Technical and Practical Implications of Thermal Field-Flow Fractionation for the Separation of Polymers

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Literature Thesis

Technical and Practical Implications of Thermal Field-Flow Fractionation

for the Separation of Polymers

By

Bram van de Put

February 2020

Student number Responsible Teacher

12258156 Dr. A. Astefanei

Research Institute Daily Supervisor Van ’t Hoff Institute for Molecular Sciences I. K. Ventouri, MSc.

Research Group Second Reviewer

Analytical Chemistry Prof. Dr. R. A. H. Peters

DSM Geleen Industry supervisor

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Abstract

Thermal Field Flow Fractionation (ThFFF) is an interesting technique in the realm of polymer analysis. It provides a means for the gentle separation of very large and labile macromolecules with little to no degradation. More importantly, it possesses unique chemical composition and architectural selectivities, which are yet to be fully exploited.

After a lengthy hiatus in the development of the technique following the passing of Calvin Giddings, the inventor of ThFFF, the technique has enjoyed a steady increase in interest since the early 2000’s, owing to the similarly increasing interest in complex polymeric materials.

This thesis aims to provide an overview on both the theoretical background of the technique as well as its recent developments and present-day capabilities.

It was found that the concurrent dependence of retention in ThFFF on translational and thermal diffusion allows for the separation based on a large variety of polymer properties including: the molecular weight, monomeric composition, blockiness, and branching. Although this can aid in the characterization of these properties, their subtle interrelations make this application challenging. Moreover, the necessity for accurate models of the flow profile, which is distorted due to the thermal gradient, further limits the accessibility of the technique.

Recent trends in literature reflect the current interest from industry in applying the technique for the characterization of complex polymers based on blockiness and chain architecture. Notably, there has been limited developments in ThFFF retention models, where most studies limit their view to the relatively well studied polystyrenes, polyisoprenes and polyacrylates.

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Introduction

Polymers (Greek: polys = many, meros = parts), are large molecules produced by the covalent linking of many smaller molecules. Polymers serve as raw materials for a vast variety of commercial products including plastics, resins, fibers, and adhesives. With the ever-increasing demand for high performance materials, the versatility of polymeric materials is often called for.

The size, structure, and composition of polymer molecules define the physical and chemical properties of the end-product. Depending on the polymerization method and conditions, these molecular characteristics can be tailored to specific applications. To obtain a fundamental understanding of the structure-property relationship and aid in the optimization of production conditions, in depth characterization of polymer structures is required. However, the complete characterization of a polymeric sample is a challenging ordeal as many properties are inherently disperse [1].

Polymer characterization is generally achieved by a separation followed by a concentration dependent detection. The separation is ideally selective to a single property, such that a pure property distribution is obtained. Additionally, property-selective detectors may be used to obtain additional information. A multitude of different separation and detection techniques are often employed to achieve a comprehensive picture of a polymer’s different property distributions. Often no technique is available that is selective to a singular property of interest, instead the obtained information is a convolution of two or more property distributions. By combining information from different analysis techniques, interfering properties can be corrected, and pure distributions can be obtained. The technique that is most commonly used to separate polymers according to their molar mass distribution is Size Exclusion Chromatography (SEC). As polymers grow larger this becomes increasingly more challenging due to slalom effects, column clogging and adsorption [2]. The bias introduced by these shortcomings, limits the use of SEC to relatively small and stable macromolecules. Less destructive techniques are thus necessary for large and unstable polymers.

Thermal Field Flow Fractionation is a separation technique that was shown to be a promising asset in the characterization of polymers, though after the passing of its inventor, Calvin Giddings, the technique had been mostly abandoned in favor of SEC, in spite of its unique composition based selectivity.

In this literature review the ability of Thermal Field-Flow Fractionation to shed light on varying polymer property distributions will be explored. A brief overview of common polymeric properties will be provided, after which common detection techniques will be explained. Considerations towards the experimental design and parameter selection will be discussed. The influence of polymer and polymer-particle properties on their retention will be outlined by presenting the corresponding studies along with the leading theory behind their conclusions. Current computational and modelling approaches that support the technique will be discussed. Lastly, a list of recent papers of particular interest along with their main features is provided.

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Theoretical description of ThFFF

Field-Flow Fractionation (FFF) is a size-based separation technique used for macromolecules and (nano-)particles. FFF was developed by Giddings in 1966[3] and is now a collective term for several sub-techniques based on the same core principle:

A laminar Field-Flow is applied to a thin ribbon-like flow channel. Due to friction with the upper and lower walls

a Poiseuille (parabolic) flow-profile is created. The flow channel is typically formed by tightening a thin spacer with a geometric cut-out between two metal walls. An external field, perpendicular to the Field-Flow concentrates analyte molecules to a collection wall where the Field-Flow is lowest. Simultaneously, diffusion drives the analytes from the analyte enriched accumulation wall towards the middle of the flow channel where flow is higher. Separation is achieved by differences in the effect of the external field on the analytes and their diffusivity.[4][5]

While polymer diffusivity (D) is solely dependent on the molar mass (M), the external field may include a multitude of factors. The influence of D and the external field on analyte retention is collectively dubbed the retention factor (λ) defined as the reduced particle distribution along the thickness of the flow-channel. λ relates to the retention time according to the following equation:

𝑡0

𝑡𝑟

= 𝑅 = 6𝜆 (coth (1

2𝜆) − 2𝜆)

In which tr is the retention time of the compound and t0 is the dead volume (often empirically defined by an

unretained component). R is dubbed the retention factor. This model only holds under the assumption of the flow profile being parabolic, which is often overlooked, corrections to this model are discussed in another section. The nature of the external field defines the sub-type of FFF being employed, an overview of the most common FFF sub-techniques and the polymer properties that contribute to their retention is given in table 1.

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Table 1 Most common FFF subtechniques, adapted from: Asten 1995 Sub-technique Analyte properties in λ applications Sedimentation FFF Effective mass

(size, mass, density) Particles and colloids

Flow FFF Diffusivity Particles, colloids

and polymers Thermal FFF Soret Factor

(Diffusivity, Thermal Diffusivity) Particles, colloids and polymers Electrical FFF Diffusivity, Electrophoretic mobility Particles and colloids

Thermal Field-Flow Fractionation (ThFFF) exploits differences in thermal diffusion behavior, providing chemical structure-based selectivity. A quantitative description of the thermal diffusion coefficient (DT) does not yet exist,

however several theoretical approaches for estimating DT have been described with varying success.

Additionally, many studies have qualitatively assessed the influence of different polymer properties on DT

including; molecular weight[6], monomeric composition[7], branching[6]and solvent-solute interactions[8][9]. The external field in ThFFF is a thermal gradient between the two walls which is applied by heating one of the walls (usually the top wall) and cooling the other. Since the flow channel is very thin (70 – 300 μm) the heat transfer capability of the thermostat is crucial in order to keep a constant temperature difference (ΔT), typically the temperature difference is limited by the cooling capacity. The maximum value for ΔT can be calculated using the following equation [4].

(Δ𝑇 𝑤)𝑚𝑎𝑥

=𝑃𝑚𝑎𝑥 𝜅𝐴

Where w is the width of the flow channel, κ is the heat transfer coefficient of the solvent, A is the area of the channel cross section, and Pmax is the maximum system power which is limited by the heat transfer efficiency of

the coolant (typically in the range of 1-3 kW).

