Pore size effect on the separation of polymers by interaction chromatography. A Monte Carlo study
Graphical abstract
Introduction
Liquid chromatography (LC) belongs to the most exploited analytical methods and finds applications in almost all chemical, pharmaceutical, biomedical and technological fields. In spite of the fact that its principles have been revealed and understood long time ago and, in the meantime, a suitable methodology has been developed and optimized, a vivid research aimed at improvement of separation efficiency, acceleration of analyses, development of better and miniaturized apparatuses, and at various targeted applications in biomedical and environmental areas still continues which is documented by a number of recently published papers [[1], [2], [3], [4], [5], [6], [7], [8], [9], [10]]. One of LC variants, called the size-exclusion chromatography (SEC) represents the benchmark method for the characterization of polymers and biopolymers. Recently, various advanced variants of SEC have been developed and used, e.g., the gradient chromatography, barrier chromatography [[11], [12], [13], [14], [15], [16], [17], [18]]. Besides experimental studies, a number of theoretical and computer-based papers devoted to the elucidation of separation principles at the molecular level and to the explanation of new effects and trends that appear in methods that combine different separation mechanisms have been recently published [[19], [20], [21], [22], [23], [24]].
The separation of components of mixtures by LC is based on their partitioning between the mobile and stationary phases which depends on their chemical nature, molar mass, etc. [[25], [26], [27], [28]]. When LC is applied to polymer samples, a porous stationary phase is commonly used, and polymers are either sterically excluded or retained in pores. Naturally, pore size is a crucial parameter that affects the efficiency of chromatographic separations. The partition coefficient, K, which characterizes the distribution of polymers between these two phases, is commonly expressed as a function of the experimentally measurable polymer elution volume or elution time. The partition coefficient, K, is related to the change in Gibbs free energy, ΔG, accompanying the transfer of a chain with a specific length from bulk to pore and can be expressed as [25].where cS and cM are the equilibrium concentrations of the solute (polymer component) in the pore and in bulk (mobile phase), respectively, kb is the Boltzmann constant, T is the temperature, and ΔH and ΔS are the corresponding changes in enthalpy and entropy. Eq (1) shows that the competition between ΔH and ΔS controls the separation of polymer chains. Because this competition and the resulting separation mechanism can be affected by the choice of stationary and mobile phases and by the experimental conditions, various advanced LC techniques have been invented. Among the LC family, the ideal size-exclusion chromatography (SEC) belongs to the category of relatively simple methods. The enthalpy contribution is zero, i.e., ΔH = 0, and the neat separation mechanism is driven only by the loss of conformational entropy, ΔS, of polymer chains, which are transferred from the bulk solvent to the porous medium [29]. The loss of conformational entropy in SEC results from the reduced number of chain conformations in a restricted pore volume. Using the appropriate theoretical approach, Casassa [30] has shown that the partition coefficient, K, decreases monotonically with the increase in coil-to-pore size ratio, λ, defined aswhere Rg,bulk is the radius of gyration of bulk polymer chains, and D is the characteristic size of the confinement, e.g., the pore diameter of a cylindrical (columnar) pore, or the width of a narrow slit. Thus, for a fixed polymer length, N, K monotonically increases as a function of D (from K = 0 for D → 0 to K = 1 for D → ∞), and this function has also been experimentally confirmed [25,31]. SEC is important method for the separation of disperse polymer samples and has become a benchmark technique for the characterization of molar mass distributions of polymers since 1950s [12,[31], [32], [33], [34], [35]].
The impact of pore size becomes more complicated in real chromatography when the nonzero enthalpy term in eq (1) affects the partitioning of chains and either partly counteracts the entropic contribution or fully dominates the separation process. The resulting effect can be exploited to optimize the separation of complex polymer samples characterized by several factors, including the molar mass distribution, the molecular topology and the chemical composition and functionality [26,27,36]. As mentioned above, a number of innovative approaches, such as interaction chromatography (IC) in porous media when∣ΔH/kbT∣ > ∣ΔS/kb∣ have appeared in recent decades [[37], [38], [39]]. The condition of equality of both terms, i.e., ∣ΔH/kbT∣ = ∣ΔS/kb∣defines the critical adsorption point (CAP), and LC performed under this particular condition, termed liquid chromatography at the critical condition (LCCC), has proven to be a suitable technique for the characterization of copolymers [[40], [41], [42], [43], [44]]. Furthermore, temperature gradient interaction chromatography (TGIC) coupled with SEC or with other chromatographic techniques [28] enables independent studies of molar mass distribution, of branching effects, and of the dispersity of individual blocks in block copolymers [40,41,45]. Despite major advances in recent years, the application potential of combined IC-SEC methods and the opportunities offered by the intricate interplay between enthalpy and entropy have not yet been fully exploited and therefore pore size effect must be investigated and understood in more detail.
