Size Exclusion Chromatography: A Teaching Aid for Physical Chemistry

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Size Exclusion Chromatography: A Teaching Aid for Physical Chemistry Howard G. Barth* Analytical Chemistry Consultants, Ltd., 207 Welwyn Road, Wilmington, Delaware 19803, United States ABSTRACT: All analytical techniques are governed by the principles of physical chemistry, and chromatography is certainly no exception. By examining the underlying concepts of size exclusion chromatography (SEC), students can gain greater appreciation of and insight into the science and utility of thermodynamics, as well as solution properties of macromolecules. In this paper we will show how the separation mechanism that governs SEC, an unusual form of chromatography, can be explained using the tenets of thermodynamics. This paper gives educators access to an HPLC technique usually not included in undergraduate chemistry curriculum. KEYWORDS: Upper-Division Undergraduate, Graduate Education/Research, Analytical Chemistry, Physical Chemistry, Inquiry-Based/Discovery Learning, Chromatography, HPLC, Thermodynamics

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process. The older name, “gel filtration chromatography”, is occasionally given when aqueous mobile phases are employed, but it is misleading since it implies a filtration process. This technique covers a molecular mass range of about 4 orders of magnitude, from small molecules (∼100 g/mol), to oligomers (∼500 to 103 g/mol), to high-molecular-weight macromolecules with an upper molecular mass limit of roughly 5 × 106, depending on polymer conformation; however, its main utility is for macromolecules, rather than small molecules. SEC employs the same instrumentation as HPLC, except that gradient elution is not used, and specialized software is required for molecular mass calibration. Details on SEC instrumentation, methodology, and data treatment can be found in refs 51−54. It should be emphasized that SEC is a relative, not absolute, method of measuring molecular size. Columns must therefore be calibrated with polymer standards of known size or mass. As an alternative to column calibration, detection systems that are specific for molecular mass or size can be used, such as online mass spectrometry, intrinsic viscosity, or light scattering.51−53,55,56 An example of an SEC separation is given in Figure 1, which shows a chromatogram of a homologous series of saturated fatty acids, ranging from C4 to C30. The C30 fatty acid (triacontanoic acid), the largest molecular-weight component, is the first oligomer to elute from the column (17.6 mL), followed by progressively lower molecular mass components, with C4 (butanoic acid) eluting last (24.4 mL). The small, negative dip occurring after C4 is residual water present in the sample, and it represents the total elution volume of the column, 26.4 mL. If we plot log molecular mass (or size) of a homologous series of polymer standards versus elution volume, we obtain a nearly linear relationship as shown in Figure 2. Macromolecules that are too large to access the pore volume are excluded from

INTRODUCTION Educators have long recognized the critical role of macromolecules in chemical sciences and the need to incorporate polymer topics into course curricula,1−15 although institutions have been surprisingly slow to address this need.1,2,5,7,8,12,13 In 2015, the ACS Committee on Professional Training (CPT) included the study of macromolecules in guidelines for certified bachelor degree chemistry programs.9 The rationale of broadening degree requirements to include macromolecules, as well as supramolecular, mesoscale, and nanoscale systems, was given by Ford,13 Pienta,11 and Wenzel and co-workers.10,12 Size exclusion chromatography (SEC), the premier method for characterizing macromolecules in terms of molecular mass and molecular mass distribution, gives educators another approach for introducing polymer science into undergraduate chemistry programs.8,9,12 A literature search of this Journal on SEC as a pedagogical approach for studying macromolecules has revealed a number of papers on the use of this technique in the undergraduate curriculum;14−50 however, there are none that address SEC theory from a thermodynamic perspective. The main emphasis of this paper is on thermodynamics, as exemplified by SEC, a topic not previously presented in pedagogical form. In addition, factors responsible for nonequilibrium behavior are examined from a pedagogical point of view, since students may encounter these situations in other types of chromatography and distribution processes. As a way of introducing macromolecules, these concepts also can be incorporated into undergraduate physical chemistry courses, and adapted into analytical chemistry modules as well. Furthermore, knowledge of this unusual chromatographic technique, found in many industrial laboratories, will help prepare students for chemistry-related careers.

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SEC OVERVIEW There are several terms used to describe size separation: The two common ones are size exclusion chromatography and gel permeation chromatography (GPC), but the former is preferable because it accurately describes the separation © XXXX American Chemical Society and Division of Chemical Education, Inc.

