Journal of Colloid and Interface Science 481 (2016) 181–193

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Journal of Colloid and Interface Science

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Regular Article

Mechanisms of chain adsorption on porous substrates and criticalconditions of polymer chromatography

http://dx.doi.org/10.1016/j.jcis.2016.07.0190021-9797/� 2016 Elsevier Inc. All rights reserved.

⇑ Corresponding author.E-mail address: aneimark@rutgers.edu (A.V. Neimark).

Richard T. Cimino a, Christopher J. Rasmussen b, Yefim Brun b, Alexander V. Neimark a,⇑aDepartment of Chemical and Biochemical Engineering, Rutgers, The State University of New Jersey, 98 Brett Road, Piscataway, NJ 08854, USAbDuPont Central Research & Development, Corporate Center for Analytical Sciences, Macromolecular Characterization, Wilmington, DE 19803, USA

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

a r t i c l e i n f o

Article history:Received 20 June 2016Revised 11 July 2016Accepted 11 July 2016Available online 15 July 2016

Keywords:Critical adsorptionPolymer adsorptionMonte Carlo simulationPolymer chromatographyNanoporous materials

a b s t r a c t

Polymer adsorption is a ubiquitous phenomenon with numerous technological and healthcare applica-tions. The mechanisms of polymer adsorption on surfaces and in pores are complex owing to a compe-tition between various entropic and enthalpic factors. Due to adsorption of monomers to the surface,the chain gains in enthalpy yet loses in entropy because of confining effects. This competition leads tothe existence of critical conditions of adsorption when enthalpy gain and entropy loss are in balance.The critical conditions are controlled by the confining geometry and effective adsorption energy, whichdepends on the solvent composition and temperature. This phenomenon has important implications inpolymer chromatography, since the retention at the critical point of adsorption (CPA) is chain lengthindependent. However, the mechanisms of polymer adsorption in pores are poorly understood and thereis an ongoing discussion in the theoretical literature about the very existence of CPA for polymer adsorp-tion on porous substrates.In this work, we examine the mechanisms of chain adsorption on a model porous substrate using

Monte Carlo (MC) simulations. We distinguish three adsorption mechanisms depending on the chainlocation: on external surface, completely confined in pores, and also partially confined in pores in so-called ‘‘flower” conformations. The free energies of different conformations of adsorbed chains are calcu-lated by the incremental gauge cell MC method that allows one to determine the partition coefficient as afunction of the adsorption potential, pore size, and chain length. We confirm the existence of the CPA forchain length independent separation on porous substrates, which is explained by the dominant contribu-tions of the chain adsorption at the external surface, in particular in flower conformations. Moreover, we

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show that the critical conditions for porous and nonporous substrates are identical and depend only onthe surface chemistry. The theoretical results are confirmed by comparison with experimental data onchromatographic separation of a series of linear polystyrenes.

� 2016 Elsevier Inc. All rights reserved.

1. Introduction

Adsorption of polymer chains on porous substrates is a complexphysico-chemical phenomenon relevant to a broad range of appli-cations such as drug delivery, filtration, and adhesion, as well asvarious separation methods including liquid chromatography ofmacromolecules. Adsorption of polymeric chains is governed bythe competing mechanisms of enthalpic attraction and entropicrepulsion [1]. At strong adsorption conditions, the enthalpy gaineddue to adsorption interaction exceeds the entropy loss due torestrictions imposed by confining pore geometry on chain confor-mations, and chains are predominantly adsorbed. The free energyof adsorbed chains decreases with the chain length and the parti-tion coefficient that determines the concentration of adsorbedchains increases. At weak adsorption conditions, the entropy pen-alty is prohibitive and chains are effectively repelled from pores.The free energy of adsorbed chains increases with the chain lengthand the partition coefficient decreases. The transition betweenstrong and weak adsorption is quite sharp, and following the sem-inal work of de Gennes [2] and others [3–5] who studied chainadsorption on plain surfaces, it is treated as a critical phenomenon,occurring at what is termed the critical point of adsorption (CPA).Skvortsov and Gorbunov (S-G) [6] demonstrated the CPA condi-tions during adsorption in pores using the ideal chain model with-out the effect of excluded volume. However, the subsequentmolecular simulation studies with more elaborate models of realchains with excluded volume [7,8] questioned the existence of CPAfor polymer adsorption on porous substrates despite the fact thatthe experimental manifestation of this phenomenon is widespreadin polymer chromatography [1,9–11]. Here, we investigate thiscontroversy by using Monte Carlo simulations and posit that thecritical conditions in polymer chromatography are related to thespecific mechanism of partial confinement of chains in so-calledflower conformations [12], when only a part of the chain is locatedinside the pore.

As noted above, the phenomenon of critical adsorption is widelyexploited in polymer chromatography. Separation of macro-molecules is typically performed on porous substrates. Whenadsorption is weak (ideally, negligibly small), steric (size exclu-sion) interactions between the polymer chains and porous sub-strate provides separation by molecular size. This separationregime is known as size exclusion chromatography (SEC) that iswidely used to measure molar mass distribution of polymers. InSEC mode, larger chains are excluded from the pore volume andelute earlier than their smaller counterparts. When adsorption isstrong, in the regime of liquid adsorption chromatography (LAC),adsorption prevails and retention increases with the molecularweight; the order of elution is opposite to that in SEC with largerchains retained more strongly than shorter chains. The intermedi-ate regime of mutual compensation between the attractive adsorp-tion and repulsive steric interactions is called liquidchromatography at critical conditions (LCCC) [9,10]. In LCCC, elu-tion of polymer chains is molecular weight-independent, enablingthe separation of polymer fractions by other structural and/orchemical factors, e.g. by the type and number of various functionalgroups in telechelic polymers [13]. The respective experimentalconditions that correspond to the CPA for a given class of polymerscan be achieved by variation of either solvent composition or tem-

perature of the polymer solution [14]. Although LCCC is widelyused for separation of complex polymers and biopolymers by dif-ference in chemical structure, and the critical adsorption condi-tions are well documented for many systems, the mechanisms ofchain adsorption on porous substrates are still poorly understoodand the conclusions derived from different theoretical models arecontroversial.