The retention factor in ThFFF includes the translational diffusion coefficient (D), thermal diffusion coefficient (DT)

and the applied temperature gradient (ΔT). The factor DT/D is often dubbed the Soret coefficient (ST).

𝜆 = 𝐷 𝐷𝑇Δ𝑇

= 1

𝑆𝑇Δ𝑇

As established early on by Schimpf and Giddings[6], DT seems to arise on the scale of a single monomer,

following the observation that molar mass and branching did not contribute to DT. The monomeric polymer

chemistry and solvent were found to strongly affect thermal diffusion. The thermal diffusion effect is also known to be proportional to the temperature, often expressed as the cold wall temperature (TC).

ThFFF is a versatile technique, allowing separation of polymers and particles too large for column-based techniques, either because of shear induced degradation, total exclusion (in case of SEC), or because they would simply clog packed columns. Furthermore, FFF based techniques can separate a very large range of molecular weights within a single measurement [10].

Like column-based polymer separations, ThFFF can be equipped with a broad variety of detectors, each with their own added perspective on the various polymer properties. Mostly the same detectors as in SEC are used.

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Polymer property distributions

The complete structure of a polymer chain determines its physicochemical properties, including melting behavior, crystallization behavior, tensile strength, reactivity, solubility, and solution state behavior. Inversely any measurement result is influenced by all aspects of a polymer structure [11]. In order to optimize production processes and separation methods, shedding light on the different property distributions that define a polymer’s structure is vital. A brief description of the main polymer property distributions is provided in this section.

Molecular Weight Distribution

The one distribution virtually all production polymers possess is a Molecular Weight Distribution (MWD) [12]. Straightforwardly, it is the number of molecules corresponding to each specific molar mass. There are several different expressions for the MWD mostly corresponding to a certain measurement technique though each expression provides a different perspective on the sample.

The simplest expression of the molar mass is the number average molar mass (Mn), the arithmetic mean of the

molar mass. Another is the weight average molar mass (Mw), where polymers of higher mass contribute more,

obtained from light scattering experiments. Yet another is the viscosity average molar mass (Mv) in which the molar

mass contribution depends on polymer-solvent interactions.

𝑀𝑛 =∑ 𝑛𝑖𝑀𝑖 ∑ 𝑛𝑖 , 𝑀𝑤 =∑ 𝑛𝑖𝑀𝑖 2 ∑ 𝑛𝑖𝑀𝑖 , 𝑀𝑣 = (∑ 𝑛𝑖𝑀𝑖 𝑎+1 ∑ 𝑛𝑖𝑀𝑖 ) 𝑎

From the combination of these molar mass values the relative width of the distribution can be expressed in different ways, often dubbed the polydispersity index or molar-mass dispersity (Ð) [13].

Ð =𝑀𝑤 𝑀𝑛

Chemical Composition Distributions

If a polymer is made out of two or more distinct monomers which can be assembled randomly it will have a chemical composition distribution (CCD)[14]. The CCD is often expressed as the mean monomer ratio along with the range of monomer ratios.

Chemical composition is also used as a colligative term for several other distributions including: the number and type of end-groups (functionality type distribution or FTD), the order of monomers (sequence distribution or SD), the length of segments of the same consecutive monomer (Block Length Distribution or BLD) and, the number of branches (Branching Distribution or BD).

Figure 2 Schematic overview of polymer property distributions. From: Uliyanchenko et al. 2012

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Detection

Differential Refractive Index

(differential) Refractive Index Detection (RID or dRI) measures the difference in refractive index between the pure solvent and the analyte containing effluent. RID is a bulk detector, in that it measures a colligative property of the separation effluent, namely the refractive index, which is an average of the refractive indices of each component in the flow cell. RID is the most widely used concentration sensitive detector in polymer separations as it is the only detector that provides direct concentration information, allowing the signal of other (feature selective) detectors to be corrected for concentration effects.

In order to obtain absolute concentration values the refractive index increment (dn/dc) needs to be known. dn/dc is known to be virtually independent of the polymer molar mass (if Mn > 5000) [15], allowing the accurate determination of concentration after a simple calibration experiment.

In case of copolymers the dn/dc is a linear addition of the response factors for each monomer of the form:

(𝑑𝑛 𝑑𝑐)𝐴−𝐵−⋯ = 𝑋𝐴( 𝑑𝑛 𝑑𝑐)𝐴 + 𝑋𝐵( 𝑑𝑛 𝑑𝑐)𝐵 + ⋯

Where XA and XB are the molar concentrations of monomer A and B respectively and dn/dcA and dn/dcB the

refractive index increment of the pure polymers.

Absorbance

UV(-VIS) absorbance spectroscopy (UV) is a technique with which the reduction in intensity of monochromatic light is measured. The wavelength of the light may correspond to an electronic transition of a chromophore in the sample causing it to be absorbed. Since absorbance requires resonance between the light and a chromophore, absorbance detection is more selective than RID.

Like RID, the response factor of absorbance detection depends on the kind and number of each different monomer, however many monomers do not contain chromophores, thus utilizing RID and absorbance detection simultaneously can aid in identification and quantification of different monomer types.

Differential Viscometry

Differential Viscometry (dVI) measures the viscosity of a solution compared with the pure solvent. A differential viscometer consists of four restriction capillaries, divided over two flow paths.[16] One of these flow paths contains a delay reservoir that contains the pure solvent. During analysis, the solvent in the reservoir is gradually displaced by the effluent of the separation, forcing the solvent through the following restriction. In the other flow path, the effluent itself is forced through the second capillary. The pressure difference between the latter capillaries (differential pressure or DP) of the two flow paths is measured. The pressure difference between the inlet and outlet is also measured (inlet pressure or IP). From the

Figure 3: Schematic representation of a differential viscometer, wavy lines are restriction capillaries, boxes are differential pressure meters, blue cylinder is delay reservoir (from: Agilent)

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9 measured values of IP and DP, the specific viscosity (ηsp) of the

effluent can be calculated using the follwing equation: 𝜂𝑠𝑝= 4𝐷𝑃/(𝐼𝑃 − 2𝐷𝑃)

If the concentration (c) is known (from eg. An RID detector), the specific viscosity at infinite dilution, otherwise known as the intrinsic viscosity ([η]) can be calculated according to:

[𝜂] = lim

𝑐→0(𝜂𝑠𝑝/𝑐)

The intrinsic viscosity is a property portraying the solvated state of the polymer. More specifically, it relates to the expansion of the polymer chain in the solvent, otherwise known as the hydrodynamic volume. Since the solution state is constant for a certain polymer-solvent system, the intrinsic viscosity can be used to translate viscometric data to molar mass values using the Mark-Houwink equation[17]:

[𝜂] = 𝐾 ∗ 𝑀𝛼

In which K and α are empirical constants, known as the Mark-Houwink constants. These are derived from the linear regression of log [η] against log M, for which the y intercept is log K and the slope is α.