Received: March 8, 2018 Revised: May 23, 2018

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DOI: 10.1021/acs.jchemed.8b00171 J. Chem. Educ. XXXX, XXX, XXX−XXX

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i.e., M ≥ M0, will have the same elution volume V0, except for some unusual circumstances.57 As the hydrodynamic volume of the standard decreases with respect to the average pore size of the packing, the polymers occupy progressively more pore volume and elution volume increases. The maximum elution or permeation volume Vt is reached when the macromolecule is sufficiently small so that it can freely access the total pore volume of the packing; under these conditions, all solutes with M ≤ Mt elute at Vt. The total elution volume of a column Vt is equal to its interstitial volume (the volume between packing particles,V0) and the pore volume Vi, such that Vt = V0 + Vi. The entire SEC process takes place within a relatively small volume, the pore volume. The equation that governs SEC elution behavior is Vr = V0 + KSECVi

(1)

where Vr is the elution volume and KSEC is the corresponding distribution coefficient of the component being analyzed.51,58 This relationship tells us that Vr is simply the sum of two volumes: the interstitial volume V0 of the column plus a fraction of the pore volume, KSECVi. The SEC distribution coefficient KSEC of a given component58 is equal to the ratio of the concentration in the pore volume ci to the interstitial volume c0

Figure 1. SEC chromatogram of a mixture of C4−C30 (88−452 g/ mol) saturated fatty acids. SEC conditions: THF mobile phase; detector, refractometer; flow rate, 1 mL/min; TSKgel G2000H, x3 columns. From the Tosoh Bioscience database, https://www. separations.us.tosohbioscience.com. Adapted and used with permission from Tosoh Bioscience.

KSEC = c i /c0

(2)

The separation mechanism becomes apparent if we consider that the interface between the pore volume and the interstitial volume is analogous to a semipermeable membrane through which macromolecules can diffuse. Macromolecules too large to penetrate the pore volume, KSEC = 0, elute within the interstitial volume, i.e., Vr = V0, where Vr is the elution volume for a given sample component. For small molecules that can freely enter the pore volume, KSEC = 1, and these elute at Vt = Vi + V0. KSEC values of different molecular weight macromolecules, shown in Table 1, range from zero to unity. If SEC conditions are not fully optimized for a given separation, the distribution coefficient of samples may exceed unity, that is, KSEC > 1, implying that intermolecular interactions exist between solutes and packing. Under these conditions the separation is no longer based on molecular size and all information regarding molecular mass is lost. In this case, mobile phase composition must be adjusted or a different type of packing must be used. Although rather rare, KSEC < 0 for ultra-high-molecular-weight components (M > 106 g/mol), or cross-linked gel-like particles.57 Typically, this material is extraneous to the sample and can be removed by centrifugation or filtration. KSEC can be determined experimentally from elution volume measurements by rearranging eq 1

KSEC = (Vr − V0)/Vi

(3)

Since KSEC is a normalized parameter that is independent of flow rate, it can be used in place of Vr for data evaluation or to construct SEC calibration plots.

Figure 2. Typical SEC calibration showing a plot of log M against elution volume Vr of a series of polymer standards of known molecular weight. M0 is the exclusion MW limit occurring at the exclusion or void volume V0; Mt is the total permeation MW limit occurring at the total permeation volume Vt (Vt = V0 + Vi, where Vi is the pore volume). Adapted and used with permission from ref 51. Copyright 1999 Springer Verlag.

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THERMODYNAMICS OF SEC The link between thermodynamics and chromatography is Nernst’s distribution law,59,60which states that the activity, a, of a given solute distributed between two immiscible liquid phases, 1 and 2, will equal a constant,51 K, at defined temperature, pressure, and phase compositions

the packing and elute at V0, the interstitial or void volume of the column. The molecular weight M0 corresponding to V0 is the minimum molecular mass in which exclusion occurs. All high-molecular-weight polymers with values greater than M0,

K = a1/a 2 B

(4) DOI: 10.1021/acs.jchemed.8b00171 J. Chem. Educ. XXXX, XXX, XXX−XXX

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Table 1. Influence of Macromolecular Size on the SEC Distribution Coefficient Relative Solute Size Large macromolecules Small molecules Intermediate macromolecules Indeterminate Indeterminate

Solute Distribution Coefficient58 KSEC = ci/c0 (eq 2)

Solute Elution Volume58 Vr = Vo + KSECVi (eq 1)

Elution Description

KSEC = 0

Vr = Vo

Solutes elute in the interstitial volume.

KSEC = 1 0 < KSEC < 1

Vr = Vt = Vi + Vo Vr is within size separation region of the column.

