of peaks for different chain lengths defined as the difference of peak positions divided by the average of peak half-widths: R = 2 (tR2 – tR1)/(w2 – w1), where subscripts 1 and 2 correspond to shorter and longer chains, respectively. Because (i) the tR vs. N curve currently decreases in SEC and increases in IC, and (ii) the slope of some tR vs. N curves changes from negative to positive values with r2, in contrast to the majority of papers, we use both positive and negative values to depict the impact of r2 on the sequence of the peaks. Fig. 5 outlines the impact of r2 on the resolution of chains differing in length. In SEC columns with narrow pores with r2 = 0 (Fig. 5a), the resolution of chains differing in length is poor, simply because KD(cM) are small and the overall separation efficiency is low, e.g., the long chains with N = 100 and 150 are eluted simultaneously. The presence of a fraction of wider pores (for r2 = 0.3) shifts the position of peak maxima and improve significantly the separation efficiency. Fig. 5b shows that the IC resolution in narrow pores is much better than that in SEC. Fig. 5c and 5d show that the impact of r2 in wide

Fig. 5. Chromatograms of polymer chains eluted from a column with lognormally distributed pore sizes with the median of the lognormal distribution D0 = 8, upper row (a) and (b), and for D0 = 30, the bottom row (c) and (d). Left column (a) and (c) is for SEC and the right one (b) and (d) is for IC. The magenta, ochre and cyan curves correspond to N = 50, 100 and 150, respectively. The dashed curves represent r2 = 0 and the solid curves r2 = 0.3. The resolution of elution curves for chains with N = 50 and 150, RS,50-150, as a function of r2 is shown in the inserts. The total number of theoretical plates is 1000 and the injection zone covers 20 plates. The normalized dispersion coefficient, k = 0.1. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

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pores on both SEC and IC resolution is mild, but the shifts in peak positions are important. Inserts in Fig. 5a–5d show the resolution of the chains with N = 50 and 150, RS,50-150, as a function of r2. The changes in IC narrow pores with D0 = 8 are the most striking. The resolution is good for both r2 = 0.0 and 0.3. However, the sequence of peaks for increasing N reverses with the increase in r2 which requires that the RS,50-150 curve passes zero, i.e., RS,50-150 decreases at low r2, and at a certain r2 value, it is completely suppressed. This finding is very important for LCCC chromatography. This chromatography variant is often used in studies of block copolymers [19,41–43], because it enables to suppress purposely the molar mass resolution of one block and investigate the dispersity of the other block. Nevertheless, thorough experimental studies revealed that the retention volume vs. molar mass is non-monotonous close to CAP [25,44], which complicates the behavior. Our simulations show that a relatively mild non-uniform porosity of the stationary phase reduces the resolution considerably, which can be exploited for tuning the conditions in experimental set-ups. 4. Conclusions In this computer study, we addressed the little investigated concentration effects and thermodynamic PSD effects on the partitioning of polymers between bulk solvent and porous stationary phase and their impact on results of chromatographic methods. By MC simulations, we generated the sets of partition coefficients depending on polymer concentration, lengths of polymer chains, average pore size and on PSD and analyzed the trends of the partitioning equilibrium. As the concentration in chromatography studies is low, experimentalists usually assume that its variations do not significantly affect the chromatography results [45]. In our recent studies [24,25], we observed that the concentration changes occurring during the passage of polymer zone through the column play important role both in SEC and IC and affect the position and shape (asymmetry) of elution curves. In this paper, we addressed this little explored topic and studied the concentration-dependent chromatographic elution in columns containing stationary phases differing in pore-size distribution by computer simulations. The study shows that not only the average pore size and interactions, but also pore size dispersity (PDS) influences considerably the chromatograms. The proof that PSD plays important role both in SEC and in IC required the development of suitable computational machinery. We significantly modified our original MC code [28] and elaborated an efficient NVT MC simulation variant, which enables simultaneous evaluation of the ensemble of partition coefficients, KD(cM), describing the partitioning of polymers at different concentrations between bulk solvent and pores differing in pore sizes. The elution curves were generated by numerical solution of the mass balance equation describing the passage of the polymer zone through the column using the set of KD(cM) as input parameters. The new original algorithm reflects the fact that polymer concentration considerably varies across the moving concentration profile and the partitioning depends both on concentration and on PSD. The simulations show that the impacts of changing concentration and that of PSD are very complex. The actual shapes of elution curves (broadness and symmetry) and their positions depend on steric confinement, i.e., on the coil-to-pore size ratio, k, and on the interaction of polymer beads with the wall, eW. In the strongly sterically constrained systems, the trends in the IC curves with increasing r2 are similar to SEC in spite that the chains are strongly attracted into narrow pores and that KD(cM) are appreciably higher than one. In less constrained systems, the trends of the chromato-