Furthermore, viscometry may be used as a detection method when there is no difference between the refractive indices of a polymer solution and the pure solvent (dn/dc = 0) and the polymer does not possess any chromophores. [18]

Static Light Scattering

Light Scattering Detection measures the amount of light diffusely scattered from the eluate after irradiation with a narrow-bundled laser beam [19]. The scattering intensity is expressed as the Rayleigh ratio (Rθ) of which the

expression is complex, though if the polymer concentration is low and the size of the molecule is below 10 nm it can be simplified greatly to the following form:

𝑅𝜃 = 𝑀𝑤 𝐾 (

𝑑𝑛 𝑑𝑐)

2

𝑐

In which Mw is the weight average molecular weight, and K a constant defined as:

𝐾 =4𝜋

2_𝑛 0 2

𝜆₀4_𝑁

In which n0 the refractive index of the solvent, λ0 the

wavelength of the light, and N Avogadro’s number. These equations directly relate Rθ to the molar mass

of the polymer with values that can be deduced from a secondary Refractive Index detector.

Scattered light is typically measured at a right angle form the incident laser beam (90°), where the measured intensity from the laser itself is minimal. In

Figure 4: Schematic depiction of intrinsic viscosity calculation form refractive index (red) and viscometry (blue) data. (Source: Waters, 2016)

Figure 5: Schematic depiction of decreasing off-angle light scattering for particles of size nearing λ. (from: Agilent)

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10 this case the polymer is assumed to be a point source of scattering, of which scattering intensity is equal in all directions (isotropic scattering). As the size of the polymer approaches the wavelength of the laser (> 1/20 λ) things start to change, the scattered light off-angle from the incident beam will become gradually more out of phase and start interfering destructively with itself. In turn, the determined molecular weight from scattered light measured at the standard 90° angle will be greatly underestimated. This effect is known as dissymmetry. A logical solution to dissymmetry would be to measure at smaller angles to the incident beam, ideally at an angle of 0°, this is not possible though, due to the interfering incident beam and the strong scattering of particulate contaminants in this region. Multiangle Light Scattering Detection (MALS) employs an array of detectors at different angles in order to predict the scattering intensity at 0° by extrapolation using one of several methods, depending on the application [20].

The magnitude of dissymmetry can however be used to estimate the radius of gyration of a particle. The radius of gyration, like the hydrodynamic volume, is a measure of particle size expressed as the root mean squared distance of all atoms in a molecule from their collective center of mass [2]. Contrary to measuring the molecular weight, dissymmetry-based radius of gyration determination can only be performed when the particle size is larger than a twentieth of the light’s wavelength.

Dynamic Light Scattering

Like static light scattering, dynamic light scattering (DLS) measures the amount of scattered light of a solution. However, DLS specifically measures the fluctuations of the scattering intensity. The frequency of these fluctuations is related to the Brownian motion of the detected molecules which is related to the diffusion coefficient. Spectrum analysis can be used to directly relate the frequency of the scattering fluctuations to the diffusion coefficient without the necessity of prior knowledge of the type of polymer or its properties. This makes it an invaluable tool in FFF, as the diffusion factor D can straightforwardly be excluded if the influence of the external field is of interest. Lastly, the combination of DLS with MALS or dVI can be used to find the empricial factors that connect the diffusion coefficient with the hydrodynamic radius.

Figure 6: Schematic representation of a DLS measurement for large and small particles. The frequency of fluctuations is higher for smaller particles.

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MALDI-TOF Mass Spectrometry

Mass Spectrometry (MS) is a family of techniques that use electric and/or magnetic fields to manipulate charged molecules or molecular assemblies in the gas phase to separate them according to their molar mass [21]. A large variety of mass spectrometer designs exist, though only one of these is well applicable to the relatively large polymers ThFFF can separate, being Matrix Assisted Laser Desorption Ionization Time of Flight mass spectrometry (MALDI-TOF).

For the conduction of a MALDI-TOF experiment, the polymer solution of interest is mixed with a certain “matrix” compound and the resulting solution spotted on a stainless-steel (target-)plate. The solution is airdried leaving a spot of polymer encapsulated in a microcrystalline structure of the matrix compound. The target plate is then transferred to the mass spectrometer and brought under high vacuum. The spot is irradiated by a pulsed laser which is strongly absorbed by the matrix molecules, in turn the polymer molecules are desorbed from the plate into the gas phase where charge transfer occurs.

The exact mechanism by which desorption and ionization occurs in the MALDI process is unknown, though it is postulated that the high internal energy of the matrix after absorption of the laser light results in tiny explosions, effectively blowing matrix and analyte into a highly energetic “plume”. During the early stages of this plume, charge transfer between molecules would be possible due to the high density and kinetic energy of the individual molecules [22].

It is known that the efficiency of analyte desorption and charging depends highly on the compatibility of the analyte-matrix-solvent system used, to ensure homogenous distribution of the analyte in the crystalline structure of the matrix [23].

For synthetic polymers, a metal-salt is often added to form charged adduct ions. Effectiveness of the salt often depends, again, on the compatibility with the system and the gas phase interactions between the metal cation and the analyte. Polar, carbonyl containing polymers bind strongest with alkali-metal cations (sodium, potassium, lithium) whereas conjugated polymers bind more strongly with transition-metal ions (most notably silver and copper) [24].

The formed ions are separated according to their molar mass using Time of Flight mass spectrometry (TOF-MS) where the ions are accelerated to equal kinetic energy using a short pulse of a strong electric field, flinging them into a field free region (drift tube) [21]. The time the ions take to arrive at the detector on the other side of the field free region is proportional to the square root of the mass to charge ratio (mi/zi)1/2 [25]. Using a

straight-forward calibration TOF-MS can obtain fast, high resolution mass spectra.

MALDI-TOF is extremely well suited for analysis of high molecular weight compounds since it lacks transfer optics and ion lenses which diminish the mass range of most other types of mass spectrometers, simultaneously, the fact both ionization and mass analysis are operated in a pulsed manner makes MALDI-TOF an effective hyphenation [26].

MWD determination with MALDI-TOF is limited to samples of low dispersity (Ð < 1.2) because of the strong dependence of solubility, co-crystallization behavior, and ionization on the molar mass [27]. Fractionation is often required to lower the dispersity of a sample before measurement, after which each fraction may be analyzed, and the data recombined to obtain the entire molecular weight distribution. If high resolution spectra are obtained, MALDI-TOF can be used for the concurrent determination of composition, end-groups and molar mass.

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Experimental factors influencing ThFFF separations (efficiency, selectivity,

retention)

The efficiency of a ThFFF system is often expressed as the amount of dispersion over a unit channel length, otherwise known as the plate-height (H) [28]. H consists of different factors which contribute additively, often split into two general factors.

𝐻 =𝜎

2

𝐿 = 𝐻𝑃+ 𝐻𝐶

Of which HP is the peak broadening due to sample polydispersity, often being the goal of an analysis, this factor

should be maximized with regards to HC, which is the column contribution to band broadening. HC can further be

divided into several velocity dependent factors [29].

𝐻𝑐= 2𝐷1 𝑅〈𝜐〉+ 𝜒 𝑤2〈𝜐〉 𝐷2 + ∑ 𝐻𝑟

The first term expresses the longitudinal diffusion, the second term non-equilibrium effects and, the last term a collection of (relatively small) band broadening effects due to the finite injection volume, channel irregularities, and relaxation-based effects. These three factors will each be discussed further.