Solutes elute in the total permeation volume. Solutes are size separated.

KSEC > 1

Vr > Vt, Vr exceeds size separation region of column.

KSEC < 0

Vr < V0, Vr is less than column pore volume, a seemingly impossible condition.

Solutes interact with packing, size information is lost. Ultrahigh MW samples or particulates within parabolic velocity profile.57

This equation means that if a given amount of solute is added to either one of the phases or to both of the phases, shaken, and allowed to come to equilibrium, the activity ratio of the solute will always adjust itself so that it reaches a constant value K that is specific for each solute, immiscible liquid phases, and temperature. Nernst’s distribution law can be applied to SEC by letting phases 1 and 2 be equal to the pore and interstitial volumes of an SEC column, respectively. Recalling that KSEC = ci/c0 from eq 2, and assuming that SEC is carried out close to equilibrium conditions, then K ≈ KSEC = c i /c0

Equation 8 tells us that all distribution processes, including chromatographic separations, are governed by both enthalpic and entropic contributions. SEC, however, is an entropy-based size separation, in which enthalpic interactions within pores are absent; that is to say, there are no measurable intermolecular interactions that occur between macromolecules and the packing. In essence, all molecules exhibit some degree of intermolecular interactions with surfaces; however, during an SEC experiment, chromatographic conditions (i.e., mobile phase composition, packing surface chemistry, and column temperature) are chosen such that the sorption energy of solute molecules is equal to or less than those of solvent molecules. As a result, there is no net solute interaction with the packing. According to eq 5, we can assume that the SEC process is close to equilibrium, such that K ≈ KSEC; thus, eq 8b becomes

(5)

Since KSEC is not necessarily at equilibrium, this parameter is referred to as a coef f icient, rather than a constant. The importance of conducting SEC analysis close to equilibrium conditions is to prevent peak distortion, peak broadening, and nonreproducible elution volumes, as discussed in the nonequilibrium section. The relationship between K and the change in Gibbs free energy ΔG° is ln K = − ΔG°/RT

KSEC = exp(ΔS°/R )

The two most important conclusions that we can draw from eq 9 are that the SEC distribution coefficient is independent of temperature and that it depends only on the conformational entropy of polymers when they diffuse from one phase to the other.

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(6)

CONFORMATIONAL ENTROPY Except for rigid polymers, most macromolecules are fairly flexible in solution: bonds between segments can rotate, bend, and twist, depending on chain stiffness. As a result, macromolecules can morph into different conformations, subject to their chemical composition and interaction with solvent molecules. Macromolecules that can diffuse into the pores of the packing experience a loss of conformational entropy. This decrease is exhibited by a reduction of KSEC, as described by eq 9. Within the pores of the packing, the number of conformational states of macromolecules are restricted by the walls of the pores; as a result, ΔS decreases and K < 1, as shown in Table 1. Very large macromolecules, with respect to the pore openings, undergo significant entropy loss; under these conditions, KSEC approaches zero. Small molecules experience a minimal loss of conformational entropy during diffusion into pores. Consequently, they exhibit a negligible entropic change, i.e., ΔS ≈ 0, and can access the entire pore volume. This behavior is summarized in Table 2 for hypothetically small, average, large, and excluded macromolecules.

where the superscript ° signifies standard state conditions, R is the gas constant (8.31 J K−1 mol−1), and T is absolute temperature. Standard state refers to components in their normal physical states at 298 K at 1 atm. For solutions, the standard state is 1 M, a hypothetical concentration used for calculating thermodynamic parameters. Change in Gibbs free energy can be expressed in terms of thermodynamic quantities, enthalpy (ΔH°) and entropy (ΔS°), using the well-known relationship ΔG° = ΔH ° − T ΔS°

(7)

In this context, enthalpy and entropy are the energetics that accompany the transfer of solute molecules from one phase to the other. The driving force is the solute concentration gradient that exists between phases. Recall that, at thermodynamic equilibrium, the changes in Gibbs free energy in the pore and interstitial volumes are equal. When solutes diffuse from one phase to the other, there is a corresponding free energy change, described by eq 7. By combining eq 6 with eq 7, we obtain the relationship between the distribution coefficient and thermodynamic parameters ln K = − ΔH °/RT + ΔS°/R

(9)

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(8a)

SEC NONEQUILIBRIUM

Concentration Effects

or K = exp( − ΔH °/RT )exp(ΔS°/R )

The influence of injection concentration on SEC elution behavior is complicated because of the many concentration-