Detailed analyses of critical adsorption on non-porous surfaceswere performed by using various theoretical and computationalmodels [3–6,12,15–20]. The underlying theory of LCCC on poroussubstrates was suggested by Skvortsov and Gorbunov (S-G) [6],who considered adsorption of ideal chains confined in slit-shapedpores. In modeling studies, the adsorption strength is controlledby the magnitude of the effective interaction energy, U. S-G definedthe CPA as the value of U = UC, at which the difference in free ener-gies between adsorbed and unconfined chains vanishes. This con-dition was interpreted as a complete mutual compensation of theeffects of enthalpic and entropic interactions so that the concentra-tion of chains confined in pores equals to the concentration ofunconfined chains in bulk solution. They showed that the CPAexists for ideal chains and corresponds to the conditions of molec-ular weight-independent elution in LCCC.

However, further modeling studies showed that the existence ofCPA for chains adsorbed in pores is limited to the ideal chains. InMonte Carlo simulations of real chains with excluded volume, Cifra& Bleha [7] and Gong & Wang [8] found that the equilibrium par-tition between the free unconfined chains in the solution andchains confined within pore channels of several shapes alwaysdepends on the chain length regardless of the adsorption potential.These authors suggested to redefine the CPA as the magnitude ofadsorption potential, U = UC, at which the partition coefficient var-ies the least with the chain length. This definition may explain theapparent observation of critical conditions in chromatographicexperiments, assuming that the deviations in the partition coeffi-cient are smaller than the limits of experimental accuracy. Thisconclusion was confirmed by Yang and Neimark [12] who usedself-consistent field theory (SCFT) to simulate adsorption of realchains inside spherical cavities.

Another mechanism that may explain the experimental evi-dence of critical conditions for chain adsorption on porous sub-strates is the presence of partially confined or ‘flower’ chainconformations [12,21,22]. Specifically, it has been shown [12] thatwhen the characteristic chain size exceeds the pore size, the por-tion of chains completely confined within pores diminishes, asthe adsorption potential reduces to its critical value (U ! UC),and the flower conformations outweigh completely confinedstates. This effect is accompanied by the adsorption of chains onthe external surface of the substrate [20], where the chain entropyis significantly increased relative to the pore [23].

In our previous work, the critical conditions of chain adsorptionwere studied with a MC simulation model of real (excluded vol-ume) chains interacting with non-porous substrates, in particular,in chromatographic columns packed with non-porous particles[20,24]. In that case, adsorption occurs exclusively on the particleexternal surface. In this work, we re-examine the phenomenon ofpolymer adsorption on porous substrates and its application topolymer chromatography using the same MC simulation modelthat was used in [20,24]. We explore the chain adsorption taking

R.T. Cimino et al. / Journal of Colloid and Interface Science 481 (2016) 181–193 183

into account three general mechanisms [12]: adsorption on theexternal surface, complete adsorption inside pores, and partialadsorption in pores in flower conformations, see Fig. 1. We deter-mine the critical conditions for this model and show that whilethe CPA does not exist for completely confined chains, it does existfor the chains which are allowed to be adsorbed partially withinthe pores and on the external surface. Moreover, in the vicinityof CPA, these two adsorption mechanisms (external adsorptionand partial confinement) dominate and the fraction of completelyconfined chains diminishes.

The remainder of this paper is structured as follows. In Section 2,we re-visit the thermodynamic definition of the critical conditionof adsorption and show that the CPA implies the equality of theincremental chemical potentials of adsorbed and free chains,rather that the equality of the respective free energies as wasassumed in earlier literature. Section 3 details our MC simulationset-up employed for modeling adsorption of real chains. In Sec-tion 4, we study chain adsorption at the plain surface, and showthe existence of CPA at a certain value of the effective adsorptionpotential U = UC. This value is then used as a benchmark to probethe existence of critical conditions for chains completely confinedto pores in Section 5. It is shown that, in contrast to ideal chains,the CPA does not exist for real chains completely confined to poresand the partition coefficient of completely confined chains pro-gressively decreases with the chain length. In Section 6, we studythe mechanism of partial confinement in flower conformationsand calculate the respective partition coefficient, which turns outto be chain length independent at the same critical adsorptionpotential U = UC, as determined for the external surface. In Sec-tion 7, these partition coefficients are combined into an apparentpartition coefficient, which is calculated and used to predict theretention volumes of chain molecules on porous adsorbents. Usinga simplistic model of a surface corrugated with an array of ink-bottle spherical pores, we calculate the retention volumes for a ser-ies of linear polystyrenes separated on a chromatographic columnpacked with porous particles. The model involves two columnparameters, the packing porosity and surface area, which areadjusted to get a quantitative agreement with the experiment.The relative weights of different mechanisms of adsorption are dis-cussed, and it is shown that adsorption on the external surface andin flower conformations are dominant. As the magnitude of theadsorption potential increases to its critical value, the chains areexpelled from the pores to the external surface, helping to explainthe existence of the CPA for porous substrates. The results are sum-marized in Section 8.

2. Critical conditions of polymer adsorption

There are several definitions for the critical conditions ofadsorption for polymer chains based on various chain properties,i.e. number of adsorbed monomers [3], radius of gyration [4], andchain free energy [6]. The latter definition implies the equality ofthe excess free energies of adsorbed and free chains,FðNÞ ¼ F0ðNÞ. This condition is correct for the models of idealchains, however, for real chains with excluded volume, it holdsonly in the limit of infinitely long chains. As shown in [12,24] inthe general case of real chains with exclusion volume, the CPA con-dition of chain length independent separation requires the equalityof the incremental chemical potentials of adsorbed and free chains:

lincrðNÞ ¼ l0incr at CPA for N > N� ð1Þ

This condition is held provided that the chain length N is suffi-ciently long, N > N⁄, (N⁄ is a certain small number (�10) of chainsegments [25]).

The incremental chemical potential defined as the differencebetween free energies of chains of length N + 1 and N,lincrðNÞ ¼ FðN þ 1Þ � FðNÞ, is a measure of the work necessary toadd or remove a segment at the end of a chain [26]. As such, thefree energy of a chain is the sum of the incremental chemical

potentials of its constituent segments FðNÞ ¼PN�1i¼0 lincrðiÞ. For a

free unconfined chain, it has been shown previously that the freeenergy F0 is a linear function of the incremental chemical potential:

F0ðNÞ ¼PN��1i¼0 l0

incrðiÞ þ ði� N�Þl0incr [26]. Similar linear dependence

holds also for the chains tethered or adsorbed to a nonporous sur-face [12,24].