Channel design

The design of the thermal FFF channel is relatively simple and well established. The channel needs to be sufficiently broad such that the inevitable parabolic flow profile at the side walls of the channel does not contribute notably to band broadening. Flow channels in ThFFF are long compared to those of other sub-techniques in the FFF family, owing to the similarly high λ and thus low retention. The most influential and optimizable parameter is the channel width, defined by (but not equal to) the thickness of the used spacer, its significance is most notable in the different causes of band broadening which will be discussed in the following subsections.

Relaxation effects

After injection of a small volume of sample, the individual molecules will traverse the flow-channel towards their steady state distribution close to the cold wall. During this traversal the molecules will experience different flow velocities as they pass the parabolic flow profile, and the injection peak will thus be distorted [10]. Other relaxation effects are caused by convection currents and inhomogeneities of the cold wall surface.

Introducing a short stop-flow just after injection allows the steady state to be formed without dispersing currents and can mostly eliminate the relaxation effect. The optimal stop-flow time, called the relaxation time (tRx), depends

on the channel width (w), the diffusion rate of the polymer (D) and the retention factor (λ) [29].

𝑡𝑅𝑥 = 𝑤2_𝜆 𝐷 ( 1 2− 𝜆 + 1 𝑒1/𝜆_{− 1})

The contribution of relaxation effects to the plate height can be expressed by considering a number of relaxation events, each broadening the elution zone over a distance h0 defined by:

ℎ0=

𝑤〈𝑣〉𝑇 𝐷𝑇𝐷𝑑𝑇/𝑑𝑥

Where 〈𝜈〉 is the cross-sectional average solvent velocity and dT/dx the mean temperature across the flow-channel.

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13 The relaxation-based contribution to the plate height Hr is:

𝐻𝑟=

17 140

𝑛ℎ02

𝐿

Where n the number of relaxation events and, L the length of the flow-channel.

The contribution of relaxation effects can be reduced by increasing the field strength, decreasing the flow, and using a thinner channel. Furthermore, relaxation effects are less significant for components with both high thermal diffusion and high translational diffusion coefficients.

Non-equilibrium effects

In any form of FFF, separation is achieved by differences in the size of particle distributions from the cold wall, where flow velocity is low, into higher flow regions further from the accumulation wall. Due to finite diffusivity, particles of the same size will not traverse the same velocity regions on average. At any single point in time, particles in the higher flow regions will travel at a different velocity than those in the low flow region and thus band broadening occurs. This kind of band broadening is called the non-equilibrium contribution to band broadening and is the most significant band broadening effect in FFF [30].

The non-equilibrium effect is a form of mass transfer comparable to that in column chromatography, increasing with linear solvent velocity and decreasing with diffusion. The expression of the non-equilibrium contribution is as follows [31].

𝐻𝑁= 𝜒

𝑤2_〈𝑣〉

𝐷

Where χ is a function of the retention parameter λ, indicating non-equilibrium effects are more prominent for analytes showing little retention. The following function is a decent approximation of χ for retention parameters commonly achieved in ThFFF.

𝜒 = 24𝜆3₍1−10𝜆+28𝜆2 1−2𝜆 )

Figure 7: Schematic representation of non-equilibrium dispersion in FFF, less retained particles are more strongly distorted than highly retained particles.

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Temperature

Increasing the overall temperature in the ThFFF channel increases DT and therefore analytes will be more strongly

retained [32]. Concurrently, the translational diffusion (D) will increase with temperature, which deters retention. The overall temperature in ThFFF is often expressed as the cold wall temperature (TC). Retention in ThFFF also

increases with increasing temperature gradient (ΔT), as it increases the contribution of the thermal diffusion effect. Since the viscosity (η) of a solvent changes with temperature

[33], The temperature gradient applied in ThFFF gives rise to non-parabolic flow profiles as solvent near the hot wall will move faster than that near the cold wall [34]. This causes significant deviations in the calculation of tr from λ or

vice versa. The solvent viscosity is inversely proportional to the temperature, however, the thermal conductivity (κ) also depends on the temperature in turn making the temperature profile along the thickness of the flow-channel nonlinear, further complicating the issue. Van Asten et al. [8] compared various methods of corrections which were found to have limited success. They found that errors of an uncorrected (parabolic) flow profile were on the order of 4% when compared to empirical values.

If a sample of high dispersity is to be measured, the field strength (ΔT) can be reduced during the measurement. If suitable parameters are chosen, field programming allows fractionation over a broad mass range with approximately the same separation power within an acceptable timeframe. Different program shapes have been utilized, including: linear [35], parabolic, exponential [36], and power [37] functions. Of these program shapes, only the power function can maintain a stable fractionation power (Fd) across the entire program and is therefore the most

used. The power function of field-strength is as follows:

Δ𝑇(𝑡) = Δ𝑇0(

𝑡1− 𝑡𝑎

𝑡 − 𝑡𝑎

)

𝑝

With: ΔT(t) the field-strength at time t, ΔT0 the initial field-strength, t1 an initial period of constant field-strength

and ta and p variable program parameters. These factors should obey the following for field decay: t ≥ t1 > ta

and p > 0.

Using field programming, the general equation of retention does not apply anymore, instead more complex models must be used for the deduction of Soret factors and thermal diffusion coefficients. Therefore, calibration using standards cannot readily be applied. Instead size selective (DLS, MALS, dVIS) detection is required to obtain useful molar mass data.

Figure 8: Velocity profiles of ethylbenzene: assuming constant κ and η (full line) and with temperature dependent κ and η. With x the height position in the flow channel, w the full height of the flow channel, ν(x) the linear velocity at point x, and <ν> the mean linear velocity across the entire channel height. (Source: Asten et al. 1994)

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Concentration

The concentration does not significantly influence retention as long as it is low enough. More specifically, the concentration must be low enough as to prevent chain entanglement and maintain ideal solution behavior. The concentration at which a solution changes from dilute (non-interacting) to semi-dilute is called the critical concentration c*, which can be calculated using the following equation [38].

𝑐∗ ₌ 𝑀

𝑁𝐴𝜌𝑅𝑔3

In which, M the molar mass, NA Avogadro’s number, ρ the density of the solvent and, Rg the radius of gyration of

the polymer.

This is especially relevant when field programming is used, since the high molar mass components will be strongly confined against the cold wall, increasing the local concentration [39]. In this case the injection concentration should not exceed c*λ.

Polymer properties influencing retention

Molar Mass

The thermal diffusion effect is independent of molar mass for polymers of appreciable size [6], leaving mass based separation purely dependent on differences in translational diffusion. It should be noted that calculation of the diffusion coefficient, D, from intrinsic parameters cannot be used for calibration in ThFFF since polymers of different molecular weight will be distributed over different temperature regions, and thus the relation between M and D will be slightly different for each [39]. Instead the following expression may be used for mass calibration:

log 𝜆Δ𝑇 = log 𝐵 − 𝑏′_{log 𝑀}

In which, λ the retention parameter, ΔT the temperature gradient, B and b’ empirical constants dependent on the polymer solvent system and, M the molar mass. An alternative equation can be used employing retention volumes instead of the retention parameter:

(𝑉𝑟− 𝑉0) = 𝐴′+ 𝑆𝑚log 𝑀

In which, Vr the retention volume, V0 the dead volume, A’ and Sm the calibration parameters. It should be noted

that for accurate determination of physicochemical properties from ThFFF data, the extra column volume should be known with precision and be subtracted from all retention volumes including V0_{. These relations only apply to}

linear homopolymers though since branching and composition inevitably change D and both D and DT respectively.