graphic behavior reverse and correspond to IC. This means that both the adsorption of chains on walls and entropy decrease due to steric constraints play important roles. The interplay is intricate and both antagonist effects almost mutually compensate. The most important results of the study (interesting for experimentalists) concern narrow pores and particularly their role in IC regime. The performed simulations confirm and elucidate the experimental fact that IC is efficient in fairly narrow pores [46,47] where SEC partition coefficients are very small and preclude efficient separation. The chains are attracted into pores, but, as mentioned above, steric constraints are important and the separation of polymers according to molar masses obeys the SEC calibration curve. The study thus reveals a new interesting chromatography regime. However, the impact of new findings exceeds the scope of chromatography. The observations obtained by MC simulations help to understand complex phase equilibria of polymers in porous media in general. Other important finding of the study concerns the positions and shapes of elution curves both in narrow and wide pores. While the hydrodynamic effects always broaden the elution curves and in case of PSD they amplify the broadening and tailing [13,14,48], they do not significantly affect the elution times. The thermodynamic PSD effect causes the narrowing or widening of elution curves (depending on conditions) and considerable shifts in position of elution peaks. The observation that the resolution is strongly influenced by PSD is also an important piece of information for LCCC chromatography. LCCC enables targeted studies of the dispersity of individual blocks in block copolymers [43,49–52], but it requires accurate tuning of conditions for masking the effect of the other block. The fact that the retention volume vs. molar mass curve is nonmonotonous close to the CAP [44] complicates the adjustment of experimental conditions in studies of molar mass dispersity in a broad range. Results of simulations show that a relatively modest PSD suppresses the resolution, which facilitates the tuning the experimental conditions. In wide pores, i.e., under mild constrains in the region of k close to one, which are relevant for practical chromatography because the KD(cM) values are large enough in both the SEC and IC regimes, the SEC curves became broader with the increase in r2, and they shift towards higher tR, while the IC curves become narrower and shift towards lower tR. This difference partially explains the observation by the Chang group concerning the comparison of SEC and IC studies of virtually monodisperse polymer standards [43]. However, as the narrower IC curves shift towards lower elution times and the wider SEC curves towards higher elution times, the resolution of peaks for chains with different lengths based on the difference between positions of peak maxima and on average peak half-widths change only little, which is, in turn, convenient from the experimental point of view. In summary, the study elucidates some not enough investigated and not yet fully understood features of SEC and IC caused by PSD and provides important hints for experimental chromatographic studies of various polymers concerning the choice of porous materials (e.g., suitable porosity of column packings, polymer-wall interactions) and helps to interpret correctly experimental data. It reveals a new, so far experimentally unobserved (probably overlooked) chromatography regime, in which almost all retention characteristics correspond to IC, but the sequence of retention volumes vs. molar masses corresponds to SEC. This interesting regime exists in a relatively narrow region of conditions and without the awareness of its existence there is practically zero chance that it can be experimentally discovered. Experimental studies under conditions corresponding to this regime would probably attribute the behavior to regular SEC.

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In this paper, we purposely studied the thermodynamic effects of PSD separately from all sources of hydrodynamic broadening [13,16]. As the behavior is complex, we do not expect that the overall effect on the shape and position of elution curves would be a straightforward linear combination of both (thermodynamic and hydrodynamic) effects. We plan a study which will address the interplay of hydrodynamic and thermodynamic effects. This will require the modification of our computational procedure, extensive series of simulations, careful analysis of results and a long and deep discussion. Therefore we plan an independent report in the near future. CRediT authorship contribution statement Xiu Wang: Methodology, Software, Validation, Formal analysis, Data curation, Visualization. Karel Procházka: Conceptualization, Validation, Writing - original draft, Writing - review & editing, Funding acquisition. Zuzana Limpouchová: Conceptualization, Validation, Visualization, Writing - review & editing. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgements This work was funded by Czech Science Foundation Grant 17-04258J and by Charles University Research Centre program No. UNCE/SCI/014. Computational resources were supplied by the Ministry of Education, Youth and Sports of the Czech Republic under the projects of CESNET (Project No. LM2015042) and CERIT-Scientific Cloud (Project No. LM2015085) provided within the program of projects of Large Infrastructures for Research, Experimental Development and Innovation of the Czech Republic. The authors acknowledge very much fruitful discussion with Prof. T. Chang. Appendix A. Supplementary material Supplementary data to this article can be found online at https://doi.org/10.1016/j.jcis.2020.01.119. References [1] J.M. Polson, D.R. Heckbert, Polymer translocation into cavities: Effects of confinement geometry, crowding, and bending rigidity on the free energy, Phys. Rev. E 100 (1) (2019) 15. [2] X. Yang, Q.H. Yang, Y. Fu, F. Wu, J.H. Huang, M.B. Luo, Study on the adsorption process of a semi-flexible polymer onto homogeneous attractive surfaces, Polymer 172 (2019) 83–90. [3] Q.P. Chen, M.R. Schure, J.I. Siepmann, Using molecular simulations to probe pore structures and polymer partitioning in size exclusion chromatography, J. Chromatogr. A 1573 (2018) 78–86. [4] J.M. Polson, Free energy of a folded semiflexible polymer confined to a nanochannel of various geometries, Macromolecules 51 (15) (2018) 5962– 5971. [5] R.T. Cimino, C.J. Rasmussen, Y. Brun, A.V. Neimark, Mechanisms of chain adsorption on porous substrates and critical conditions of polymer chromatography, J. Colloid Interface Sci. 481 (2016) 181–193. [6] A. Muralidhar, K.D. Dorfman, Kirkwood diffusivity of long semiflexible chains in nanochannel confinement, Macromolecules 48 (8) (2015) 2829–2839. [7] X.Y. Wang, M. Tang, Y.W. Wang, Equilibrium distribution of semiflexible polymer chains between a macroscopic dilute solution phase and small voids of cylindrical shape, Macromol. Theory Simul. 24 (5) (2015) 490–499. [8] A. Muralidhar, D.R. Tree, K.D. Dorfman, Backfolding of wormlike chains confined in nanochannels, Macromolecules 47 (23) (2014) 8446–8458. [9] L.I. Klushin, A.A. Polotsky, H.P. Hsu, D.A. Markelov, K. Binder, A.M. Skvortsov, Adsorption of a single polymer chain on a surface: effects of the potential range, Phys. Rev. E 87 (2) (2013).

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