The CPA condition (1) follows from the fact that the chain par-tition coefficient depends on the difference of the excess free ener-gies of adsorbed and free chains, FðNÞ � F0ðNÞ ¼ DFðN), which mustbe chain length independent, but not necessarily null, at the CPA.The latter implies that

dDFðNÞ=dN ¼ 0 at CPA ð2Þ

and therefore, lincr ¼ dFðNÞdN ¼ l0

incr ¼ dF0ðNÞdN , confirming Eq. (1). A gen-

eral derivation of the critical condition (1) is given the SupportingInformation.

In order to demonstrate the existence of critical conditions onporous substrates, MC simulations were undertaken to calculatethe incremental chemical potential of chains of different lengthat various adsorption states shown in Fig. 1. The incrementalchemical potentials were summed to calculate the chain free ener-gies, which were utilized to determine the respective partitioncoefficients for the three different mechanisms of chain adsorption.The details of the simulation procedure are outlined below.

3. Chain model and simulation methodology

3.1. Chain and adsorbent models

The polymer molecules are modeled as freely jointed chains ofspherical beads of diameter b. Adjacent beads are connected by aharmonic potential Ub = k(r12-b)2 where k is the harmonic springconstant, equal to 50kBT/b2 and r12 is the center-center distancebetween the beads. In this model, there is no stiffness or limitationon bond angles or torsions and non-neighbor beads interact via thestandard Lennard Jones potential ULJ = e((b/r12)12-(b/r12)6), where eis the bead-bead interaction energy, equal to 0.125kBT. The numberof beads in the chain varies from N = 1 to N = 200 as in our previousstudies [20,24]. These chains have corresponding gyration radii ofRG = 1–10b in free solution, calculated during simulation by ensem-ble average. The free chains approach thermodynamically goodsolvent conditions, i.e. random-coil conformation with excludedvolume interaction, by setting reduced temperature T⁄ = kBT/e = 8,near the athermal limit [27]. The adsorption interaction betweenthe beads and the surface is modeled by a square-well potentialof width b. The same adsorption potential is applied to both theplain surface and the spherical pore walls. The magnitude of theadsorption potential, U, is varied from 0 to �1kBT to capture the fullrange of adsorption and exclusion conditions [20]. All energiesmentioned below are in kBT units.

3.2. Simulation methodology

The chains were equilibrated by a series of 400 million MC dis-placement, insertion, removal, and configurational bias regrowth[28] moves using the standard Metropolis algorithm. The incre-mental chemical potential of the chains was determined via theincremental gauge cell method [25] and averaged over 500 millionMC production moves. Free chain radii of gyration were likewise

Fig. 1. Characteristic conformations of polymer chain interacting with nanoporous substrate: 1 –unconfined chain in bulk solution, 2 – chain adsorbed at the external surface,3 – chain completely confined within the pore, 4 – partially confined flower conformation.

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averaged over 500 million MC production moves, after equilibra-tion. Simulation details and the method for determining the criticalconditions for different adsorption mechanisms (external surface,complete confinement, and partial confinement) are given below.

4. Determination of the critical point of adsorption on anonporous surface

The simulation method for determining the CPA follows theprocedure outlined in our previous paper [20]. Chains are anchoredby their terminal monomers at distance z = 0. . . L from the surfaceof the planar adsorbent measured in b units, where L is sufficientlyfar from the surface to approximate a free chain. For every z, theincremental chemical potential is measured for each chain lengthN = 1–200 (corresponding to gyration radii RG � 1. . .10b in freesolution) using the incremental gauge cell method [25]. The differ-ence between the incremental chemical potential of the adsorbedchain and a free chain, Dl ¼ lincr � l0

incr acts as an indicator ofthe regime of adsorption.

The plots in Fig. 2 show the incremental chemical potential dif-ference Dl as a function of chain length N for chains anchored atdifferent distances z from the surface. For the case of zero adsorp-tion potential (U = 0, Fig. 2A), Dl is positive reflecting effectiverepulsion from the surface due to entropic restrictions. As chainsbecome longer or are anchored farther away from the surface,the surface effect diminishes and the incremental chemical poten-tial tends to that of a free chain. The critical conditions are foundfor U = �0.725 (Fig. 2B). At the CPA, the difference in chemicalpotentials Dl ¼ 0 for all anchoring distances z for chains longerthan a certain length N⁄ = 20. The chemical potential for chainsclose to the surface is either slightly positive (for very short chains,which are unlikely adsorbed) or negative (for longer chains, for

which the probability of adsorption in higher). Beyond the criticalpoint (Fig. 2C and D), the incremental chemical potential of theadsorbed chains is negative, since the majority of chains arestrongly adsorbed. Interestingly, long chains (N > 100) tethered atdistances z = 10–20b do come into contact with the adsorption well(see Fig. 2D), indicating the existence of ‘‘stretched” stateswith chain gyration radii exceeding the maximum gyration radiusRG(N = 200) � 10b of free chains. These ‘‘stretched” states (Fig. 1.2)are entropically less favorable than free chains (Fig. 1.1), but theirfree energy is partially compensated for by the strong adsorptioninteraction. Notably, the CPA determined by this method isidentical to that found by the geometrical method whereby chainstethered directly to the surface are exposed to an increasingpotential field [24].

The transition from weak to strong adsorption conditions andthe existence of a well-defined CPA are reflected in the dependenceof the Henry adsorption coefficient [12,20] on the adsorptionpotential (see Fig. 3). The Henry coefficient, which represents theratio of the surface excess concentration of chains cs to their bulkconcentration c0, KH = cs/c0, is calculated using the following equa-tion [12]:

KHðNÞ ¼Z L

z¼0exp � F N; zð Þ � F0ðNÞ

� �h i� 1

h idz ð3Þ

Notably, the Henry coefficient KH has the dimension of length,presented in Eq. (3) and below in terms of Kuhn segments, b, sincez is also measured in b. Eq. (3) states that the Henry coefficient isthe integral of the Boltzmann weighted difference in the freeenergy of anchored F(N,z) and free F0 (N) chains. The plot of theHenry coefficient KH as a function of the adsorption potential illus-trates several key features of adsorbed chains. For weak potentials,|U| < |UC|, the Henry coefficient is negative, reflecting effectiveentropic repulsion and respectively, negative excess adsorption.

