Journal of Colloid and Interface Science 474 (2016) 25–33

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Journal of Colloid and Interface Science

journal homepage: www.elsevier .com/locate / jc is

Critical conditions of polymer adsorption and chromatographyon non-porous substrates

http://dx.doi.org/10.1016/j.jcis.2016.04.0020021-9797/� 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 14 January 2016Revised 30 March 2016Accepted 1 April 2016Available online 2 April 2016

Keywords:Critical conditionsPolymer chromatographyAdsorptionMonte Carlo simulation

a b s t r a c t

We present a novel thermodynamic theory and Monte Carlo simulation model for adsorption of macro-molecules to solid surfaces that is applied for calculating the chain partition during separation on chro-matographic columns packed with non-porous particles. We show that similarly to polymer separationon porous substrates, it is possible to attain three chromatographic modes: size exclusion chromatogra-phy at very weak or no adsorption, liquid adsorption chromatography when adsorption effects prevail,and liquid chromatography at critical conditions that occurs at the critical point of adsorption. The mainattention is paid to the analysis of the critical conditions, at which the retention is chain length indepen-dent. The theoretical results are verified with specially designed experiments on isocratic separation oflinear polystyrenes on a column packed with non-porous particles at various solvent compositions.Without invoking any adjustable parameters related to the column and particle geometry, we describequantitatively the observed transition between the size exclusion and adsorption separation regimesupon the variation of solvent composition, with the intermediate mode occurring at a well-defined crit-ical point of adsorption. A relationship is established between the experimental solvent composition andthe effective adsorption potential used in model simulations.

� 2016 Elsevier Inc. All rights reserved.

1. Introduction

Polymer adsorption is the key phenomenon occurring innumerous practical applications such as colloidal stabilization,adhesion, painting, coating, and liquid chromatography, among

26 R.T. Cimino et al. / Journal of Colloid and Interface Science 474 (2016) 25–33

others. From the theoretical point of view, this phenomenon is aspecial case of interfacial phase transitions in polymer chains inter-acting with heterogeneous systems [1]. Polymer adsorption is gov-erned by a competition between enthalpic attraction and entropicrepulsion. At weak adsorption energy and high temperature,entropy penalty is prohibitive and chains are effectively repelledfrom the surface. The free energy of adsorbed chains increases withthe chain length and the partition coefficient, which determinesthe concentration of adsorbed molecules, decreases. At strongeradsorption energy and low temperature, enthalpy gain exceedsentropy loss and chains are predominantly adsorbed. The freeenergy of adsorbed chains decreases with the chain length andthe partition coefficient increases. The transition from weak tostrong adsorption regimes upon variation of adsorption strengthor temperature is quite sharp. Following the seminal work ofDeGennes [2], it is treated as a critical phenomenon occurring atthe so-called critical point of adsorption (CPA).

The aforementioned regimes of adsorption are realized in threemodes of liquid chromatography of polymers: weak or ideally noadsorption– in size exclusion chromatography (SEC), strong adsorp-tion – in liquid adsorption chromatography (LAC), and an intermedi-ate regime corresponding to the CPA – in liquid chromatography atcritical conditions (LCCC) [3]. Critical conditions (i.e. CPA) are exper-imentally found for a large number of polymers, and LCCC hasbecome a very popular technique for polymer characterization,complimentary to SEC and LAC [4]. It is noteworthy that LCCC is usedas a first step (first dimension) in the majority of reported 2-dimentional chromatographic separations of copolymers and othercomplex polymers [3]. The key advantage of LCCC is that the parti-tion coefficient at the CPA is chain length-independent and the sep-aration occurs with respect to the chemical (composition, end-groups, microstructure, and topology) differences in polymerchains, rather than to their size or molecular weight.

As with any mode of polymer chromatography, LCCC is usuallyperformed on columns packedwith porous particles [3,4]. However,the presence of pores is themain factor in the shortcomings of LCCCsuch as low mass recovery, peak splitting and distortion, andreduced efficiency due to dynamic effects caused by a substantialincrease of chain equilibration time especially in a case of narrowpores [4,5]. In SEC and LACmodes, the presence of pores is presumedto be essential: in SEC separation occurs as a result of partition ofpolymer chains between the pores and interstitial volume outsidethe particles; in LAC the internal (pore) surface increases columnloading capacity. However, for LCCC the necessity of porous sub-strates is not obvious, assuming that the CPA exists also in the caseof non-porous substrate. Here, we demonstrate for the first timeboth experimentally and theoretically all three modes of polymerchromatography on non-porous substrates, including the existenceof the CPA. The ability to perform LCCC on non-porous columnsmayimprove efficiency and mass recovery of the separations withoutany of the shortcomings of the porous substrates.

