# Adverse-Mode FFF: Multi-Force Ideal Retention Theory

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## Abstract

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## 1. Introduction

**Figure 1.**Schematic of the concept behind adverse-mode field-flow fractionation (FFF). Two strong forces (${f}_{1}$ and ${f}_{2}$), which scale differently with solute size, oppose one another. At small sizes, the force with the smaller scaling (say ${\alpha}_{1}$) dominates and the solute undergoes steric-mode at the accumulation (bottom) wall. At the specific size ${\tilde{r}}_{\text{L}}$, the two forces are equal, and the solute is buoyant, such that it diffusively samples all sterically available flows equally. At larger sizes, the adverse force dominates, and the solute is pushed against the opposite (top) channel wall (what was previously the depletion wall).

## 2. Multi-Force Ideal Retention Theory

#### 2.1. Retention Ratio of the Multi-Force Ideal Retention Theory

#### 2.2. Higher Moments of the Multi-Force Ideal Retention Theory

**Figure 2.**First three cumulants of the velocity distribution for a single external field characterized by ${\Lambda}_{1}={10}^{-5}$ and ${\alpha}_{1}=3$ as a function of solute size. The functions are determined analytically via Equation (11). The mean (retention ratio) is given by Equations (16)-(17), the variance by Equations (18)-(19) and the skewness by Equations (20)-(21).

## 3. Adverse-Mode FFF: Benefit of Two Opposing Forces

#### 3.1. Adverse-Mode: Retention Ratio Peak

**Figure 3.**Retention ratio as a function of solute size for two constant opposing fields (adverse-mode FFF). Moderate opposing fields are shown. The force toward one wall scales as ${\alpha}_{1}=1$ (as in flow-FFF), while the other scales as ${\alpha}_{2}=3$ (sedimentation-like fields). Solid lines show the full theory (Equation (5)), while dashed lines show the near-peak approximation (Equation (26)). Dots mark ${\tilde{r}}_{\text{L}}$ from Equation (25).

#### 3.2. Adverse-Mode: Retention Ratio Peak Width

**Figure 4.**Same as Figure 3, but stronger opposing fields are applied.

**Figure 5.**Full-width half-maximum (FWHM) of peaks as a function of ${\Lambda}_{1}$ (field strength) for fixed values of peak position ${\tilde{r}}_{\text{L}}$. FWHM is non-dimensionalized by channel height. The opposing field ${\Lambda}_{2}$ is set by Equation (25) for fixed ${\tilde{r}}_{\text{L}}$.

#### 3.3. Adverse-Mode: Ideal Variance Peak

**Figure 6.**Ideal variance as a function of solute size for adverse-FFF. (

**a**) Moderate opposing fields; (

**b**) strong opposing fields.

## 4. Implementation and Feasibility

- The retention ratio peaks can be quite sharp for experimentally-achievable fields and;
- the peak position ${\tilde{r}}_{\text{L}}$ can be simply predicted.

- Apply an external field with ${\alpha}_{1}=1$ (such as flow-FFF, electrical-FFF or thermal-FFF) to an eluting sample. This field should have a small enough ${\Lambda}_{1}$ that elution occurs in the steric-mode regime of FFF.
- Apply a field with ${\alpha}_{2}=3$ and ${\Lambda}_{2}$ (such as centrifugal-FFF) to oppose the first field. ${\Lambda}_{2}$ must be smaller than ${\Lambda}_{1}$ for adverse-FFF to be successful.
- Incrementally decrease ${\Lambda}_{2}$ from an initial large value, shifting ${\tilde{r}}_{\text{L}}$ to smaller values.
- Identify the abrupt increase in retention ratio (plummet in retention time) when ${\tilde{r}}_{\text{L}}\left({\Lambda}_{2}\right)\approx \tilde{r}$.
- Determine solute size through Equation (25).
- Having found ${\tilde{r}}_{\text{L}}$, simultaneously decrease the pair of device retention ratios ${\Lambda}_{1}$ and ${\Lambda}_{2}$ to hone the peak and improve the measurement accuracy.

- For thermal-FFF, ${\Lambda}_{1}\sim -{10}^{-4}$ $\text{K}/\Delta T$, where $\Delta T\sim 30$ K is the temperature difference between the accumulation and depletion walls. This estimate assumes a characteristic Soret coefficient of ∼${10}^{-7}$$\text{c}{\text{m}}^{2}{\text{s}}^{-1}{\text{K}}^{-1}$ [74].
- For flow-FFF, ${\Lambda}_{1}\sim -{10}^{-7}$ ${\text{s}}^{-1}$ ${V}_{0}/{\dot{V}}_{\text{C}}$, where the channel volume is ${V}_{0}\sim {10}^{-6}$ ${\text{m}}^{3}$ and the cross-flow rate ${\dot{V}}_{\text{C}}\sim 1$ $\text{mL/min}$ [58].