If any Chemical Composition Distributions are present, the molar mass will need to be determined using multi-detector approaches.

Figure 9: Molecular mass independence of the thermal diffusion coefficients of different polymer-solvent systems. (Source: Schimpf, Giddings 1989)

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16 For many oligomers and small polymers, the thermal diffusion coefficient is known to depend strongly on the molecular weight [40]. DT increases sharply with growing chain length until it levels off at a certain molecular

weight depending on the type of polymer (typically around 10 kDa). Stadelmaier and Köhler showed that the molecular weight at which DT becomes constant as well as the ultimate magnitude of DT are related to the length

of the Kuhn segment of that polymer, a measure of the chain stiffness [41].

Monomeric composition

As mentioned before, the retention in ThFFF is highly dependent on the chemical composition of the polymer in question. However, in contrast to the molar mass, composition-based separation is mostly allocated to differences in DT. The exact solvent-solute interactions involved in the prevalence of DT are yet unknown, though several studies

have linked DT to polymer solubility parameters [42][43][44]. Still, practical values for DT can only be obtained

empirically.

In the case of copolymers, DT provides a useful measure for composition. Especially for random copolymers

where the observed DT is a linear dependence of monomeric composition [7].

𝐷𝑇(1,2, … 𝑖) = 𝑋1𝐷𝑇(1) + 𝑋2𝐷𝑇(2) … + 𝑋𝑖𝐷𝑇(𝑖)

Where Xi and DT(i) are the molar fraction and thermal diffusion coefficient of monomer i respectively.

This relationship has held up for several polymer-solvent systems including: styrene-isoprene, styrene-butyl acrylate and styrene-methyl acrylate copolymers [45], though whether this relationship also applies to more polar polymers is not well documented.

Ponyik et al. determined the composition distribution of a Polystyrene, Polymethyl methacrylate, Poly(tert-butyl acrylate) terpolymer using ThFFF-DLS-RI-UV [46]. Since only styrene was UV-active and the chemical composition dependencies are additive for both DT, and dn/dc, they could quantify each monomer by applying a

straightforward linear system of equations.

Figure 11: Composition dependence of DT in styrene-acrylic copolymers,

monomeric fraction of Poly(n-butyl acrylate) (PBA) and poly(methyl acrylate) (PMA), in random copolymers with polystyrene.

(Source: Runyon et al., 2011)

Figure 10: ternary composition diagram as derived by multidetector ThFFF. (Source: Ponyik et al., 2013)

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Solvation

The solvated state is a measure of chain expansion and is often reflected in the hydrodynamic radius, Rh. Rh is

related to the translational diffusion coefficient through the Stokes-Einstein equation:

𝐷 = 𝑘𝐵𝑇 6𝜋𝜂𝑟

Where kB is Boltzmann’s constant, T the temperature, η the solvent viscosity and r the radius of a spherical particle,

which through calibration can be related to Rh.

When a “good” solvent is used, the polymer chains are extended to the largest degree, this in turn increases the hydrodynamic radius and decreases the translational diffusion. However, it was observed Addition of a “poor” solvent to the mobile phase increases retention significantly. Since addition of a poor solvent causes the polymer chains to be more compact, the increase in retention cannot be concluded from D as diffusion increases with decreasing hydrodynamic radius. Therefore, the increased retention was logically thought to be caused by an increase in DT [47].

It was later found that this effect depends hugely on the solvent combination that is employed [9]. Regarding the thermal diffusion effect of the solvents themselves, it was found that when the “good” solvent had a higher thermal diffusion coefficient than the “poor” solvent, retention was greatly increased in a binary solvent approach. Inversely, when the “poor” solvent shows stronger thermal diffusion, retention decreases when the poor solvent was added. It was found by Asten et al. that for combinations of good solvents, DT was the solvent fraction weighted

average of the DT values in the respective pure solvents [48].

Rue and Schimpf [49] found that partitioning of the solvents according to their thermal diffusion coefficient occurred across the width of the flow-channel. From their findings it was concluded that the solvent gradient could either act with or against the thermophoretic mobility of the polymer, depending on the orientation of the solvent gradient. When the good solvent was enriched at the cold wall, the polymer would be confined closer to the cold wall, effectively increasing retention.

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Sequence Distribution

The sequence distribution or blockiness of co-polymers is relevant only under certain conditions. If solvation state is non-selective for the different monomers, DT is the

concentration weighted average of the monomers, similar to the case of random copolymers [50]. In cases where the solvent is selective such that one monomer is more soluble than the other, the thermal diffusion of the better solvated (more extended) chain is ruling and blockiness is of importance.

The exact relationship between the sequence distribution and ThFFF retention is rather convoluted due to the concurrent effects of solvation on D and DT since the

monomeric composition, number of blocks, size of blocks and location of blocks are all of influence.

The dependence of DT on block type distributions was well

illustrated by Schimpf et al. who compared DT of di-block

copolymers of polystyrene (PS) and polyisoprene (PI) in THF and cyclohexane which are respectively: non-selective for the polymers in question and selective for PI [51]. ThFFF was coupled online to nuclear magnetic resonance spectroscopy (NMR) by Hiller et al. who analyzed polystyrene, polyisoprene and polymethyl methacrylate homopolymers as well as their block copolymers [52]. Since NMR can accurately quantitate the individual monomers, it was proposed that it can be used to study and measure differences in polymer architecture by excluding the composition-based effect, however this was not yet performed. Additionally, ThFFF-NMR can be used to directly determine the chemical composition distribution and the molecular weight distribution as long as the relationship between composition and the diffusion coefficient is known.

Muza, Greyling and Pasch thoroughly studied self-assembling block copolymers of PS and polybutadiene (PB). They also found that DT was purely determined by the composition of the more expanded outer shell, while D was

dependent on the size of the self-assemblies [53]. After determination of the Molecular Weight Distribution using ThFFF-MALS-RI-DLS, the Chemical Composition Distribution was obtained using an FTIR coupling. They later presented the influence of the core microstructure on the radius of hydration and critical micelle concentration using this approach [54].

Figure 13: DT Dependence on solvent selectivity of PS-PI

di-block copolymers (blue) compared to the ideal linear relationship (orange). Top: cyclohexane (selective solvent), bottom: THF (non-selective solvent).

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19

Branching

Schimpf and Giddings explored the effect of branching on retention in ThFFF in one of their exploratory studies on the, then novel, technique[6]. They measured a variety of polystyrene samples including linear, star and comb shaped architectures with Mw values in the range of 55,000 to 5,700,000 Da.

When comparing retention with molecular weight (figure 14, left) one would assume branching does cause differences in thermal diffusion behavior. However, it should be noted that the size, and therefore, translational diffusion (D) of polymers is dependent on branching. When comparing retention with D (figure 14, right) it is evident the differences in retention are directly related to the diffusion coefficients of the different polymers and thus cannot be caused by a change in DT.