The phenomenon of critical adsorption on planar non-poroussurfaces has been extensively studied in the literature by usingvarious theoretical and simulation methods (see reviews [2,6b,c]). DiMarzio and McCrackin [6a] performed some of the earliestMonte Carlo simulations of adsorbed polymer chains, and noteda transition from weak to strong adsorption at a specificadsorption potential. In the ensuing decades, among the mostnotable advances were the grand canonical formulation ofBirshtein [7] and the scaling formulation of Eisenriegler et al. [8].With the emergence of high speed computing, there was renewedinterest in studies of polymer adsorption. Off-lattice, real chains(i.e. chains with excluded volume effects) were studied extensivelyand compared with earlier scaling results for ideal (Gaussian)chains, with general agreement [9]. However, these studies weremainly concerned with the geometrical transformations of chains

at critical conditions and the respective scaling relationships anddid not focus directly on the adsorption thermodynamics thatdetermines the chromatographic separation. The authors inter-ested in chromatographic separation, e.g. [10] among the others,established the CPA from the condition of length independenceof the excess free energy F(N) of the tethered chains (bound tothe surface by one end) of N monomer units, which was calculateddirectly using various random walk or Monte Carlo simulationmodels. In our recent work [11], we suggested a thermodynamicdefinition of CPA based on the notion of the incremental chemicalpotential (ICP), which represents the difference of excess freeenergy of chains of size N and N + 1 monomer units, respectively[12]. Drawing on an example of real chains tethered to planar sur-faces, it was shown that the CPA condition may be derived from thecondition of equality of the incremental chemical potentials ofchains in the adsorbed and free (non-adsorbed) states [11]. Therespective calculations of the free energies of adsorbed chains wereperformed with the original incremental gauge cell Monte Carlosimulation technique [12]. However, these results cannot bedirectly applied to the calculation of partition coefficients, whichgovern polymer separation, since tethered chains do not representall possible conformations of adsorbed chains. Here, we extend thismethodology to the case of untethered chains allowing us to deter-mine the partition coefficient between adsorbed (retained) andfree (unretained) chains following a rigorous adsorption theory.The partition coefficient is controlled in simulations by an effectiveadsorption interaction potential U between the chain segmentsand the surface. We show that at a specific value of this potential,U = UCPA, the incremental chemical potential of the retained chainshappens to be equal to that of unretained chains, and this condi-tion corresponds to the chain length-independent separation atthe CPA observed in the chromatographic experiments. The calcu-lated partition coefficient is further used to predict the elution of aseries of linear polystyrenes upon chromatographic separation on acolumn packed with nonporous particles to match the respectiveexperiments. In the experiments, the partition coefficient is con-trolled by varying the solvent composition at constant tempera-ture, which corresponds to varying the model adsorptionpotential U. Without invoking any adjustable parameters relatedto the column structure, we are able to describe quantitativelythe observed transition from SEC to LAC regimes of separationupon the variation of solvent composition, with the intermediateLCCC mode occurring at a well-defined CPA. Therewith, we estab-lish a relationship between the experimental solvent compositionand the effective adsorption potential used in simulations.

The rest of the paper is structured as follows. In Section 2, wediscuss the link between the Gibbs adsorption theory and the def-initions of the retention volume and partition coefficient adoptedin the chromatographic literature. We suggest to define theretained analyte through the Gibbs excess adsorption quantifiedby the respective Henry constant and show the relevance of theHenry constant to the retention volume and partition coefficient.In Section 3, we establish the CPA condition as the equality ofthe incremental chemical potentials of retained and unretainedmacromolecules. We also discuss the incremental gauge cell MCsimulation for calculating the chain free energy and Henry con-stant. The simulation model and details of the simulation tech-nique are given in Section 4. The results of calculations of theincremental chemical potentials and Henry coefficients for thechains of varying length at different adsorption potentials are pre-sented in Section 5. The experimental data on separation of linearpolystyrenes on a column packed with non-porous particles isgiven in Section 6, and the correlation between the experimentaland modeling results is presented in Section 7. Section 8 discussespossible hydrodynamic effects during polymer separation on non-porous columns. Brief conclusions are summarized in Section 9.

R.T. Cimino et al. / Journal of Colloid and Interface Science 474 (2016) 25–33 27

2. Retention volume, partition coefficient, and Henry constant

In liquid chromatography, two chromatographic phases areintroduced to differentiate between the two states of the solute(analyte): retained and unretained. The phase with retained ana-lyte is called the stationary phase and the phase with unretainedanalyte – mobile phase. The experimentally measured quantity isretention time tR (time required for a chromatographic peak toelute from the column following sample injection) or retention vol-ume VR ¼ _v0 � tR where _v0 is volumetric flow rate, so that VR is avolume of liquid passing with the analyte. The molecules of thesolvent (liquid) used to carry the analyte along the column areunretained by definition and occupy the entire mobile phase, whiletheir concentration in the stationary phase is assumed to be zero.As such, the volume of mobile phase VM is set equal to the reten-tion volume of the solvent, which with a good approximationequals to the liquid volume of the column VL. The retention ofthe analyte is considered relative to the carrying solvent, so thatin the case of retained analyte, VR > VL. In order to eliminate theeffect of column geometry, the so-called retention factor [13]k0 = VR/VL – 1 is introduced as a parameter characterizing the reten-tion of the analyte on a column packed with particles with specificsurface chemistry at selected chromatographic conditions.

In liquid chromatography, retention is usually considered as aresult of distribution (partition) of the analyte between its retainedand unretained states [13], and the retention volume is related tothe volumes of mobile and stationary phases, VM and VST,

VR ¼ VM þ K VST ð1ÞThis equation assumes thermodynamic equilibrium between

the mobile and stationary phases, respectively. The partition (dis-tribution) coefficient, K, is defined as the ratio of equilibrium con-centrations of the analyte in the retained (stationary) andunretained (mobile) phases. Note that Eq. (1) is applicable to vari-ous mechanisms of interaction between the analyte and the sta-tionary state, including adsorption and the chromatographicpartition [13]. In case of polymer chromatography, adsorptionmechanism is more common due to a significant size of the poly-mer analyte compared to the size of the bonded phase, and thismechanism of retention will be assumed in the rest of the paper.