## 5. Conclusions

## Acknowledgements

## Author Contributions

## Conflicts of Interest

## References

- Reschiglian, P.; Zattoni, A.; Roda, B.; Michelini, E.; Roda, A. Field-flow fractionation and biotechnology. Trends Biotechnol.
**2005**, 23, 475–483. [Google Scholar] [CrossRef] [PubMed] - Kowalkowski, T.; Buszewski, B.; Cantado, C.; Dondi, F. Field-Flow Fractionation: Theory, Techniques, Applications and the Challenges. Crit. Rev. Anal. Chem.
**2006**, 36, 129–135. [Google Scholar] [CrossRef] - Ratanathanawongs Williams, S.; Lee, D. Field-flow fractionation of proteins, polysaccharides, synthetic polymers, and supramolecular assemblies. J. Sep. Sci.
**2006**, 29, 1720–1732. [Google Scholar] [CrossRef] - Roda, B.; Zattoni, A.; Reschiglian, P.; Moon, M.; Mirasoli, M.; Michelini, E.; Roda, A. Field-flow fractionation in bioanalysis: A review of recent trends. Anal. Chim. Acta
**2009**, 635, 132–143. [Google Scholar] [CrossRef] [PubMed] - Qureshi, R.; Kok, W. Application of flow field-flow fractionation for the characterization of macromolecules of biological interest: A review. Anal. Bioanal. Chem.
**2011**, 399, 1401–1411. [Google Scholar] [CrossRef] [PubMed] - Rambaldi, D.; Reschiglian, P.; Zattoni, A. Flow field-flow fractionation: Recent trends in protein analysis. Anal. Bioanal. Chem.
**2011**, 399, 1439–1447. [Google Scholar] [CrossRef] [PubMed] - Runyon, J.R.; Ulmius, M.; Nilsson, L. A perspective on the characterization of colloids and macromolecules using asymmetrical flow field-flow fractionation. Colloids Surf. A: Physicochem. Eng. Asp.
**2014**, 442, 25–33. [Google Scholar] [CrossRef] - Baalousha, M.; Stolpe, B.; Lead, J. Flow field-flow fractionation for the analysis and characterization of natural colloids and manufactured nanoparticles in environmental systems: A critical review. J. Chromatogr. A
**2011**, 1218, 4078–4103. [Google Scholar] [CrossRef] [PubMed] - Stolpe, B.; Guo, L.; Shiller, A.M.; Aiken, G.R. Abundance, size distributions and trace-element binding of organic and iron-rich nanocolloids in Alaskan rivers, as revealed by field-flow fractionation and ICP-MS. Geochim. Cosmochim. Acta
**2013**, 105, 221–239. [Google Scholar] [CrossRef] - Koopmans, G.; Hiemstra, T.; Regelink, I.; Molleman, B.; Comans, R. Asymmetric flow field-flow fractionation of manufactured silver nanoparticles spiked into soil solution. J. Chromatogr. A
**2015**, 1392, 100–109. [Google Scholar] [CrossRef] [PubMed] - Tanase, M.; Zolla, V.; Clement, C.C.; Borghi, F.; Urbanska, A.M.; Rodriguez-Navarro, J.A.; Roda, B.; Zattoni, A.; Reschiglian, P.; Cuervo, A.M.; et al. Hydrodynamic size-based separation and characterization of protein aggregates from total cell lysates. Nat. Protoc.
**2015**, 10, 134–148. [Google Scholar] [CrossRef] [PubMed] - Zattoni, A.; Roda, B.; Borghi, F.; Marassi, V.; Reschiglian, P. Flow field-flow fractionation for the analysis of nanoparticles used in drug delivery. J. Pharm. Biomed. Anal.
**2014**, 87, 53–61. [Google Scholar] [CrossRef] [PubMed] - Yang, J.S.; Lee, J.Y.; Moon, M.H. High Speed Size Sorting of Subcellular Organelles by Flow Field-Flow Fractionation. Anal. Chem.
**2015**, 87, 6342–6348. [Google Scholar] [CrossRef] [PubMed] - Nilsson, L. Separation and characterization of food macromolecules using field-flow fractionation: A review. Food Hydrocoll.
**2013**, 30, 1–11. [Google Scholar] [CrossRef] - Moquin, A.; Neibert, K.D.; Maysinger, D.; Winnik, F.M. Quantum dot agglomerates in biological media and their characterization by asymmetrical flow field-flow fractionation. Eur. J. Pharm. Biopharm.