One can still expect ThFFF to be a viable technique for branching characterization by this difference in D, though it should be noted that this effect is very much dependent on the length of the branches and does not directly relate to the number branching in the sample. Furthermore, this effect is not unique to ThFFF but is present in every diffusion driven separation method, including all other modes of FFF and SEC [55]. A general decrease in the diffusion coefficient is not directly indicative of branching, which could be due to a lower molar mass as well. The use of MALS and viscometric detection provide supporting evidence of branching in the form of the

contraction factors g and g’ respectively, defined as:

𝑔 = 𝑅𝑔2_𝐵𝑅𝐴/𝑅𝑔2_𝐿𝐼𝑁, 𝑔′ = [𝜂]𝐵𝑅𝐴/[𝜂]𝐿𝐼𝑁

Where RG and [η] are the radius of gyration and intrinsic viscosity of a branched and a linear sample at equal

molar mass [56]. The magnitude of these contraction factors indicates the number of branches, or the branching density.

Similar to the contraction factors obtained from viscometry and light scattering detection, Runyon et al. proposed the use of ThFFF derived Soret contraction factors for the quantification of branching in yet unpublished work, defined as: 𝑔′′_{= 𝑆}

𝑇𝐵𝑅𝐴/𝑆𝑇𝐿𝐼𝑁. Since the Soret coefficient includes both D and DT, g’’ can be determined directly

from the ThFFF retention using only a concentration detector.

Figure 14: Influence of branching architecture on ThFFF retention, left: logarithmic plot of retention parameter against molecular weight, right: retention parameter versus diffusion coefficient. (Source: Schimpf and Giddings 1987)

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20 Greyling et al. applied the Soret contraction factor to calculate the number of end groups of star polystyrenes [57]. After measurement of monodisperse star and linear polystyrenes of similar mass the Soret coefficients were directly derived from the retention times according to:

𝑆𝑇= 6𝑡𝑟 Δ𝑇𝑡0

From the obtained Soret coefficients g’’ was calculated. The functionality could be derived from a previously established linear relation between g’’ and the number of branches. Additionally, it was found that branched polymers showed a different relation between temperature and hydrodynamic radius than their linear counterparts (thermoresponse), an effect which diminished as the solvent strength was decreased. It was found that a thermodynamically good solvent was required for accurate functionality determination.

Ponyik et al. measured the Soret factors of polyacrylates with linear, star and bottlebrush architectures and compared these with theoretically established values [58]. Whereas the Soret factor of the linear polymer was accurately predicted by the model, the Soret factors of the star and bottlebrush polymers were distinctly different due to their thermoresponsive nature, allowing architectural characterization.

Smith et al. related Soret contraction factors to the degree of branching of silyl end-capped polyesters, where the number of silyl groups quantitatively corresponded with the functionality of the polymer [59]. In this study, branching is not the direct reason for the Soret contraction that is presented. Rather the compositional selectivity of ThFFF is exploited by the introduction of chemically different end-groups [60]. The previously discussed study used bottlebrushes where the entirety of the branches were of a different polymer chemistry than the backbone [58].

The latter of the presented examples could still be considered for polymers without distinct end-group chemistry though, since end-group derivatization could introduce similar Soret contractions. Allowing the indirect determination of branching distributions in homopolymers where the difference in the thermoresponsive nature between linear and branched structures is not present or insufficient.

Figure 15: Dependence of the thermal diffusion coefficient on the degree of branching of silyl end-capped polyesters using cyclohexane (black) which is selective for the silyl end-groups, and THF (blue) which is non-selective. The independence of DT

on the number of silyl groups in the selective solvent indicates the purely chemical dependence of DT. (Source Smith et al. 2019)

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Thermal Field Flow Fractionation of particles

Compared to dissolved polymers the behavior of polymer colloids and emulsions is somewhat different. Due to the environmental relevance of polymer nano-particle analysis and the ever-rising industrial interest in the characterization of complex particle assemblies, the implications of ThFFF in these fields will be briefly discussed. ThFFF has successfully been applied to a broad spectrum of aqueous suspensions of polymers including: Polystyrene (PS), polybutadiene (PB), polymethylmethacrylate (PMMA) [61] and Polyvinyl acetate (PVA) microgels [62], as well as inorganic particles such as silica [63] and different metals [64]. Applicable sizes have been reported ranging between 30 nm [63] and 10 μm [65].

Like for dissolved polymers, the retention of particles in ThFFF is based on the balance between translational and thermal diffusion. However, the thermal diffusion coefficient of particles is dominated by the composition of the surface. Furthermore, the translational diffusion coefficient can straightforwardly be calculated using the Stokes-Einstein equation from the radius for (spherical) particles.

Compared to homogeneous polymer solutions, secondary interactions such us wall adhesion and particle-particle interactions tend to be much more pronounced for solid dispersions. In most cases the concentration ranges used in ThFFF are low enough that particle-particle interactions are negligible. Wall adhesion can mostly be prevented by addition of a surfactant to the carrier solvent.

Because of the relatively large size of particles, steric effects start to play a role in retention. Larger particles cannot approach the channel walls as closely as their smaller counterparts, such that they protrude on average into higher velocity flow streams than expected, and thus the retention will have a secondary dependence on size. A retention equation that corrects for these steric effects is as follows [66]:

𝑡0 𝑡𝑟 = 𝑅 = 6(𝛼 − 𝛼2_{) + 6𝜆} 𝑖𝑑(1 − 2𝛼) (coth ( 1 − 2𝛼 2𝜆𝑖𝑑 ))

Where α the particle size to channel width ratio (d/w) and λid the retention parameter for the particle at

infinitesimal size, which is the aforementioned function of D and DT.

Figure 16: Effect of NaCl concentration on PS particle behavior in ThFFF. Left: retention for particles of different diameters, indicating an increase in retention with higher salt concentrations. Right: relative peak area as a function of the temperature drop for a 0.22 μm particle using, indicating a loss in recovery for higher salt concentrations. (Source: Shiundu 2003)

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22 The effect of different solvent compositions and surfactant types on particle retention in ThFFF was assessed by Mes et al. [61]. They found that the composition of the solvent influenced both retention and selectivity. Addition of acetonitrile resulted in reduced retention for small particles, while retention for large particles increased. The kind of surfactant used was found to be critical; where a cationic surfactant (CTAB) caused an increase in wall adsorption, an anionic (SDS) and neutral (Brij 35) surfactant were found to effectively reduce wall adsorption, though the anionic surfactant caused a decrease in retention. Shiundu et al. later found that particle retention increased significantly with increasing ionic strength of the carrier solution. above a certain ionic concentration however, the addition of more salt caused an increase in wall adsorption and band broadening.

Polymer emulsions are stabilized dispersions of solid polymer, that was previously dissolved in a good solvent, and then crashed out of solution in a poor solvent in the presence of a surfactant. Latexes, Another kind of polymer emulsion, are produced by polymerization of pre-emulsified monomer [67]. The interface between an emulsified polymer particle and the bulk solvent consists of a film of surfactant which stabilizes the particle and traps the polymer within. Parts of the trapped polymer can tread into the surfactant layer; these parts are referred to as mers. Thermophoresis of polymer emulsions greatly depends on the mer concentration in the surfactant layer [68]. The addition of ample amounts of salt to the carrier liquid is often required to obtain appreciable retention for emulsions [69].