The use of Eq. (1) requires the definition of the volume of thestationary phase, which is a subject of the long-lasting discussionsin the chromatographic literature [14,15] As related to liquid chro-matography of polymers, which, as liquid chromatography of anyanalytes, is commonly performed on porous substrates, the sizeof macromolecules could be comparable with the pore dimensions.For this reason, the volume of stationary phase is typically associ-ated with the volume of pores, and the volume of mobile phase –with the interstitial volume outside the porous substrate [3,4,16].In this case, the partition coefficient Kpore describes the equilibriumdistribution between the free macromolecules in the bulk solutionwithin the interstitial volume and the macromolecules confinedwithin the pores. This approach was first introduced by Casassa[17] to analyze the elution in SEC, where the polymer partitionbetween the pores and the interstitial volume is due to stericinteraction inside pores, and then extended to the adsorbingmacromolecules in LAC and LCCC [3,4,13c,18]. The partition coeffi-cient between the pores and the interstitial volume depends on thestrength of adsorption interaction and sizes of the analyte andpores. For the non-adsorbing solvent, Kpore = 1, for non-adsorbingmacromolecules in SEC regime Kpore < 1, for strongly adsorbingmacromolecules in LAC regime, Kpore > 1. Therewith, the criticalcondition of adsorption was assumed to correspond to Kpore,CPA = 1,so that retention time of polymer analytes in LCCC does notdepend on their molar mass and equals to that of solvent [3]. Such

definition of LCCC was theoretically justified for ideal chains[16] and it was experimentally verified in various applications[3,4].

However, the representation of the stationary phase as com-prised of the pore volume, accepted in the liquid chromatographyof polymers, ignores the possibility of partitioning within the inter-stitial volume due to interaction of the analyte with the externalsurface of the substrate. This deficiency becomes especially obvi-ous in case of the columns packed with non-porous particles con-sidered in this work. To remedy this situation, we will use thefundamental concepts of adsorption theory [19] to describe parti-tioning of polymer analytes between the bulk solution and theadsorbent surface.

We present the retention volume through the Henry constantKH defined according to the Gibbs adsorption theory as the ratioof the excess adsorption of the analyte per unit surface area ofpacked particles to its bulk concentration c0 in the solvent. A sim-ilar thermodynamic approach was used earlier for the theoreticaldescription of the chromatographic separation of low-molecularweight analytes, for example by Riedo and Kovats [20], Kazakevich[13b,14a,18], and Yun et al. [21], yet it has not been applied sys-tematically to polymer adsorption in liquid chromatography.

The excess adsorption is defined as the difference between theequilibrium amount of analyte in the adsorption system and theamount of analyte in the system of comparison. The latter repre-sents the homogeneous analyte solution of concentration c0 inthe volume equaled to the volume of the mobile phase, i.e. the vol-ume of liquid VL. As such, the total amount of analyte Ntot in thecolumn is presented as the amount of analyte in the bulk solutionof volume VL plus the excess adsorption that is proportional to thesurface area S of the solid phase, Ntot ¼ c0VL þ KHc0S. Assuming thatthe amount of the retained analyte equals the excess adsorption,the retention volume can be presented as

VR ¼ VL þ KH S ð2ÞThis thermodynamic approach naturally relates the experimen-

tally measurable retention volume with the Henry constant andthe geometrical characteristics of the column. For the packing ofnon-porous spherical particles of effective radius RP, the adsorbentsurface area per unit volume of liquid is S/VL = (3/RP) ð1� �Þ=�.Here, � ¼ VL=Vcol is the column porosity, defined as the ratio ofthe liquid volume VL to the total column volume, Vcol. Hence, Eq.(2) can be re-written in terms of the capacity factor k0 as

VR ¼ ð1þ k0ÞVL; k0 ¼ KHð3=RPÞð1� �Þ=� ð3ÞAlternatively, one can use the partition coefficient K defined in

the spirit of adsorption practice [19] through the excess adsorptionexpressed per unit volume of the solid phase VS (adsorbent) ratherthan per its unit area. In this case, Eq. (2) can be re-written as

VR ¼ VL þ KH S ¼ VL þ KVS; where K ¼ KHðS=VSÞ¼ KHð3=RPÞ ð30 ÞNote that the above equation is equivalent to the traditional Eq.

(1), if the volume of solid phase VS is formally considered as thevolume of a stationary phase. The volume of solid phase VS doesnot correspond to the definition of the stationary phase tradition-ally used in the chromatographic literature as a volume in whichthe retained analyte is located. The advantage of Eq. (30) with VS

playing a role of the stationary phase volume is that this volume,VS = Vcol – VL, is clearly defined and can be used in practice for pre-dictions of the retention volume. In the following discussion, wewill use the notion of the partition coefficient K implied by Eq.(30), to describe the thermodynamic equilibrium between theretained and unretained analyte.