(8b) C

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Macromolecular Crowding

Table 2. SEC Thermodynamic Behavior of Different Sizes of Solute Molecules As Exemplified with Equations 1 and 9 Relative Solute Sizea

Hypothetical Entropy Changeb

SEC Distribution Coefficient, KSEC

small average large very large

0 −0.7 −3 −5

1.00 0.50 0.05 0.00

There is an interesting solution property involving macromolecules that have relatively large hydrodynamic volumes. At a critical concentration, chain segments of macromolecules will begin to overlap with one another. When this occurs, the conformational entropy of overlapping macromolecules is reduced causing polymers to elute later than expected,see eq 9. This phenomenon, known as macromolecular crowding,51 will cause KSEC to be overestimated.

Elution Volumec Vr Vr Vr Vr

= = = =

Vt = Vi + V0 0.5Vi + V0 0.05Vi + V0 V0

a

Relative solute size with respect to average pore size (see Figure 1): small solute molecules elute within the total elution volume of the column; average-size solute molecules elute within the center of the SEC calibration; large solute molecules elute close to the void volume of the column; very large solute molecules elute within the void volume of the column. bArbitrary values with respect to ΔS°/R. cVr is the solute elution volume. Vt is the total permeation volume. Vi is the column interstitial volume. V0 is the column pore volume.

Viscous Fingering

dependent mechanisms that can occur simultaneously. In this section we will try to unpack these phenomena, with emphasis on those effects related to thermodynamic equilibrium.

Another unusual concentration effect, termed viscous fingering,51 is caused by the disruption of the parabolic profile of a flowing stream. Under laminar conditions, a flowing stream forms a parabolic-shaped profile, in which center velocity streamlines are moving faster than those closer to the surface of the packing or walls of the column. These complex patterns are easily disrupted if the relative viscosity of an injected solution is only marginally greater than that of the mobile phase, resulting in highly distorted peaks. This effect increases with molecular mass of the sample and injection concentration.

Infinite Dilution

Column Overloading

At thermodynamic equilibrium, the changes in Gibbs free energy content of the pore and interstitial volumes are equivalent and should remain so throughout the separation; if not, the separation will be thermodynamically unstable, and elution volume and peak shape may vary depending upon injection concentration. To attain equilibrium, solute−solvent intermolecular interactions must be comparable to solvent− solvent interactions. This can be accomplished using a solvent that has a chemical composition similar to that of the solute. (The terms solvent and mobile phase are used interchangeably, since it is common practice in SEC to use the mobile phase to prepare samples for injection.) Equilibrium between phases also requires minimum solute− solute intermolecular interactions, which is achieved by injecting dilute solutions. The limiting concentration is known as inf inite dilution. Infinite dilution is obtained by plotting the thermodynamic function under consideration versus solute concentration and extrapolating to zero concentration. When the slopes of these plots reach zero, infinite dilution is assumed. In SEC, however, this is not possible since there are many other processes that can affect KSEC, as discussed below. In general, dilute solutions imply that concentrations should be less than 100 mM for nonelectrolytes, and significantly lower for polyelectrolytes because of intense ionic interactions.59

Overloading can occur if the injected sample amount exceeds the pore-volume capacity of a column. When this happens, excess sample spills over into the interstitial volume, resulting in broadened and skewed peaks. This effect can be relieved by reducing the injected amount, or by using columns with increased diameters. The physical chemistry of column overloading can be expressed in the form of an adsorption isotherm, in which solute amounts are used. The onset of overloading can be determined experimentally utilizing a relationship that can be derived from eq 2, which can be rewritten as w1,i V KSEC = w1,i /Vi w1,o/Vo = × o w1,o Vi

(10)

where the numerator is the amount of component 1 (w1,i) in the pore volume Vi, and the denominator is the amount of component 1 (w1,o) in the interstitial volume V0. For a given column, V0/Vi, the phase ratio, can be expressed as a constant k KSEC = (w1,i /w1,o)k

(11)

Letting k′ = KSEC/k, we obtain k′ = w1,i /w1,o

(12)

At thermodynamic equilibrium, a plot of w1,i against w1,o will give a constant slope, k′. At the onset of overloading, the slope will become nonlinear, and the method will no longer be at equilibrium.