Schimpf and Semenov postulated that the addition of salt increased the mer solubility in the surfactant layer surrounding latex particles through the established “salting in” effect, which causes an increase in hydrocarbon solubility following the addition of salt.

As mentioned before, steric effects reduce the observed retention as particles increase in size. For ever larger particles, typically in the micrometer range, this causes an inversion of the retention mechanism. Due to the slow translational diffusion of such large particles, thermal diffusion confines them very closely to the cold wall. The mean layer thickness is now on the same order as the particle diameter causing larger particles to be in higher velocity streamlines. In this so-called steric mode of FFF high resolution separations of micrometer sized particles can be achieved.

The steric mode of FFF has led many to believe FFF in general is much like hydrodynamic chromatography, a purely steric based technique, but with an external field for added retention. In the normal mode of FFF this is certainly a misconception, as steric effects hardly play a role there.

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Determination of D and D

T

The ability to accurately determine the thermal diffusion coefficient (DT) is instrumental in relating it to polymer

properties of interest such as the chemical composition [70]. Since accurate values for DT can so far not be predicted

from first principles, it is required to dissect the Soret coefficient (as obtained from ThFFF retention) into translational diffusion (D) and DT. This is done by obtaining a value for D from one of the many size dependent

detection techniques.

Only one method provides the means to directly measure the diffusion coefficient, being Dynamic Light Scattering (DLS). DLS measures the frequency of fluctuations in Brownian motion, seen as fluctuations in scattering intensity over time. This frequency is a direct consequence of the diffusion coefficient, which can be extracted using spectrum analysis [71]. Since DLS measures spectral fluctuations over time, longer acquisition times result in higher quality data, though nowadays flow-through type DLS is easily applied to low flow systems, including ThFFF [45]. Other separation methods, that separate based on molecular mass may also be employed to measure D. This has the added advantage that composition distributions can be determined independent of molecular weight, at the cost of increased complexity and development time [72]. If the translational diffusion D is to be determined from the molecular weight as found through Size-Exclusion Chromatography (SEC), Hydrodynamic Chromatography (HDC) or Mass Spectrometry (MS), an empirical relation between the molar mass and diffusion must be established. A practical implication with good agreement is of the form [73]:

𝐷 = 𝐴 ∗ 𝑀−𝐵

Where A and B are the empirical factors to be determined.

An offline coupling between SEC and ThFFF was used by Asten et al. to measure the change in composition of styrene and acrylic polymers as a function of molecular weight [72]. Several fractions were collected across the SEC elution profile which were then reinjected on a ThFFF system. Polymer concentrations of the (up to 9) individual SEC fractions were high enough that single fractions could be used without evaporation. Diffusion coefficients were calculated from the elution time of each individual SEC fraction, after which thermal diffusion coefficients could be derived.

Since a well quantified relationship between the hydrodynamic radius and translational diffusion is known, size based detectors such as MALS and viscometry can be used to determine D at any datapoint [74][50]. The expression comes in the form of the well-known Stokes-Einstein equation [75]:

𝐷 = 𝑘𝐵𝑇 6𝜋𝜂𝑟ℎ

With kB Boltzmann’s constant, T the temperature, η the solvent viscosity and rh the hydrodynamic radius.

Application of the Stokes-Einstein equation assumes spherical particles which more often than not causes deviations in the estimated values of D, unless some empirical correction is performed.

ThFFF-MALS was employed by Mes et al. to measure the compositional drift of styrene acrylonitrile copolymers. They related the hydrodynamic radius with the radius of gyration, as obtained by MALS, using a solvation state dependent proportionality constant which they obtained through separate SEC-MALS measurements.

When the molecular weight distribution of a sample is known from a separate analysis with SEC or DLS, the influence of translational diffusion on the retention should be predictable, allowing relative values of DT to be

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Modelling of flow, dispersion and thermal diffusion

Accurate flow profiles

Deriving physical properties from ThFFF data, depends on the correctness of the relation between the retention factor (R) and the retention parameter (λ). As mentioned before, the common FFF description of a parabolic flow profile does not hold in ThFFF due to differences in thermal conductivity (κ), temperature (T), and viscosity (η) along the height of the flow channel. In order to correct for this discrepancy, a number of solvent parameters need to be known and quantitatively described.

Firstly, the dependence of the solvent thermal conductivity on temperature needs to be determined, which has been approximated to be linear with good agreement for pure, non-polar solvents [76].

𝑑𝜅 𝑑𝑇=

𝜅 − 𝜅𝑐

𝑇 − 𝑇𝑐

Where κ, the thermal conductivity at temperature T, κc the thermal conductivity at the cold wall temperature Tc,

and dκ/dT the linear temperature dependence of the thermal conductivity near the cold wall.

Lei et al. presented an alternative strategy for determining the temperature dependence of thermal conductivity [77], which showed excellent agreement for polar solvents and solvent mixtures. Though applying this method to ThFFF would require several physical tests of the solvents in question and cumbersome numerical integrations. However, for more in-depth characterization of the thermal diffusion effect such methods should be adopted. The temperature gradient can then be evaluated using a very lengthy quantitative derivation as described by Gunderson et al. which can be simplified by applying a Taylor’s expansion, with a higher order being more accurate [78]. The first order Taylor’s derivative is as follows:

𝑑𝑇 𝑑𝑥= 𝑞 𝜅𝑐 = 𝑆 𝑤

Where q is a measure of heat flux, which needs to be measured empirically and S, a constant needed for the higher order approximations.

The temperature dependence of solvent fluidity (1/η) can be derived from a cubic fit of temperature versus viscosity data of the form:

1

𝜂= 𝑎0+ 𝑎1𝑇 + 𝑎2𝑇 2_{+ 𝑎}

3𝑇3

Employing the regression parameters (a0 – a3) from this fit, the velocity profile can be expressed with great

accuracy. The full derivation is described by Gunderson et al. [78]. As outlined here, the workflow requires empirical thermal conductivity and fluidity data to be obtained which can be rather time consuming, although for many solvents this data is well reported [79].

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Deconvolution of dispersion

In order to obtain accurate molecular weight or compositional distribution information, the width of the recorded peak must be caused exclusively by the distribution of interest. Increased band broadening will cause the width of the distribution to be overestimated proportionally. Therefore, the system dispersion is typically minimized by using thin flow channels and low flowrates to minimize non-equilibrium effects, the main source of band broadening. Even with proper FFF practice of minimizing system dispersion, the capability of accurately measuring (especially narrow) distributions is limited. The necessity for low flowrates also makes FFF a slow technique.

Since system dispersion (as non-equilibrium effects) in ThFFF is well understood, it can iteratively be predicted and deconvoluted from the recorded fractogram allowing more accurate determinations of narrow distributions or the use of higher flowrates without loss in accuracy of the measured distributions [80]. Furthermore, the increase in accuracy of this method also indicates that elution profiles as predicted from models based on non-equilibrium effects can be used to approach the shape of experimental ThFFF elution profiles.

Prediction of D

T

Although a quantitative description of the thermal diffusion effect does not exist, several attempts have been made to predict DT for various polymer-solvent combinations, with mixed results. Two theories are most notable

for their ability to predict DT before any ThFFF measurement [45], one as reported by Schimpf and Semenov [44],

and one as described by Mes et al. [43].