28 R.T. Cimino et al. / Journal of Colloid and Interface Science 474 (2016) 25–33

The thermodynamic definition of excess adsorption naturallyincludes the situation of analyte repulsion from the adsorbent sur-face, e.g. by entropy-driven steric interaction responsible for size-exclusion effects in SEC. In the case of repulsion, the analyte isexcluded from the mobile phase. This effect takes place even inthe case of non-porous columns, but only in the interstitial volumebetween the particles, which usually is ignored in conventionalmodels of SEC [13a]. The excess adsorption of non-adsorbing orweakly-adsorbing macromolecules and respective Henry constantare negative, so that the retention volume of the analyte issmaller than that of the solvent. In this case, the analyte excessadsorption is negative and, respectively, k0 and K < 0.

It should be emphasized that utilization of Henry constant KH inEqs. (3) and (30) to describe the adsorptionmechanism of chromato-graphic retention is more practical as compared to the traditionaldescription of stationary phase in Eq. (1), employed in liquid chro-matography [14b]. It does not require the introduction ofthe adsorption layer or any stationary phase volume and uses theeasily measurable total liquid volume of the column VL as thereference for comparison. Below, we show that by using the Henryconstant as the thermodynamic parameter describing the partitionof macromolecules through Eqs. (3) and (30), it is possible todescribe in a unified fashion the chromatographic separation onnon-porous substrates in both LAC and SEC regimes as well as inLCCC mode.

Fig. 1. Schematics of different conformations of chains near the surface used forMonte Carlo calculation of the chain free energies. The polymer molecules aremodeled as freely jointed chains of beads, representing Kuhn segments of size b.The dark strip represents a square-well adsorption potential of width b. To accountfor the entropic restrictions of allowed conformations, the chains are distinguishedby the position z of the end segment (gray beads). Adsorbed conformations include(A) the chains end-anchored at the surface (tethered chains) and (B and C) thechains anchored beyond the adsorption well with at least one segment locatedinside the well. The free chain reference (D) is anchored at sufficient distance z > Nto exclude possible interactions with the surface.

3. Critical conditions of polymer adsorption and Henry constant

In this paper, we employ the thermodynamic definition of thecritical conditions of polymer adsorption, which is directly relatedto the chromatographic measurements: at CPA, the partition coef-ficient K between the retained and unretained analyte is chainlength (i.e. molecular weight) independent provided that the chainlength N is not too short, i.e. beyond a certain small number [22] ofchain segments N⁄

at CPA ! dK=dN ¼ 0 for N > N� ð4ÞThe molecular weight independence of the partition coefficient

for chains N > N⁄ implies that all such chains have the same prob-ability to be in the mobile or stationary phases and thus cannot beseparated based solely on their molecular weight. This definitionimplies that at CPA, the partition coefficients for chains of differentlength must converge to the same constant value K = KCPA. Note-worthy, the magnitude of KCPA is not pre-defined. Let us remindthat in our approach, the amount of retained analyte is definedthrough the Gibbs excess adsorption and the partition coefficientK is proportional to the Henry constant KH via Eq. (30). Qualita-tively, KCPA is expected to be around 0, since it separates the SECand LAC regimes characterized, respectively, by negative and pos-itive KH. This differs from the common assumption of Kpore,CPA = 1used in previous works, which considered the chain partitionbetween the interstitial and pore volumes [10,16].

Assuming a thermodynamic equilibrium during the chromato-graphic process, the partition coefficient for the chain of lengthN, K (N), is determined by the difference between the excess Helm-holtz free energy of the retained (adsorbed) and free macro-

molecules, respectively, DF = FadsðNÞ � F0ðNÞ, and is proportionalto the Boltzmann factor of DF: K � exp[�DF/kBT], were kB and Tare Boltzmann constant and temperature, respectively. From hereon, all energy terms are given in kBT units. To formulate a thermo-dynamically consistent criterion of the CPA, we invoke the defini-tion of the chain incremental chemical potential lincrðiÞ as thedifference between the excess free energy of chains of length N+ 1 and N: lincrðNÞ ¼ FðN þ 1Þ � FðNÞ ¼ dF=dN [12]. The chainexcess free energy is determined by the summation of the

incremental chemical potentials of its constituent segments

FðNÞ ¼ PN�1i¼0 lincrðiÞ. Note that lincrð0Þ represents the excess chem-

ical potential of a monomer (single segment). According to thechain increment ansatz [23], the incremental chemical potentialof free unconfined chains is independent of the chain length,l0

incrðiÞ ¼ l0incr ¼ const, provided that the chain is sufficiently long,

i > N⁄. Respectively, the free energy of unconfined chains is a linearfunction of the chain length, F0ðNÞ ¼ F0ðN�Þ þ ðN � N�Þl0

incr . TheCPA condition (4) implies that dDF/dN = 0 and respectively, theincremental chemical potential of adsorbed chains at the CPA mustbe chain length-independent and equal to the incremental chemi-cal potential of free chains,

lincr ¼ l0incr ð5Þ

As such, the thermodynamic condition of the equality of incre-mental chemical potentials is equivalent to the chromatographicdefinition of the CPA as the condition of chain length independentelution. Note that this CPA condition is different from the condition

of equality of the chain excess free energy, Fads ¼ F0, used in theearlier works [24].

In our previous work [11], we verified the CPA condition (5) foradsorbing chains tethered to a non-porous surface and showedthat this thermodynamic condition is consistent with classicalscaling relationships which exist at the CPA for the fraction ofadsorbed monomer segments, chain radii of gyration and chainfree energy [2,8]. Here, we consider a general case of chains whichare not tethered, yet interact with the surface through an adsorp-tion potential. Characteristic conformations of chains interactingwith the surface are shown in Fig. 1.