**2015**, 89, 290–299. [Google Scholar] [CrossRef] [PubMed] - Moore, L.; Williams, P.; Nehl, F.; Abe, K.; Chalmers, J.; Zborowski, M. Feasibility study of red blood cell debulking by magnetic field-flow fractionation with step-programmed flow. Anal. Bioanal. Chem.
**2014**, 406, 1661–1670. [Google Scholar] [CrossRef] [PubMed] - Rameshwar, T.; Samal, S.; Lee, S.; Kim, S.; Cho, J.; Kim, I. Determination of the Size of Water-Soluble Nanoparticles and Quantum Dots by Field-Flow Fractionation. J. Nanosci. Nanotechnol.
**2006**, 6, 2461–2467. [Google Scholar] [CrossRef] [PubMed] - Stolpe, B.; Hassellöv, M. Changes in size distribution of fresh water nanoscale colloidal matter and associated elements on mixing with seawater. Geochim. Cosmochim. Acta
**2007**, 71, 3292–3301. [Google Scholar] [CrossRef] - Von der Kammer, F.; Legros, S.; Hofmann, T.; Larsen, E.; Loeschner, K. Separation and characterization of nanoparticles in complex food and environmental samples by field-flow fractionation. TrAC Trends Anal. Chem.
**2011**, 30, 425–436. [Google Scholar] [CrossRef] - Ratanathanawongs Williams, S.; Runyon, J.; Ashames, A. Field-Flow Fractionation: Addressing the Nano Challenge. Anal. Chem.
**2011**, 83, 634–642. [Google Scholar] [CrossRef] [PubMed] - Giddings, J.; Myers, M. Steric Field-Flow Fractionation: A New Method for Separating 1 to 100 µm Particles. Sep. Sci. Technol.
**1978**, 13, 637–645. [Google Scholar] [CrossRef] - Giddings, J.C.; Williams, P.S. Multifaceted analysis of 0.01 to 100 µm particles by sedimentation field-flow fractionation. Am. Lab.
**1993**, 95, 88–95. [Google Scholar] - Berg, H.; Purcell, E.; Stewart, W. A method for separating according to mass a mixture of macromolecules or small particles suspended in a fluid. II. Experiments in a gravitational field. Proc. Natl. Acad. Sci. USA
**1967**, 58, 1286–1291. [Google Scholar] [CrossRef] - Lee, S.; Kang, D.; Park, M.; Williams, P. Effect of Carrier Fluid Viscosity on Retention Time and Resolution in Gravitational Field-Flow Fractionation. Anal. Chem.
**2011**, 83, 3343–3351. [Google Scholar] [CrossRef] [PubMed] - Berg, H.; Purcell, E. A method for separating according to mass a mixture of macromolecules or small particles suspended in a fluid. III. Experiments in a centrifugal fluid. Proc. Natl. Acad. Sci. USA
**1967**, 58, 1821–1828. [Google Scholar] [CrossRef] - Mélin, C.; Perraud, A.; Akil, H.; Jauberteau, M.O.; Cardot, P.; Mathonnet, M.; Battu, S. Cancer Stem Cell Sorting from Colorectal Cancer Cell Lines by Sedimentation Field Flow Fractionation. Anal. Chem.
**2012**, 84, 1549–1556. [Google Scholar] [CrossRef] [PubMed] - Caldwell, K.; Kesner, L.; Myers, M.; Giddings, J. Electrical Field-Flow Fractionation of Proteins. Science
**1972**, 176, 296–298. [Google Scholar] [CrossRef] [PubMed] - Gigault, J.; Gale, B.; Le Hecho, I.; Lespes, G. Nanoparticle Characterization by Cyclical Electrical Field-Flow Fractionation. Anal. Chem.
**2011**, 83, 6565–6572. [Google Scholar] [CrossRef] [PubMed] - Vickrey, T.; Garcia-ramirez, J. Magnetic Field-Flow Fractionation: Theoretical Basis. Sep. Sci. Technol.
**1980**, 15, 1297–1304. [Google Scholar] [CrossRef] - Williams, P.; Carpino, F.; Zborowski, M. Characterization of magnetic nanoparticles using programmed quadrupole magnetic field-flow fractionation. Philos. Trans. R. Soc. A
**2010**, 368, 4419–4437. [Google Scholar] [CrossRef] [PubMed] - Huang, Y.; Wang, X.; Becker, F.; Gascoyne, P. Introducing dielectrophoresis as a new force field for field-flow fractionation. Biophys. J.