Fig. 2. Chain length dependence of the incremental chemical potential difference Dlðz;UÞ of anchored chains at several U. (A) U = 0. (B) U = �0.725 (CPA), (C) U = �1. (D)Zoom of case C for large N. The range of anchoring distances z varies from 0.5 to 200.

Fig. 3. The Henry coefficient KHðNÞ computed for a series of characteristic chainlengths (N = 15–200) as a function of the adsorption potential U. The intersectionpoint corresponds to the CPA at U = UC = �0.725. Here, KH is given in units of b.

R.T. Cimino et al. / Journal of Colloid and Interface Science 481 (2016) 181–193 185

As a consequence, weak potentials produce a significant spread ofthe Henry coefficient values across different chain lengths. Thechains are mostly repelled from the surface in proportion to their

size and KH decreases with chain length. As the adsorption poten-tial approaches the critical condition, U � UC = 0.725, the Henrycoefficients for all chain lengths intersect at the critical valueKH � 1.25. At these conditions, which are identical to the case ofa tethered chain [20,24], all chains are equally likely to adsorb orto remain free, so the Henry coefficient is chain length-independent. Noteworthy, the critical value of KH is not zeroreflecting the CPA condition of the equality of the incrementalchemical potentials rather than chain free energies. Above the crit-ical potential |U| > |UC| there is a sharp transition wherein thechains become strongly adsorbed to the surface. Smaller chainsare less likely to adsorb than larger ones, and so the Henry coeffi-cient increases with chain length.

5. Probing the existence of critical conditions for completelyconfined chains

The value UC of the critical potential on the nonporous surfaceacts as a benchmark to probe the critical point on porous surfaces.In this and the following section, the existence of critical condi-tions is investigated using simple model pores under two differentgeometric constraints: complete or partial confinements. It will beshown that, under specific circumstances, critical conditions existon porous surfaces and the critical adsorption potential on a por-ous surface is equal to the critical potential on a nonporous surface.

Adsorption in pores is modeled by confining chains to sphericalpores of various sizes. We explore the range of pore radii

186 R.T. Cimino et al. / Journal of Colloid and Interface Science 481 (2016) 181–193

Rpore = 1.5–6b that corresponds to the conditions of strongRG/Rpore � 10 to moderate RG/Rpore � 2 confinement for the largest(N = 200) chains considered. The adsorption potential at theinternal surface of pores is assumed to have the samesquare-well form as the potential at the external surface.

To demonstrate the effect of pore size on adsorption behavior,we present in Fig. 4 the results of two computational experiments.In the first experiment, chains of lengths N = 1–200 were com-pletely confined within pores of a range of pore sizes Rpore = 2, 3,4, 5 and 6b at U = UC, and the chains were allowed to equilibrate.The incremental chemical potential of these chains was measuredand plotted as a function of the chain length (see Fig. 4A). It is clearfrom Fig. 4A that the incremental chemical potential of confinedchains is a monotonically increasing function of chain length dueto the increasingly prohibitive entropic penalty to insertion ofadditional beads in a pore volume. The incremental chemicalpotential intersects the unconfined (free) chain value for somechain length N = N⁄, (equivalently, gyration radius RG = RG

⁄),referred to here as the critical chain length and critical gyrationradius, respectively. At N⁄ (RG

⁄), the enthalpic and entropicinteractions are exactly balanced. The values of the critical chainlength/gyration radius are dependent on the pore size Rpore anddecrease progressively as Rpore gets smaller. The dependence ofN⁄ and RG

⁄ on Rpore is plotted below in Fig. 4B.The critical chain length N⁄ is quadratic in Rpore (N⁄ � Rpore

2)(equivalently, RG � N0.5) indicating that chains of length N < N⁄

behave as free chains in the pore when the adsorption potentialis at its critical value, U = �0.725. However, any chain larger thanN⁄ will not hold to this scaling, due to the penalty imposed bythe positive incremental chemical potential. The dependence ofthe critical gyration radius RG

⁄ on the pore size approaches a linearasymptote (RG

⁄ � Rpore) (Fig. 4B, solid line) for large pores(Rpore P 4), indicating that as pore size increases, it is possible todirectly predict the maximum chain size that will adsorb in thepore. There is deviation from the asymptote for small poresRpore 6 4, which is explained by the increasing effect of entropicconfinement.

In the second experiment, the incremental chemical potential ofthe chains was measured as a function of the adsorption potentialU, for a single pore size, RP = 4b (see Fig. 4C). At this pore size,chains of N < 50 beads (RG 6 4 in free solution) fit comfortablywithin the pore when exposed to adsorption potential U corre-sponding to critical conditions on a plain surface U = �0.725 (asper Fig. 4B). However, beyond N = 50, the incremental chemicalpotential increases rapidly with N for all adsorption potentials con-sidered. In the limit of an extremely strong potential, it is conceiv-able that any chain consisting of N < 200 beads could adsorb withinthe pore. However, the monotonically increasing chemical poten-tial indicates that longer chains (N� 200) would be totallyexcluded from this pore.

Fig. 4C clearly demonstrates the absence of the critical adsorp-tion conditions for this pore. This situation is applied in generalcase: critical conditions (Eq. (1)) do not exist for real chains con-fined to pores of comparable or smaller size than that of the freechain. The results illustrated in Figs. 4 confirm the earlier findingsfor excluded volume chains, which were reported for various poregeometries [8,29]. To further illustrate this point, the difference infree energy D F = F(N) – F0(N) and the partition coefficient KP = exp[�DF] for chains confined to pores of sizes Rpore = 1.5b and 4b areplotted as functions of N in Fig. 5 below.

The difference in free energy D F of confined chains plotted inFig. 5A and C shows several interesting features. First, in theabsence of adsorption (U = 0), the free energy is monotonicallyincreasing with N regardless of the pore size. This is consistent withthe effect of a pore being gradually filled up with the chain of beads– as the pore fills, the work required to add another bead to the

chain increases due to strictly entropic forces. In the presence ofan adsorption potential, the shape of the free energy dependencechanges dramatically. Instead of being monotonically increasing,the free energy experiences a minimum [30], after which the freeenergy becomes progressively positive as N increases. This mini-mum is more pronounced for the larger (4b) pore (Fig. 5C) thanthe smaller pore (Fig. 5A), and the chain length NM at the pointof minimum varies with U and Rpore.