As defined above in Eq. (30), the partition coefficient is directlyproportional to the adsorption Henry constant, KH(N), that is theratio of the excess adsorption per unit surface area and the bulkconcentration of the chains with N segments. The Henry constantKH(N) at a plane surface can be presented as the integral alongthe z-direction perpendicular to the adsorbing surface of the ratioof the Boltzmann factors of the chains located at distance z fromthe surface and free chains [22].

Fig. 2. Incremental chemical potential as a function of N for different anchoringdistances z at the critical value of the adsorption potential UCPA = �0.725. All linesconverge to the dotted line lincr ¼ l0

incr as N becomes sufficiently large.

R.T. Cimino et al. / Journal of Colloid and Interface Science 474 (2016) 25–33 29

KHðNÞ ¼Z 1

z¼0exp � FðN; zÞ � F0ðNÞ

� �� �� 1

h idz ð6Þ

Here, FðN; zÞ is the excess free energy of chains anchored by theend segment at the distance z to the surface. For sufficiently largez, FðN; zÞ ! F0ðNÞ. The ratio of Boltzmann factors expð�FðN; zÞÞ= expð�F0ðNÞÞ represents the ratio of concentrations ofchains anchored at distance z and free chains. A detailed derivationof Eq. (6) is given in Supporting Information. In order to calculate theHenry constant, KHðNÞ, one has to compute the excess free energy,FðN; zÞ, as a function of the anchoring distance z. This is done belowusing the incremental chemical potential representation of the free

chain energy FðN; zÞ ¼ PN�1i¼0 lincrði; zÞ and employing the incremen-

tal gauge cell MCmethod [12]. A respective schematic of this proce-dure is shown in Fig. SI.1 in the Supporting Information.

4. Chain model and simulation methodology

To compute the chain free energy and analyze its dependenceon the chain length and adsorption potential, we employ themethodology developed in our previous works [11,12,25]. The sim-ulation set-up is the same as before [11], so that the resultsobtained there for tethered chains are used as references. The poly-mer molecules are modeled as freely jointed chains of beads con-nected by harmonic springs. In this simplistic model, the beadsrepresent Kuhn segments of length b, and there is no chain stiff-ness or limitations on bond angles or torsions. Exclusion volumeeffects are introduced via Lennard-Jones (LJ) interactions betweennon-bonded beads. Simulations were performed at dimensionlesstemperature T⁄ = 8 to ensure ‘‘good” solvent conditions in the bulk[26]. Adsorption at the solid surface is modeled by a square-wellinteraction between beads and the adsorbing surface of widthequal to the bead diameter b. The magnitude of this potential, U,is varied from 0 to 1 to capture the whole range adsorption, exclu-sion, and critical conditions.

Independent simulations were performed for chains anchoredat discrete distances z, measured in units b, by varying the chainlengths from N = 1 to N = 200 as illustrated in Figure SI.1. Freeenergy minimization is accomplished by the Metropolis methodand the incremental chemical potentials lincrði; zÞ are measuredvia the incremental gauge cell method [12]. Each simulation con-sisted of 400 million MC moves per chain (displacement, inser-tion/deletion and configurational bias regrowth [27]) toequilibrate chain conformations, followed by 500 million movesto sample the conformational space and to compute the incremen-tal chemical potential lincrði; zÞ. The incremental chemical poten-tials lincrði; zÞ of chains of length i from 1 to N are summed to

compute the chain excess free energy FðN; zÞ ¼ PN�1i¼0 lincrði; zÞ. The

free chain excess free energy F0ðNÞ is computed by anchoring thechain at z = N to exclude a possibility of its interaction with thesurface. Our calculations showed that FðN; zÞ converges to F0ðNÞat distances z exceeding the calculated radius of gyration RG(N)of the free chain of length N. The Henry constant KHðNÞ was thencomputed with Eq. (6).

5. Determination of the critical point of adsorption in MonteCarlo simulations

In Fig. 2, the incremental chemical potentials lincr of the chainsanchored at different distances z are plotted as a function of thechain length N at the adsorption potential U = UCPA = �0.725. Asshown before [11], this value corresponds to the CPA in the caseof tethered chains. At this value of the adsorption potential, thetwo key features of the incremental chemical potential areapparent. First, for all chains of N > 10 and any z, the incremental

chemical potential is practically constant, i.e. chain length-independent. Second, the incremental chemical potentialapproaches that of the free chain, fulfilling the CPA condition (5):lincr ¼ l0

incr . We also have demonstrated (not shown) that atz > RG, the incremental chemical potential is constant and equalsto l0

incr for any N, which is a logical result for the short-range squarewell adsorption potential utilized, i.e. as the anchoring distanceincreases, the number of configurations affected by the surfacedecreases. Notably, CPA in the general case of adsorbing yetuntethered chains is the same as for the tethered chains.

After summation of the incremental chemical potentials andcalculation of the free energies, the Henry constant was obtainedusing Eq. (6), and plotted as a function of adsorption potential(Fig. 3, left) for a series of characteristic chain lengths N, and as afunction of N for various adsorption potentials (Fig. 3, right).