**1997**, 73, 1118–1129. [Google Scholar] [CrossRef] - Shim, S.; Gascoyne, P.; Noshari, J.; Stemke Hale, K. Dynamic physical properties of dissociated tumor cells revealed by dielectrophoretic field-flow fractionation. Integr. Biol.
**2011**, 3, 850–862. [Google Scholar] [CrossRef] [PubMed] - Semyonov, S.; Maslow, K. Acoustic field-flow fractionation. J. Chromatogr. A
**1988**, 446, 151–156. [Google Scholar] [CrossRef] - Budwig, R.; Anderson, M.; Putnam, G.; Manning, C. Ultrasonic particle size fractionation in a moving air stream. Ultrasonics
**2010**, 50, 26–31. [Google Scholar] [CrossRef] [PubMed] - Giddings, J. Field-Flow Fractionation. Chem. Eng. News Arch.
**1988**, 66, 34–45. [Google Scholar] [CrossRef] - Kononenko, V.; Giddings, J.; Myers, M. On the possibility of photophoretic field-flow fractionation. J. Microcolumn Sep.
**1997**, 9, 321–327. [Google Scholar] [CrossRef] - Giddings, J.; Yang, F.; Myers, M. Theoretical and experimental characterization of flow field-flow fractionation. Anal. Chem.
**1976**, 48, 1126–1132. [Google Scholar] [CrossRef] - Cumberland, S.; Lead, J. Particle size distributions of silver nanoparticles at environmentally relevant conditions. J. Chromatogr. A
**2009**, 1216, 9099–9105. [Google Scholar] [CrossRef] [PubMed] - Wahlund, K.; Giddings, J. Properties of an asymmetrical flow field-flow fractionation channel having one permeable wall. Anal. Chem.
**1987**, 59, 1332–1339. [Google Scholar] [CrossRef] [PubMed] - Yohannes, G.; Jussila, M.; Hartonen, K.; Riekkola, M.L. Asymmetrical flow field-flow fractionation technique for separation and characterization of biopolymers and bioparticles. J. Chromatogr. A
**2011**, 1218, 4104–4116. [Google Scholar] [CrossRef] [PubMed] - Thompson, G.; Myers, M.; Giddings, J. An Observation of a Field-Flow Fractionation Effect with Polystyrene Samples. Sep. Sci.
**1967**, 2, 797–800. [Google Scholar] [CrossRef] - Runyon, J.; Williams, S.R. A theory-based approach to thermal field-flow fractionation of polyacrylates. J. Chromatogr. A
**2011**, 1218, 7016–7022. [Google Scholar] [CrossRef] [PubMed] - Cölfen, H.; Antonietti, M. Field-Flow Fractionation Techniques for Polymer and Colloid Analysis. In New Developments in Polymer Analytics I; Schmidt, M., Ed.; Springer: Berlin, Germany; Heidelberg, Germany, 2000; Volume 157, pp. 67–187. [Google Scholar]
- Messaud, F.; Sanderson, R.; Runyon, J.; Otte, T.; Pasch, H.; Williams, S.R. An overview on field-flow fractionation techniques and their applications in the separation and characterization of polymers. Prog. Polym. Sci.
**2009**, 34, 351–368. [Google Scholar] [CrossRef] - Fraunhofer, W.; Winter, G. The use of asymmetrical flow field-flow fractionation in pharmaceutics and biopharmaceutics. Eur. J. Pharm. Biopharm.
**2004**, 58, 369–383. [Google Scholar] [CrossRef] [PubMed] - Sant, H.; Gale, B. Microscale Field-Flow Fractionation: Theory and Practice. In Microfluidic Technologies for Miniaturized Analysis Systems; Hardt, S., Schönfeld, F., Eds.; Springer: New York, NY, USA, 2007; pp. 471–521. [Google Scholar]
- Wahlund, K.G. Flow field-flow fractionation: Critical overview. J. Chromatogr. A