The partition coefficient KP for chains completely confined topores is defined in terms of the exponent of the difference in freeenergy of confined and unconfined chains KP = exp[�DF]. The plotsof KP as a function of chain length (Fig. 5B and D) further illustratethe absence of critical conditions for finite length chains, regardlessof pore size. At zero adsorption potential, the partition coefficientmonotonically decreases as chain length increases, reflecting theincreasing difficulty to force larger excluded volume chains intothe pores. In the presence of adsorption potential, the partitioncoefficient is non-monotonic in a manner similar to the freeenergy. The partition coefficient increases with chain length forchains, which can fit in the pore with minimal restriction(N < NM). The length NM of the chains ‘‘most likely” residing insidethe pore shifts to slightly larger values as adsorption potentialincreases. However, for N > NM, the partition coefficient rapidlydecreases with the chain length for all adsorption potentials dueto severe entropic limitations, with the smaller (1.5b) pore(Fig. 5B) accommodating roughly 10 beads (RG = 1.6b in free solu-tion), and the larger (4b) pore (Fig. 5D) up to 40 beads (RG = 4b infree solution). The non-monotonic behavior of the partition coeffi-cient reinforces the conclusion that critical conditions do not existfor the real chains completely confined to pores, when pore sizeRpore is comparable to chain size RG. This effect would not beobserved for ideal chains, for which the CPA does exist [31]. Below,we show that the critical behavior observed in experiments is dueto the adsorption mechanisms involving the external surface of thesubstrate.

6. Partially confined chains

The characteristic chain length NM mentioned above is of spe-cial interest. Provided the pore has an opening, chains of lengthN > NM tend to escape from the pore, at least partially, formingso-called flower conformations [12]. Depending on the pore size,these partially confined states have been shown to make up a sig-nificant portion of chain’s conformations [12]. A flower conforma-tion consists of a ‘‘root” or trans section of the chain residing insidethe pore and a ‘‘stem” or cis section hanging out of the pore. Thestem of the flower conformation may either be adsorbed, if theadsorption potential U is strong, or protrude into the bulk solution,if U is weak, see Fig. 6(A and B). Partially confined flower chainshelp us to explain the presence of critical conditions on poroussubstrates where the pores are of smaller size than the polymerchains in free solution Rpore < RG.

The free energy of partially confined chains was a subject ofdetailed studies related to the translocation of chain molecules intonanopores [30,32]. The free energy of a partially confined chain oflength N is presented as a function of the degree of translocation s– the number of monomers inside the pore. The free energy F (N, s)is composed of the free energies of cis FC(N-s) and trans FT(s) sub-chains: F (N, s) = FC(N-s) + FT(s), where cis denotes beads outsidethe pore, and trans – inside the pore (see Fig. 6B). The sub-chainfree energies are themselves composed of the sum of the incre-mental chemical potentials for cis and trans chains of length N-sand s respectively. For details of the free energy calculations offlower conformations, see Supporting Information.

The cis- and trans- sub-chains are simulated independently, in amanner similar to that used for externally adsorbed or completely

Fig. 4. Incremental chemical potential as a function of N for chains of length N = 1–200. (A) Chains confined to pores of size Rpore = 2, 3, 4, 5 and 6b at U = �0.725. Black linerepresents the incremental chemical potential of unconfined (‘free’) chain. (B) Chain length N⁄ (open circles) and gyration radii RG⁄ (closed circles) for which lincr = lincr

0 , bothas a function of Rpore for U = �0.725. Solid black line – asymptotic dependence of RG

⁄ vs Rpore for larger pores. Dashed line – parabolic dependence of N⁄ on Rpore. (C) Chainsconfined to a pore of size Rpore = 4b for various adsorption potentials U spanning from 0 to �1. Black line represents the incremental chemical potential of unconfined chains.

Fig. 5. The chain length dependences of the free energy difference D F = F(N) – F0(N) (A, C) and the respective partition coefficient KP (B, D) for the chains completely confinedto a pore of Rpore = 1.5b (top) and 4b (bottom) for a series of adsorption potentials U = 0 to �1.

R.T. Cimino et al. / Journal of Colloid and Interface Science 481 (2016) 181–193 187

confined chains. In each case, sub-chains are tethered by a terminalmonomer with a center distanced 0.5b from the surface separatingcis and trans conformations (denoted by the dashed line in Fig. 6).Upon equilibration, the incremental chemical potentials of the cisand trans sub-chains are measured and the free energies of cisFC(N-s) and trans FT(s) subchains are calculated. The sum of sub-chain free energies gives the free energy F(N,s) of the partiallytranslocated chains with given degree of translocation.

In order to determine the free energy F(N) of the ensemble offlower chains with different degrees of translocation s = (1. . .N),one has to sum the Boltzmann-weighted free energies F(N,s) ofthe chains with given degree of translocation [12]:

FðNÞ ¼ � lnXNs¼1

expð�FðN; sÞÞ !

ð4Þ

It is important to note that each state F(N,s) is itself an ensembleaverage of the chain free energy representing the free energy of themost favorable conformations at given degree of translocation.Fig. 7A and B below shows the incremental chemical potentialand free energy difference DF = FðNÞ � F0ðNÞ as functions of thechain length N for chains partially confined to a small pore(Rpore = 1.5b) and exposed to different adsorption potentials.

The effect of the pore is evident from the behavior of the incre-mental chemical potential of the partially confined chains. For all

Fig. 6. Partially confined chains in two conformations: (A) strong adsorption (|U| > |UC|) and (B) weak adsorption (|U| < |UC|). Dashed line represents delineation between cisand trans states.

Fig. 7. (A) Incremental chemical potential, and (B) free energy difference as functions of chain length N for chains partially confined to a pore of radius 1.5b for severaladsorption potentials U = 0 to �1. Partition coefficient KF for chains partially confined to a pore of radius Rpore = 1.5b as (C) a function of adsorption potential U for several Nand (D) for N = 1–200 for several U.

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but the smallest chain lengths, the incremental chemical potentialis a constant in chain length and resembles the behavior of an end-tethered chain [24]. For strong potentials |U| > |UC|, whereUC = �0.725 is the CPA found previously for non-porous surface,the incremental chemical potential is more negative than that ofthe free chain, indicating that the cis sub-chain is mostly adsorbed(Fig. 7A). The incremental chemical potential for weak potentials |U| < |UC| however approaches the free chain value, indicating that

most cis beads are not adsorbed (see Fig. 7B). Noteworthy, atU–UC, the incremental chemical potential equals that of anunconfined chain, fulfilling the CPA condition Eq. (1).