The Henry constant is given naturally in the units of length. Forthe adsorption potentials weaker than the critical potential, theHenry constant is negative, indicating negative excess adsorptionand depletion from the surface (Fig. 3, left). There is a significantspread of the Henry constants at zero adsorption potential, indicat-ing increased selectivity of separation in SEC mode. Then, thespread of KH becomes narrower as the critical potential isapproached. At the CPA, the Henry constant is the same for allchain lengths (intersection point in Fig. 3, left) – a consequenceof its relation to chain excess free energy. The Henry constant atthe CPA is close to 0 yet is slightly positive KHðNÞCPA = 1.25 ± 0.45;its magnitude rapidly increases as the strength of the adsorptionpotential deviates from the CPA value. Above the critical potential,the Henry constant (and excess adsorption) is positive and chainsare separated in the reverse order to that of the weak potentials.

Likewise, the behavior of the Henry constant as a function of thechain length (Fig. 3, right) at various values of adsorption potentialU confirms the existence of critical conditions at UCPA = �0.725 aspredicted by the incremental chemical potential. Here, it is shownthat for U > UCPA, KHðNÞ is a decreasing function of N, implyingshorter chains are more likely to be adsorbed than longer ones.At the other end of the spectrum, when U < UCPA, KHðNÞ increaseswith N, indicating longer chains have a higher probability to beadsorbed. At U = UCPA, the Henry constant is constant for chainswith N > 10, which is consistent with the definition of critical con-ditions in Eq. (4).

In the chromatographic experiments, the retention factor k0 orthe partition coefficient K = k0VS/VL can be estimated from thedirectly measurable retention volume VR.. In order to directly com-pare the experiments to the theoretical predictions, it is necessaryto convert the thermodynamically derived Henry constant KH(N)into the partition coefficient K from Eq. (30). To accomplish this

Fig. 3. (Left) Henry constant KHðNÞ computed for a series of characteristic chain lengths (N = 20–200) as a function of the adsorption potential U. The intersection pointcorresponds to the CPA at U = UCPA = �0.725. (Right) Henry constant as a function of N for different values of U. At UCPA = �0.725, KHðNÞ is constant (LCCC mode), decreasingKHðNÞ values correspond to SEC mode at U > UCPA, increasing KHðNÞ values correspond to LAC mode at U < UCPA.

30 R.T. Cimino et al. / Journal of Colloid and Interface Science 474 (2016) 25–33

conversion, oneneeds toknow the followingparameters of the chro-matographic column: the column porosity �, total liquid volume VL

and effective particle radius RP. The comparison between the exper-imental data and the theoretical prediction is illustrated below.

6. Experimental confirmation of the existence of CPA inchromatographic separation on a non-porous column

A series of isocratic chromatographic experiments was per-formed using a column packed with non-porous particles. Separa-tion was completed using Waters Corporation (Milford, MA, USA)Alliance� 2695 chromatography system coupled with Waters2489 UV/Vis dual-wavelength absorbance on-line detector. Imtakt(Portland, OR) 4.6 mm ID � 150 mm Presto� FF-C18 column packedwith 2 lm diameter non-porous C18-bonded silica particles wasused for separation. The mobile phase was comprised of mixturesof two HPLC-grade solvents, tetrahydrofuran (THF) and acetonitrile(ACN), both obtained from J.T. Baker (Phillipsburg, NJ) and usedwithout further purification. The percentage of ACN ranged from0% to 56% by volume to cover the full range of elution modes asshown previously for a similar porous substrate [28]. Column tem-perature was kept constant at 35 �C and mobile phase flow rate _v0

was 0.25 ml/min. The series of narrow polydispersity linearpolystyrenes (PS) ranging in peak molar mass M from 1350 to186,000 g/mol was purchased from Waters. Retention time tRwas measured using the UV/Vis detector. Toluene was used tomeasure unretained liquid volume. Retention volume VR for eachindividual polystyrene at each solvent composition characterizedby the vol% ACN, X, was assessed using the standard formulaVR ¼ _v0 � tR and is presented in Fig. 4(left) as a function of Log M.The data clearly demonstrates the transition from the SEC modeof separation at lower ACN concentrations (X < 54%) to the LACmode at higher concentrations (X > 54%). The LCCC mode isobserved at X = 54%, when the retention volume is chain lengthindependent within the experimental error.

7. Correlation between the experimental and modeling results

For the column employed in experiments, RP (the average parti-cle radius) is �1 lm, as reported by the vendor. It is assumed thatthe retention volume measured by UV detector of a tracer mole-cule (toluene), VR = 1.088 ml, is an approximation of the total liquidvolume VL. Taking into account the total column volumeVcol = 2.49 ml, the column porosity is estimated as � ¼ VL

Vcol¼ 0:44.

Using these parameters in Eq. (30), we calculated the partition

coefficients for several chains of different lengths N (Fig. 5). Thesechain lengths correspond directly to the molecular weights of theexperimental polystyrene chains described in Fig. 4(left), usingthe scaling b = 2 nm, corresponding to the Kuhn segment lengthfor polystyrene in a thermodynamically good solvent [29].

The partition coefficient K in Fig. 5 clearly illustrates the transi-tion from SEC to LAC modes with the decrease of the adsorptionpotential. For weak potentials U > �0:725, the partition coefficientis negative indicating effective ‘repulsion’ from the stationaryphase. Here, the shorter chains have smaller partition coefficientsthan larger chains, constituting the size-exclusion mode. Con-versely, at stronger negative adsorption potentials U < �0:725, Kis positive and the order of partitioning with respect to chainlength is reversed. Finally, in the vicinity of UCPA ¼ �0:725, the K-values for all chains intersect, indicating the position of the criticaladsorption potential.