Section snippets
Motivation of the simulation study
In the IC regime, in porous media, attractive interactions between polymers and the porous stationary phase exceed those at critical adsorption point (CAP). In practice, experimentalists adjust the composition of eluents and change the temperature of the column to acquire the IC state. The IC method shows the inverted sequence of elution characteristics from that obtained in SEC, i.e., the longest chains (with the highest molar mass) elute last in IC separations, whereas they elute first in the
Simulation methods
We used a twinbox model described in our earlier studies [19,22,57]. For the comfort of readers, we present its geometry in the Supporting Information. The large cubical volume and the square channel represent the bulk solution (mobile phase) and the porous medium (stationary phase), respectively. The dimensions of the bulk compartment are 200 (x-axis) × 100 (y-axis) × 100 (z-axis), and the dimensions of the square pore are 200 (x-axis) × D (y-axis) × D (z-axis), where D is the variable pore
Results and discussion
In order to assess and to elucidate the pore size effect observed by experimentalists, we performed a series of MC simulations of the partitioning of self-avoiding (SAW model) and interpenetrating (RW model) chains between pores and bulk for both good and poor solvents (particularly for the Θ-solvent). The curves of the partition coefficient, K, of the self-avoiding chains in a good solvent as a function of the square pore size, D, ranging from 8 to 60, at different strengths of attractive
Summary and conclusions
Pore size plays a key role in liquid chromatography of polymers because it controls both the entropy and enthalpy (except in ideal SEC) of chains in pores. Thus, we performed a series of MC simulations using a square pore model, and our simulation data confirm the experimental observations. For strong attractive polymer-wall interactions, i.e., in the IC regime, the partition coefficient, K, sharply increases with D in fairly narrow pores, peaks and then decreases in broad pores. Conversely, in
Acknowledgement
This work was funded by Czech Science Foundation Grant 17-04258J, and by Charles University Research Centre program No. UNCE/SCI/014. Computational resources were supplied by the Ministry of Education, Youth and Sports of the Czech Republic under the Projects CESNET (Project No. LM2015042) and CERIT-Scientific Cloud (Project No. LM2015085) provided within the program Projects of Large Research, Development and Innovations Infrastructures. The authors thank Prof. Taihyun Chang for the highly
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2022, Journal of Colloid and Interface ScienceCitation Excerpt :Note that the impact of internal energy is the same as that of enthalpy in experimental studies. The performed computer study confirms the experimental observation that the interaction chromatography (IC) which employs attractive interaction between polymeric units and porous media is a more robust method for studying dendrimers and other branched flexible polymers than the size-exclusion chromatography (SEC) [14,15,18,19,50,52,59,60]. However, great care has to be devoted to the selection of the column fillings with respect to their porosity and the interaction strength between dendrimer units and pore walls.
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2020, Journal of Colloid and Interface ScienceCitation Excerpt :This in turn provides more space in the central part of the pore for the entrance of more chains and the overall concentration inside the pores increases. However, in wide pores, the ratio of the volume close to walls, in which the chains are affected, to the inner volume decreases with D and therefore KD(cM) also decreases with D [25]. This explains the shifts in maxima positions towards lower tR.
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2019, PolymerCitation Excerpt :The equilibria and the separation processes were studied in eluents (mobile phases) that varied in the solvent quality described by interaction parameter εpp = 0, −0.1 and −0.2. More detailed discussion of the impact of various parameters on the polymer partitioning between the bulk solvent and the pores can be found in our earlier studies [38,42]. The equilibration of polymer chains between mobile and stationary phases was studied by modified configurational-bias lattice Monte Carlo (CBMC) simulations [43].