Theta Conditions

Additional complications arise when analyzing macromolecules. Because of the finite chain length of macromolecules in solution, intramolecular interactions can occur within the same chain, as well as intermolecular interactions among chains. To reduce this type of nonideal behavior, thermodynamic equilibrium is achieved by adjusting solvent composition and temperature such that polymer−polymer interactions are equivalent to polymer−solvent and solvent−solvent intermolecular interactions, known as theta conditions.60−63 At this stage, all interactions are balanced, and we arrive at thermodynamic equilibrium.64 Nevertheless, theta conditions can be unstable; a slight change of temperature or solvent composition can promote chain−chain interactions, and the polymer may come out of solution.

Local Nonequilibrium

During elution, solutes travel longitudinally within the interstitial volume, with lateral excursions into pores of the packing. Since it takes a finite amount of time for solutes to diffuse into pores, where they become stationary, and then back into the interstitial volume, where they resume their journey, the concentration profiles between the two phases are offset by a time delay. The time lag increases as the solute moves through the column. This process, called local nonequilibrium, causes the distribution coefficient to cycle about its true value, resulting in symmetrically broadened peaks.65,66 D

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To reduce local nonequilibrium, the linear velocity through the column must be decreased to give macromolecules sufficient time to diffuse into and out of pores. In addition, the path length taken by macromolecules when they diffuse from one phase to the other must also be reduced by employing small particle size packings. Another approach to relaxing local nonequilibrium is by increasing the diffusion coefficient of macromolecules. This is accomplished by increasing column temperature or reducing the viscosity of the mobile phase.51,66,67

It is instructive to note that the hydrodynamic volume of polymers at theta conditions usually expands slightly with temperature because of weak polymer−solvent interactions.61 In general, though, the effect of temperature on polymer size is not a concern since SEC is typically carried out under nontheta conditions. When developing an SEC method, column packing, mobile phase composition, and column temperature are carefully chosen so that all enthalpic interactions with the packing are absent. Even though SEC does not depend upon temperature, columns should be thermostated, nevertheless, to provide flowrate stability and to reduce detector noise and drift. Furthermore, operating at increased column temperature maintains solubility of marginally soluble polymers and, as discussed in the previous section, increases the diffusion rate of samples.

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EFFECTS OF TEMPERATURE ON SEC An interesting feature of eq 9 is that, unlike all other chromatographic techniques, KSEC is independent of column temperature, which implies that SEC can be operated at any temperature without significantly affecting elution volume. This property is demonstrated by injecting three different poly(ethylene glycol) mixtures (A, B, and C) with average molecular weights of 200, 300, and 500 at 25 and 60 °C, as shown in Figure 3. As indicated, elution volumes remain fairly

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CONCLUSIONS We have shown that SEC can be used as a pedagogical platform for demonstrating the principles of thermodynamics, specifically Nernst’s distribution law and the influence of nonequilibrium on chromatographic elution behavior. Concepts such as conformational entropy of macromolecules are introduced as they apply to SEC separation mechanism. The unusual solution properties of polymers are highlighted with respect to injection anomalies that lead to distorted peaks. Moreover, this article provides sufficient background material in support of SEC laboratory experiments. The minimum requirement for demonstrating the principles and applications of SEC is the availability of HPLC instrumentation with UV detection. An appropriate SEC column and a set of polymer standards of known molecular mass are also needed. Since SEC software may not be available, the molecular mass distribution and average molecular masses of polydisperse samples can be computed manually using procedures outlined in refs 51 and 52.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Figure 3. SEC chromatograms of three different poly(ethylene glycol) mixtures of increasing average molecular weights: (A) 200, (B) 300, and (C) 500. Column temperatures are 25 and 60 °C for the upper and lower chromatograms, respectively. SEC conditions: aqueous mobile phase, refractometer, 1.0 mL/min, and TSKgel G-Oligo-PW x2 columns. From the Tosoh Bioscience database, https://www. separations.us.tosohbioscience.com. Adapted and used with permission from Tosoh Bioscience.

ORCID

Howard G. Barth: 0000-0003-1542-8723 Notes

The author declares no competing financial interest.

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constant, even with a 35 °C increase, a drastic temperature difference, demonstrating the entropic behavior of SEC. Although not readily apparent in Figure 3, increased column temperature decreases peak broadening, a separate mechanism related to column efficiency, and more prevalent in HPLC of small molecules.67 Intuitively it would appear that the hydrodynamic volume of macromolecules would expand with increasing temperature. When a polymer is dissolved in a good solvent, however, its hydrodynamic volume is already extended in solution because of strong polymer−solvent interactions.61 It turns out that the molecular size of most polymers is only slightly influenced by temperature.68 E

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DOI: 10.1021/acs.jchemed.8b00171 J. Chem. Educ. XXXX, XXX, XXX−XXX