The difference in free energy (Fig. 7B) as a function of the chainlength is a clear indicator of the existence of critical conditions forpartially confined chains. For adsorption potentials stronger thanUC, DF decreases with chain length, indicating that longer chainsare more likely to be adsorbed than shorter ones. For potentials

R.T. Cimino et al. / Journal of Colloid and Interface Science 481 (2016) 181–193 189

weaker than UC, DF increases with chain length, meaning shortchains are more likely to be adsorbed. At the critical conditions,the free energy difference becomes constant, and notably, this crit-ical condition UC = �0.725 is the same as for the externallyadsorbed chains.

In a manner similar to completely confined chains, we definethe partition coefficient for flower conformations as the exponentof the difference in free energy between a confined and unconfinedchain: KF = exp[�DF]. The partition coefficient for chains partiallyconfined to a pore of size Rpore = 1.5b is given in Fig. 7C and D asa function of adsorption potential U and chain length N, respec-tively. The partition coefficient of partially confined chains demon-strates several key features, which differentiate it from the case ofcomplete confinement. First, for potentials stronger than the criti-cal conditions |U| > |UC|, the partition coefficient is monotonicallyincreasing in chain length. (Fig. 7C, |U| > 0.725; 7D, yellow1, graylines). These potentials (|U| > |UC|) indicate enhanced adsorption oflonger molecules (LAC conditions). Second, there exists a character-istic potential (the critical point, UC = �0.725), where the partitioncoefficient switches from a decreasing function of N to an increasingone (Fig. 7C intersection point, 7D red line). At the critical potentialUC, the partition coefficient does not depend on N. This potential isidentical to the critical potential of chains adsorbed at a non-porous surface, which was described above. For potentials weakerthan that corresponding to the critical condition |U| < |UC|, KF

decreases with chain length, indicating enhanced partition of smallmolecules (SEC conditions) (Fig. 7C, |U| < 0.725, 7D, green, purplelines). The transition from SEC to LAC through the CPA is very sharp– adsorption is greatly enhanced (relative to the weak adsorptioncase) for all molecules beyond the critical conditions at |U| > |UC|.

The ability of partially confined chains to experience criticaladsorption conditions illustrates the importance of accountingfor the mechanisms which include external surface adsorption inthe case of porous substrates. Indeed, the mechanism of partialconfinement offers an explanation for experimental observationsof critical conditions. In the final section below, we explore theapplication of external surface adsorption and in particular, thecontribution of partial confinement to chain retention in polymerchromatography on porous substrates.

7. Chain retention on porous substrates

Assuming the thermodynamic equilibrium between unretainedand retained analyte (chains), the retention volume VR is defined asthe ratio of the total quantity Ntot of analyte in the column to thebulk concentration c0 of the unretained analyte in the interstitialvolume, VR = Ntot/c0. Accounting for all three possible mechanismsof retention, Ntot can be expressed as

Ntot ¼ c0 VI þ KHSext þ KPVP þ KFVO½ � ð5ÞEq. (5) states that the total amount of analyte in the column is

divided into four contributions from chains located: (1) in theinterstitial volume VI (unretained), (2) on the external surface ofthe stationary phase particles of area Sext, (3) completely withinthe pores of accessible volume VP, and (4) at the external surfaceof stationary phase particles in flower conformations, where VO isthe volume of surface pore openings. Normalizing Eq. (5) by thebulk concentration c0, one arrives at the equation of the retentionvolume

VR ¼ VI þ KHSext þ KPVP þ KFVO ð6Þ

1 For interpretation of color in Fig. 7, the reader is referred to the web version ofthis article.

In the spirit of the Gibbs adsorption theory [33] the sum of thelast three terms in the right-hand side of Eq. (5) represents theexcess adsorption Nex ¼ c0 KHSext þ KPVP þ KFVO½ � of analyte in thecolumn, as compared to the amount of analyte in the interstitialvolume at the bulk concentration, N0 = c0VI. Provided the excessadsorption is attributed to the retained analyte, the overall parti-tion coefficient K can be defined as the ratio of the excess adsorptionconcentration, cex = Nex/VS, per unit volume of the porous solid par-ticles of volume VS = Vcol – VI and the bulk concentration c0 in thesolvent, K = cex/c0 [20]. Note that the volume VS includes the vol-ume of pores inside the particles. As such, the retention expression(6) may be rewritten as VR = VI + KVS, with the overall partitioncoefficient

K ¼ KHSext þ KPeP þ KFeS ð7ÞHere, Sext = Sext/VS is the specific external surface area per unit

volume of solid phase, eP = VP/VS is the accessible porosity of thesubstrate, and eS is the surface porosity. The surface porosity is

defined as eS = nS vO Sext; nS is the number of pore openings per unitof external surface area and vO – the volume of one pore opening.These parameters are determined by the column packing geometryand particle structure. Eq. (7) represents the extension of the def-inition of the partition coefficient for a nonporous surface [20] toa porous substrate.

The chain model employed is described in Section 3 with themonomer size scaled to represent the Kuhn segment of polystyr-ene at good solvent conditions (molar mass 832 Da, b = 2 nm)[34]. The chain length in the simulations varies from N = 2 to 200that corresponds to the variation of molecular mass M of polystyr-ene fractions from 1664 to 166,400 Da. In Fig. 9, chain length N isconverted to molecular mass M via the Kuhn segment mass.

As a structural model of porous substrate, we use the surfaceperforated by ink-bottle pores with cylindrical pore openings andspherical pore bodies, Fig. 8. The diameter and length of the poreopening are set equal to the bead diameter b, so that vO = pb3/4.The size of pore bodies is set to Rpore = 1.5b, reflecting the nominalpore width of 6 nm of NovaPak� C18 substrate from Waters Tech.(Milford, MA), used in [11]. Assuming a dense packing of sphericalpores, the surface pore density is set to nS = 0.13/b2 = 3.2�1012/cm2.The fixed density nS corresponds to a surface porosity of eS = 0.1and an accessible volume porosity of eP = 0.09. This model impliesthat the retention volume (6) depends only on two columnparameters, the packing porosity eC = VI/Vcol, where Vcol is the totalcolumn volume, and substrate external surface area Sext:

VR ¼ eCVcol þ Sext½KH þ 1:8bKP þ 0:1bKF � ð8ÞEq. (8) is used directly to calculate the theoretical retention

plotted in Fig. 9B. To illustrate the influence of multiple retentionmechanisms – and in particular the role of partial confinement(flower conformations) – on chain adsorption and partitioning,we have applied our model to a well-characterized example of iso-cratic elution of polystyrene (PS) fractions in binary mixtures ofTHF and ACN on a chromatographic column packed with porousparticles [11].