The calculated values of partition coefficients K were next usedto calculate directly the retention volume of the chains from Eq.(30) to compare the experimental retention volumes with thosepredicted by simulations. The results of these calculations are pre-sented in Fig. 4(right) alongside the experimental retention vol-umes in Fig. 4, (left). There is a striking quantitative similaritybetween the theoretical and experimental data that allows us toconclude that the effective adsorption potential U used in the sim-ulations depends on the solvent composition X with the respectiveCPA values of XCPA = 54% and UCPA = �0.725. Using the theoreticaland experimental values of the retention volumes, one can con-struct a correlation between U and X using an interpolationscheme. At Fig. 6, such a relationship is shown in the correspondingdimensional units. This correlation may then be used to predict theretention behavior at an arbitrary solvent composition, or for gra-dient elution separation in a similar system [28].

The reduced adsorption potential U0 = (U � UCPA)/UCPA andreduced solvent composition X0 = (X–XCPA)/XCPA presented inFig. 6 are normalized by the critical values UCPA ¼ �0:725 andXCPA = 0.54, respectively. In these coordinates, the size-exclusionorder of elution corresponds to negative values of U0 and X0,and the adsorption mode - to positive. At the critical conditions,U0 = X0 = 0. The transition from SEC to LAC modes occurs within avery narrow range of mobile phase compositions, which is alsofound to be true for the adsorption potential. The correlationshown in Fig. 6 may serve as a justification for various solventstrength models discussed in the HPLC literature [13a,14].

The experimental retention diagram shown in Fig. 4(left) illus-trates for the first time all three modes of polymer chromatographyon a non-porous column, including the existence of critical

Fig. 4. (Left, triangle symbols) experimental and (right, square symbols) theoretical retention volumes for a series of polystyrenes as a function of vol% ACN, X = 0–56%, in theeluent (left) and the adsorption potential, U = 0–0.875 in kBT units (right). CPA corresponds to X = 54% and U = �0.725, respectively.

Fig. 5. Partition coefficient K as a function of the adsorption potential U for chainsof length N between 2 and 200.

Fig. 6. Correlation between the reduced adsorption potential U0 = (U � UCPA)/UCPA

and reduced solvent composition X0 = (X � XCPA)/XCPA. The respective CPA values areXCPA = 54% and UCPA = �0.725.

R.T. Cimino et al. / Journal of Colloid and Interface Science 474 (2016) 25–33 31

conditions, i.e. molecular weight independent elution at eluentcomposition close to 54% ACN. Such a conclusion was possiblebecause of including into consideration both size-exclusion(entropy-driven) and adsorption (enthalpy-induced) interactions

occurring in the interstitial volume between particles, which arecommonly ignored in both size-exclusion and interaction chro-matography [3]. The SEC-type behavior of the chromatographicsystem is observed at lower concentration of ACN (below 54%)(Fig. 4, left). Thus, in pure THF (purple triangles), the differencein retention volumes for the large and lowmolecular weight chainsdiffers by >10%, and the largest chains elute well before an injectedflow marker (toluene, VR = 1.088 ml). As the column used is non-porous, only interstitial volume can contribute to such size-exclusion type of elution. For large chains, the transition fromSEC to LAC is very sharp, so that at 55% ACN and higher the reten-tion of large chains far exceeds those of the smallest molecularweight.

The correspondence of the experimental data to the results ofsimulation shown in Fig. 4 is remarkable: almost quantitativeagreement is found without invoking any adjustable parametersrelated to the column geometry. The thermodynamic model pro-posed for the partition coefficient involves known geometricalparameters of the experimental column and its packing particles.The effective adsorption potential then is mapped to the experi-mental solvent composition.

8. Effect of hydrodynamic separation

In the theoretical consideration above, we took into account onlythe thermodynamicmechanismof separation ofmacromolecules asit is related to the interplaybetween steric (entropic) andadsorption(enthalpic) interaction. Such an approach is well accepted for theanalysis of chromatographic separation of macromolecules insidepores, but never was used for description of the separation outsidepores. It is a general consensus [3,13a] to ignore both adsorptionand steric interactions in interstitial volume and to consider the lat-ter as a source of possible hydrodynamic (flow-induced) separationinhydrodynamic chromatography (HDC) [30]. Inhomogeneityof theflowfield as a potentialmechanismof separation by size in the chro-matographic column packed with porous particles has been intro-duced by DiMarzio and Gutman [31] at the same time as Cassasaoffered his thermodynamic, entropy-driven mechanism [32,17].The commonlyaccepted theory currently is that the thermodynamicmechanism of separation is dominant only inside pores, while theseparation (if any) outside pores occurs by flow-induced forces[13c]. This theory is based on the assumption that liquid insidepores represents so-called stagnant mobile phase, where masstransfer occurs predominately by molecular diffusion, while themass transfer in the interstitial volume is described by the laminarflow convection accompanied by diffusion. Usually, this laminar

32 R.T. Cimino et al. / Journal of Colloid and Interface Science 474 (2016) 25–33

flow is described by a parabolic (Poiseuille) flow velocity profileusing the analog of an open tube channel [30] (see also SupportingInformation). The separation occurs due to a distribution of theresidence time of chains of difference size as a function of their dis-tance from the wall. The parabolic flow velocity profile allows for asmall solute to be close to thewallswhere theflow is stagnant,whilethe largermolecules remainnearer the center of the tubewhereflowis fastest. Qualitatively, the result of such flow-induced separation isthe sameas of the steric effect in SEC: the larger analytes elute earlierthan the smaller ones. Quantitatively, the standard HDC model forpolymers in a packed bed [30] only superficially underestimatesthe spread of the retention times for the system considered in thiswork, as shown in the Supporting Information where the results ofseparation in the aforementioned experimental system (polystyre-nes in THF) are modeled using both the hydrodynamic and thermo-dynamic theories.