Schimpf and Semenov described the thermal diffusion effect as arising from an osmotic pressure gradient around each monomer, caused by the temperature dependence of solvent parameters, specifically the monomer-solvent interaction potential, temperature coefficient, thermal expansion, and compressibility. The monomer solvent interaction potential describes dipole-dipole interactions and can be calculated, whereas the others need to be established empirically.

Mes et al. calculated DT by regarding it as an enthalpic unmixing effect in the presence of a temperature gradient.

The model requires the temperature dependence of the Flory-Huggins interaction parameter and the partial molar volumes of solvent and monomer. The temperature dependence of the Flory-Huggins parameter is not easily measured, but instead, was estimated by employing Hildebrand’s solubility parameters [81].

The performance of both theories was tested by Runyon et al. who compared the predicted DT values for

polystyrene in different solvents with those obtained from ThFFF [82]. Both theories systematically underestimated the DT values with the Mes approach performing well

enough to be used for solvent selection. The predictions by the Mes method were then compared with experimental values for DT of PS, PBA, PMA, and PMMA

in THF, resulting in deviations lower than 25% for all except PMA, for which deviations were around 50%. Though these theoretical approximations of DT are rough at best, their use and proliferation will most likely be imperative for the usability of ThFFF in the future, as empirical testing of each solvent-polymer system is unlikely to become routine due to the vast time requirement of performing such tests. The applicability of the Mes method to narrow down the solvent selection for empirical screenings should be noted.

0 0.5 1 1.5 2 0.5 0.7 0.9 1.1 1.3 1.5 1.7 1.9 Predi ct ed D T (x10 7cm 2/s K ) Experimental D_T(x107_cm2_/sK)

Predictive capability of DT

calculations

Schimpf-Semenov Mes ThFFF

Figure 18: Overview of the experimental versus predicted values for DT as obtained with the Schimpf-Semenov and Mes approaches. The diagonal (experimental against experimental) is displayed in black with error bars. Although the predicted DT values are systematically lower, the results obtained

using the Mes approach correlates well, and thus can serve as preliminary solvent selection. (Adapted from: Runyon and Williams, 2011)

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Contemporary ThFFF methods for polymer characterization

The following is a list of notable ThFFF applications including the used (supporting) techniques and the information derived from the data. A-b-B denotes a block copolymer of A and B.

Polymer type(s) molar mass

range Applied techniques Obtained information Source PS,

1,4-PBd25-PS34

98 kDa –

250 kDa UV, MALS, dVIS, RID, DLS Mn, Rg, [η], Mark-Houwink const., conformation, composition U. L. Muza, H. Pasch. 2015 [83] PBiBEM-PBA (grafted bottlebrushes) 90 kDa –

9,6 MDa MALS, DLS, RID Mn, architecture C. A. Ponyik et al. 2015 [58]

PS 11 kDa –

575 kDa UV, MALDI-TOF

Mn, Molecular Weight

Distribution G. E. Kassalainen, S. K. Williams 2003

[84]

Poly(Styrene-Co-Isoprene) 38 kDa UV, RID Mw, composition, dVIS M. E. Schimpf 1995 [50]

PS, PMMA, PtBA PS-PtBa-PMMA (triblock)

18 kDa -

64 kDa MALS, DLS, RID, UV (NMR for validation)

Composition C. A. Ponyik, D. T. Wu, S. K. Williams

2013 [46]

Silyl-endcapped-polyesters 30 kDa – 153 kDa MALS, RID Branching (from number of silyl groups) W. C. Smith et al. 2019 [59] PS, PMMA, PI, PS-b-PI, Pi-b-PMMA, PS-b-PMMA 59 kDa –

357 kDa UV, NMR Composition, Molecular Weight Distribution, Block Length Distribution W. Hiller, W. van Aswegen, M. Hehn, H. Pasch 2013 [52] PS 2.6 kDa -

11.2 kDa UV, Maldi-TOF Molecular Weight Distribution, Retention at very low MW G. E. Kassalainen, S. K. Williams [85]

PS-b-PEO 90 kDa –

2.3 MDa UV, RI, MALS, DLS Molecular Weight Distribution, block conformation in different systems

N. Ngaza, M. Brand, H. Pasch, 2015 [86]

PS (linear and star) 34 kDa –

305 kDa RI, MALS Architecture: differentiation between linear and star G. Greyling, A. Lederer, H. Pasch, 2018

[57] PBA,

PS-b-PMA 29.4 kDa – 415 kDa RI, MALS, DLS Molecular Weight Distribution, Composition Distribution J. Runyon, S. K. Williams, 2011 [45] PVAc, PMMA 30 kDa –

350 kDa

RI, MALS Molecular Weight Distribution, radius of gyration D. lee, S. K. Williams, 2010 [62] PMMA (isotactic, syndiotactic and atactic) 131 kDa –

139 kDa RI, MALS, DLS Tacticity (from diffusion coefficient) G. Greyling, H. Pasch, 2015 [87]

PS-b-PB Self

assemblies 1.52 MDa - 28.3 MDa RI, MALS, DLS, FTIR Molecular Weight Distribution, Chemical Composition, Dh

U. Muza, G. Gteyling, H. Pasch, 2018 [54]

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Conclusions

Thermal field-flow fractionation is a versatile method, allowing separation based on size, as well as composition and chain architecture. The linear dependence of retention on composition is rather unique in the field of macromolecular characterization. Furthermore, its very low induced shear stresses and lack of packing allows the analysis of very large and labile polymers and particles.

A great strength of the technique is that the retention can be quantitatively described, and even predicted, in terms of the translational and thermal diffusion coefficients. However, this is consecutively its downfall since one cannot be obtained without information of the other requiring either much prior knowledge of the sample, or the use of (expensive) dedicated equipment and complicated correction strategies.

In recent years, research on ThFFF has been conducted mainly by the groups of Harald Pasch at Stellenbosch University and of Kim Williams at the Colorado School of Mines. Their research has mostly focused on gaining insights in the structure of complex polymers with blockiness and/or branching using multi-detector setups. Recent interests in higher order structural characterization of synthetic polymers has sparked new life into the technique, though the fundamental background still needs to be resolved before it can be applied effectively in a routine setting.

Interestingly, many dependencies of the magnitude of the thermal diffusion coefficient appear to be caused by self-interaction. This is evidenced by the significantly weaker thermal diffusion of polar macromolecules (proteins, polysaccharides, polyesters), as well as for polymers with shorter Kuhn segments (more flexible chains). The rather one-sided view of fundamental studies to non-polar polymers (most notably polystyrene and polyisoprene) seems to have biased the perception of thermal diffusivity to fit a model where interaction between monomers does not have a significant contribution. Admittedly, models that include (polar) self-interactions would be much harder to formulate and implement, however, with the current and rising interest in complex polyester and polyamide materials this would be a necessary development.

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Acknowledgements

I would like to thank Iro Ventouri for providing me with this project and for her great guidance throughout. Great gratitude goes to DSM and Dr. Harry Philipsen in particular for developing the concept in the first place, providing feedback and insight in the progression and providing funding for the project. I would further like to thank Dr. Alina Astefanei and Prof. Dr. Ron Peters for reviewing this thesis and providing feedback where necessary.

Technical and Practical Implications of Thermal Field-Flow Fractionation for the Separation of Polymers

Literature Thesis