The diagram in Fig. 9A shows the experimental correlationbetween the chain molecular weight and the retention volumefor a range of solvent compositions in terms of volume fraction Xof ACN recalculated from the data presented in [11]. As clearlyseen, the composition of XC = 0.52 corresponds to the criticalconditions of molecular weight independent elution with the CPAretention volume VCPA = 2.35 ml. Compositions X < XC correspondto the SEC order of elution with larger chains eluted first, mostprominent for the limiting case of pure THF (X = 0). At thismobile phase composition, (X = 0) the chain retention exhibits acharacteristic shape that points toward different states of polymer

Fig. 8. Model porous substrate:plane surface perforated by ink-bottle pores with spherical bodies and cylindrical openings. (A) Top-view and (B) side-view.

Fig. 9. (A) Molecular mass M elution profiles for the isocratic separation of polystyrene [11]. The solvent composition X of the experimental data is given in terms of thevolume fraction of acetonitrile, X = 0–0.6. Three chromatographic modes, SEC, LCCC (elution at CPA) and LAC are clearly distinguished by the shape of elution profiles. (B) Thesame experimental data (triangle symbols) alongside theoretical retention volumes (square symbols) calculated using the retention Eq. (8) in conjunction with the partitioncoefficients derived from simulations. In both graphs, solid vertical line denotes CPA retention volume, VCPA = 2.35 ml, dashed line – retention volume of largest chains,N = 200 in SEC mode (X = 0), fitted by adjusting the column porosity to eC = 0.53, corresponding to interstitial volume VI = 1.9 ml. (C) Correlation between effective adsorptionpotential U and solvent composition X in terms of dimensionless deviation U0 , X0 from the critical values, UC and XC, for a nonporous column.

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chains depending on molecular weight. Larger chains are excludedfrom the surface and it is less probable for them to enter thepores of the substrate than for smaller chains. Conversely, it ispossible for smaller chains to penetrate into the pores of thesubstrate and the retention of these chains may be influencedby the presence of partially confined (flower) conformations.Above the CPA (X > 0.52), the elution volume increases with the

molecular weight that implies the LAC regime, with chainsstrongly attracted to the substrate and the reverse sequence ofelution. Larger chains have a higher probability to interact withthe substrate than smaller ones, and therefore are morestrongly retained. When chains are strongly adsorbed, theenthalpic effects favor the chains to penetrate into the poresand therefore one may expect a larger contribution of flower

R.T. Cimino et al. / Journal of Colloid and Interface Science 481 (2016) 181–193 191

conformations for chains of all molecular weights than is presentin the SEC regime.

To describe the chain retention in this system, we made the fol-lowing assumptions. First, we associated the SEC conditionsobserved in pure THF (X = 0) with the negligible adsorption poten-tial, U = 0, and the CPA conditions at XC = 0.52 with the criticaladsorption potential U = UC = -0.725. Next, we determined thepacking porosity eC, and the external surface area, Sext, which arethe only two adjustable parameters in our model. The packingporosity eC = 0.53 (corresponding to VI = 1.9 cm3, for a column ofvolume Vcol = 3.58 cm3) was adjusted to fit the theoretical retentionvolume of chains of length N = 200 (VR = 1.6 ml, Fig. 9B, dashedline) for the case of zero adsorption potential (U = 0) to the exper-imental retention of the largest (166,400 Da) chains in pure THF(X = 0). Sext = 4 � 105 cm2 was chosen to best reproduce the criticalretention volume VCPA = 2.35 ml (Fig. 9A and B solid line). The valueof Sext is larger than the estimate made by assuming smooth, spher-ical particles of nominal radius RP = 2 lm11, which may imply a sig-nificant degree of surface roughness and particle sizeheterogeneity. With these parameters, we predicted the retentionbehavior at different molecular weights and selected adsorptionpotentials (U = 0, �0.7, �0.725, �0.875, and �1).

The experimental and theoretical retention are overlaid inFig. 9B. The square symbols correspond to the theoretical retentionof chains exposed to a range of adsorption potentials U = 0 to �1.The experimental and theoretical retention volumes at SEC (X = 0,U = 0) and CPA (X = 0.52, U = UC = �0.725) conditions are in goodagreement, since the selected points at these conditions were cho-sen for the fitting of model parameters. For potentials weaker thanUC, retention is a decreasing function of molecular weight. Forpotentials stronger than the critical value, the theoretical retentionmimics qualitatively the behavior of chain retention in the LAC

Fig. 10. Contributions of adsorption mechanisms in SEC, LCCC, and LAC regimes: (A) U = 0– complete confinement. Filled triangles –flower conformations. (D) Comparison of contwith the decrease of adsorption potential and increase of the chain length.

regime. Note that the shown experimental and theoretical depen-dences deviate because the chosen values of the adsorption poten-tial do not match the experimental compositions.

In order to match the effective adsorption potential U employedin simulations and the solvent composition X, we employ the cor-relation derived in our previous work [20] for chain separation on anon-porous column, Fig. 9C. This correlation relates the dimension-less deviations of the potential U0 = (U-UC)/UC and composition X0 =(X-XC)/XC, both reduced to their respective critical values. Usingthis correlation, the potentials employed in calculations U = (0,�0.7, �0.725, �0.875, and �1) correspond respectively to the sol-vent compositions X = (0, 0.38, 0.52, 0.537, 0.58). (See legend inFig. 9.) This correlation leads to logical results: theoretical andexperimental retention volumes are properly ordered and theoret-ical data for X = 0.38, 0.537, and 0.58 are located to the left ofexperimental data at X = 0.5, 0.54, and 0.6, respectively. Notewor-thy, the quantitative agreement found with the correlation derivedfor a non-porous column confirms the robustness of our model andtransferability of obtained parameters.