It is hypothesized that a discrepancy between the HDC modeland experimental results described here can be attributed to thenon-Poiseuille nature of the packed bed flow profile, which is char-acterized by a pronounced stagnant (boundary layer with zero orlow flow velocity) zone in the mobile phase near the surface ofthe particles, and a plug-flow velocity profile within the channelsbetween particles, outside the stagnant zone [33]. Which mecha-nism really controls the separation in the interstitial volumedepends on the size of the stagnant zone, which is affected by elu-ent viscosity, particle geometry, flow rate and diffusion coefficientof the analyte. If the stagnant zone is narrower than the size(radius of gyration RG) of a polymer chain, then the hydrodynamic(flow) effects prevail (Fig. 7, left). In the opposite case of a widestagnant zone (Fig. 7, right), all chains that interact with the sur-face are located within the stagnant zone and are effectivelyshielded from the ‘‘moving” part of the mobile phase flow. In thiscase, the separation is controlled by the thermodynamic effectswith competing steric and adsorption interactions, and the hydro-dynamic effects have less impact. As shown in Supporting Informa-tion, the estimation of the width of the stagnant layer in theaforementioned experimental system [34], leads to a value farexceeding 10 Kuhn segments (radius of gyration of the largestchain considered), which justifies the use of the thermodynamicapproach in simulation.

Fig. 7. Effect of stagnant zone in the interstitial volume on retention. (Left) Stagnantzone (width denoted by black arrow) is smaller than chain’s radius of gyration,hydrodynamic effects prevail. (Right) Stagnant zone is larger than radius ofgyration, thermodynamic effects govern the separation.

9. Conclusions

This work presents a novel thermodynamic method todescribe the macromolecule adsorption on nonporous surfacesand to examine the critical conditions of adsorption using theincremental gauge cell MC method. The developed theoreticalapproach is applied to chromatographic separation of polymeranalytes on non-porous column. The proposed approach impliesthe thermodynamic equilibrium between unretained and retainedanalyte treated in terms of the Gibbs adsorption theory. Theamount of retained analyte is defined through the Gibbs excessadsorption that is quantified by the respective Henry constantKH. Therewith, the partition coefficient K between retained andunretained analyte is introduced through the Henry constantKH. Such definition implies that (i) at weak or no adsorption char-acterized by negative KH (partition coefficient K(N) < 0 and pro-gressively decreases with N), the chains are effectively repelledfrom the surface and the chain elution proceeds in the SEC modewith larger chains eluted first; (ii) at strong adsorption character-ized by positive KH (K(N) > 0 and progressively increases with N),the chains are adsorbed at the surface and the chain elutionproceeds in the LAC mode with smaller chains eluted first; (iii)the critical point of adsorption (CPA), which separates SEC andLAC modes, is experimentally defined by the simultaneous elu-tion of chains regardless of their length, and corresponds tothe conditions of chain length independence of KH (K (N)= const = KCPA � 0).

From the thermodynamic standpoint, the CPA is defined by theequality of the incremental chemical potentials of adsorbed(retained) and free (unretained) chains. Using the incrementalgauge cell MC method, we calculated the free energies and therespective Henry constants and partition coefficients of theadsorbing chains of different lengths at given adsorption potential.Upon the increase of the adsorption potential, we traced the tran-sition from the SEC to LACmodes of separation with clearly definedintermediate mode corresponding to CPA. The theoretical resultswere compared with the specially designed chromatographicexperiments with linear polystyrenes separated on a columnpacked with nonporous particles. For the first time, all three modesof elution were demonstrated on a non-porous column, bothexperimentally and theoretically. We found that by choosing theappropriate conversion factors derived entirely from the columnparameters, it is possible to qualitatively reproduce the retentionbehavior of a series of linear polystyrenes across a broad range ofsolvent compositions, encompassing the SEC, LAC, and LCCC condi-tions. The comparison of the experimental and calculated retentionvolumes allowed establishing the relationship between the solventconcentration and the effective adsorption potential used insimulations.

The simulation model employed here is one of the mostsimplistic models of the polymer chains with excluded volume.However, it captures the main competing mechanisms of polymerseparation: entropic repulsion leading to size-exclusion mode ofseparation and enthalpic attraction due to adsorption. Theadvantage of this model is the absence of adjustable parametersexcept for the effective adsorption potential that allowed for adirect mapping of the theoretical results to the experimental data.This model can be further elaborated to take into account morecomplex chain topologies and microstructures, like specifics ofend groups, star polymers, block- and statistical copolymers, etc.

Acknowledgements

This work was supported by the NSF GOALI Grant Nos. 1064170and 1510993.

R.T. Cimino et al. / Journal of Colloid and Interface Science 474 (2016) 25–33 33

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.2016.04.002.

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