In Fig. 10, we present the calculated contributions of differentadsorption mechanisms in retention of polymer chains of differentlength at three different adsorption potentials U = 0, �0.725 and�1, which have been shown to correspond to the SEC, LCCC andLAC conditions from the experiments. We quantify the contribu-tion of each mechanism to theoretical retention using the compo-nents of Eq. (8) attributed to external surface, completeconfinement to pores, and flower conformations, respectively:

VR;ext ¼ KHSext;VR;pore ¼ KP1:8bSext;VR;fl ¼ KF0:1bSext ð9ÞThe contributions (9) in cm3 are plotted as functions of the

chain length. For the SEC regime (U = 0), the (exclusion) contribu-tion from the external surface VR,ext dominates (Fig. 10A); it is

, (B) U = UC, and (C) U = �1. Symbols for A–C: Stars–external adsorption. Open circlesribution of complete confinement across adsorption potentials – rapidly diminishes

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negative due to the excluded volume effects prominent in theabsence of adsorption potential. The complete confinement contri-bution is negligible and the flower conformations contribute onlyinto the retention of smaller chains. At the critical conditions(U = UC), the contributions of external adsorption and flower con-formations are comparable and constant for larger chains(Fig. 10B). The contribution from the complete adsorption is nota-ble only for the smallest chains and rapidly decreases with thechain length. Even at strong adsorption (U = �1), external adsorp-tion and flower conformations dominate for all but the smallestchains (Fig. 10C). Noteworthy, the contribution from completeadsorption progressively vanishes with the increase of the chainlength and the decrease of adsorption potential, see Fig. 10D.

This analysis indicates that adsorption on the external surfaceand partial confinement in pores represent the major factors deter-mining chain partition. The mechanism of partial confinement(flower conformations) helps one explain the presence of criticalconditions observed in polymer chromatography with porous sub-strates and resolve the controversy between the experimentalobservations and prior molecular simulations [7–8] which focusedentirely on the behavior of chains completely confined in poresneglecting the external surface effects.

8. Conclusion

Using a simple excluded volume chain model and MC simula-tions of thermodynamic equilibrium between adsorbed and freechains, we examine mechanisms of chain adsorption on a modelporous substrate searching for conditions of the critical point ofadsorption (CPA), at which the partition coefficient in the processof chromatographic separation is chain length independent. Wedistinguish three adsorption mechanisms depending on the chainlocation: on external surface, completely confined in pores, andalso partially confined in pores in so-called ‘‘flower” conforma-tions. The chain adsorption is treated in terms of the Gibbs excessadsorption theory [33] developed earlier for non-porous surfaces[20]. The thermodynamic condition of the CPA is defined fromthe equality of the incremental chemical potentials of adsorbedand free chains [24]. The free energies of different conformationsof adsorbed chains are calculated by the incremental gauge cellMC method [25] that allows one to determine the partition coeffi-cient as a function of the adsorption potential, pore size, and chainlength. We confirm the existence of the CPA for the chain lengthindependent separation on porous substrates, which is explainedby the dominant contributions of the chain adsorption at the exter-nal surface, in particular in flower conformations. The CPA, knownin chromatography as the LCCC regime, separates the regime ofweak adsorption with SEC sequence of elution, with longer chainsspending shorter time in the column, and the LAC regime of strongadsorption with the reverse sequence of elution. We show that asthe magnitude of adsorption potential becomes stronger andapproaches the CPA, the chains completely confined in pores areexpelled to the external surface forming flower conformations.This mechanism was neglected in previous studies of excluded vol-ume chains in pores, which questioned the very existence of theCPA for porous substrates. Moreover, we show that the critical con-ditions for porous and nonporous substrates are identical anddepend only on the surface chemistry.

Thus, we suggest that the experimentally observed criticaladsorption on porous substrates may be explained by the presenceof external adsorption and partially confined or ‘flower conforma-tion’ chains, and it was shown that flower conformations exhibitthe same critical point as externally adsorbed chains.

The theoretical results are confirmed by comparison withexperimental data on chromatographic separation of a series of

linear polystyrenes with variable solvent composition [11]. Theoverall partition coefficient, which included contributions fromall three adsorption mechanisms, was calculated for chains adsorb-ing to a model porous substrate perforated by ink-bottled pores.Although this model does not imitate the pore structure of realchromatographic substrates, it captures the major physicalmechanisms in a coherent fashion. The advantage of this modelis a minimal number of adjustable parameters, the column packingporosity and packing external area, which were determined to getthe best fit to the experimental data on the retention volumes atthe non-adsorption surface (SEC regime) and at CPA. The presenceof molecular weight independent elution at CPA and SEC-, LAC-likebehavior was confirmed by varying the adsorption potential. Theretention volume contributions of each adsorption mode werecalculated and compared at characteristic adsorption potentials,corresponding to SEC, LCCC, and LAC conditions. It was found thatexternal adsorption and flower conformations dominate over themechanism of complete confinement, providing an explanationfor the disagreement between the results of previous simulationsand experimental observations. Finally, using a correlationbetween the effective adsorption potential in our model and thesolvent compositions previously determined for separations onnonporous substrate [20], the magnitude of the effectiveadsorption potential on the porous substrate model was relatedto the solvent composition reported in [11] that led to quantitativeagreement with the experimental observations.

In conclusion, we developed a thermodynamic framework formodeling chain adsorption on porous substrates that is based onthe Gibbs excess adsorption theory and the establishment of theequilibrium conditions from the equality of excess free energiesof the adsorbed and free chains. The critical conditions of chainlength independent separation are defined by the additional condi-tion of equality of the incremental chemical potentials of theadsorbed and free chains. Using a simple yet instructive model ofporous substrate, we show the existence of the CPA for excludedvolume chains and describe the SEC, LCCC, and LAC regimesobserved for separation of linear polystyrenes on non-porous andporous chromatographic columns in a unified fashion. A quantita-tive agreement was achieved with transferring the adsorptionpotential – solvent composition correlation determined for thenon-porous column to the porous column. This confirms therobustness of the proposed methodology and its potential applica-bility, with proper extensions to more complex systems includingblock-copolymers, star polymers, end-functionalized chains, aswell as to the challenging problem of separation of polymer graftednanoparticles.

Acknowledgements

This work was supported by the NSF GOALI grants No. 1064170‘‘Multiscale Modeling of Adsorption Equilibrium and Dynamics inPolymer Chromatography” and No. 1510993 ‘‘Theoretical Founda-tions of Interaction Nanoparticle Chromatography”.

Appendix A. Supplementary material

Supplementary data associated with this article can be found, inthe online version, at http://dx.doi.org/10.1016/j.jcis.2016.07.019.

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