**2013**, 1287, 97–112. [Google Scholar] [CrossRef] [PubMed] - Martin, M.; Beckett, R. Size Selectivity in Field-Flow Fractionation: Lift Mode of Retention with Near-Wall Lift Force. J. Phys. Chem. A
**2012**, 116, 6540–6551. [Google Scholar] [CrossRef] [PubMed] - Westermann-Clark, G. Note on Nonisothermal Flow in Field-Flow Fractionation. Sep. Sci. Technol.
**1978**, 13, 819–822. [Google Scholar] [CrossRef] - Belgaied, J.; Hoyos, M.; Martin, M. Velocity profiles in thermal field-flow fractionation. J. Chromatogr. A
**1994**, 678, 85–96. [Google Scholar] [CrossRef] - Pasol, L.; Martin, M.; Ekiel-Jeżewska, M.; Wajnryb, E.; Bławzdziewicz, J.; Feuillebois, F. Motion of a sphere parallel to plane walls in a Poiseuille flow. Application to field-flow fractionation and hydrodynamic chromatography. Chem. Eng. Sci.
**2011**, 66, 4078–4089. [Google Scholar] [CrossRef] - Shendruk, T.; Tahvildari, R.; Catafard, N.; Andrzejewski, L.; Gigault, C.; Todd, A.; Gagne-Dumais, L.; Slater, G.; Godin, M. Field-Flow Fractionation and Hydrodynamic Chromatography on a Microfluidic Chip. Anal. Chem.
**2013**, 85, 5981–5988. [Google Scholar] [CrossRef] [PubMed] - Shendruk, T.; Slater, G. Hydrodynamic chromatography and field flow fractionation in finite aspect ratio channels. J. Chromatogr. A
**2014**, 1339, 219–223. [Google Scholar] [CrossRef] [PubMed] - Slater, G.; Shendruk, T. Can slip walls improve field-flow fractionation or hydrodynamic chromatography? J. Chromatogr. A
**2012**, 1256, 206–212. [Google Scholar] [CrossRef] [PubMed] - Martin, M.; Feuillebois, F. Onset of sample concentration effects on retention in field-flow fractionation. J. Sep. Sci.
**2003**, 26, 471–479. [Google Scholar] [CrossRef] - Sant, H.; Gale, B. Characterization of a microscale thermal-electrical field-flow fractionation system. J. Chromatogr. A
**2012**, 1225, 174–181. [Google Scholar] [CrossRef] [PubMed] - Johann, C.; Elsenberg, S.; Schuch, H.; Rösch, U. Instrument and Method to Determine the Electrophoretic Mobility of Nanoparticles and Proteins by Combining Electrical and Flow Field-Flow Fractionation. Anal. Chem.
**2015**, 87, 4292–4298. [Google Scholar] [CrossRef] [PubMed] - Kato, H.; Nakamura, A. Separation of nano- and micro-sized materials by hyphenated flow and centrifugal field-flow fractionation. Anal. Methods
**2014**, 6, 3215–3218. [Google Scholar] [CrossRef] - Shendruk, T.; Slater, G. Operational-modes of field-flow fractionation in microfluidic channels. J. Chromatogr. A
**2012**, 1233, 100–108. [Google Scholar] [CrossRef] [PubMed] - Ratier, C.; Hoyos, M. Acoustic Programming in Step-Split-Flow Lateral-Transport Thin Fractionation. Anal. Chem.
**2010**, 82, 1318–1325. [Google Scholar] [CrossRef] [PubMed] - Wang, X.B.; Vykoukal, J.; Becker, F.F.; Gascoyne, P.R. Separation of Polystyrene Microbeads Using Dielectrophoretic/Gravitational Field-Flow-Fractionation. Biophys. J.
**1998**, 74, 2689–2701. [Google Scholar] [CrossRef] - Yang, J.; Huang, Y.; Wang, X.B.; Becker, F.F.; Gascoyne, P.R.C. Cell Separation on Microfabricated Electrodes Using Dielectrophoretic/Gravitational Field-Flow Fractionation. Anal. Chem.
**1999**, 71, 911–918. [Google Scholar] [CrossRef] [PubMed] - Wang, X.B.; Yang, J.; Huang, Y.; Vykoukal, J.; Becker, F.F.; Gascoyne, P.R.C. Cell Separation by Dielectrophoretic Field-flow-fractionation. Anal. Chem.
**2000**, 72, 832–839. [Google Scholar] [CrossRef] [PubMed] - Giddings, J. Nonequilibrium Theory of Field-Flow Fractionation. J. Chem. Phys.
**1968**, 49, 81–85. [Google Scholar] [CrossRef] - Williams, P. Retention ratio and nonequilibrium bandspreading in asymmetrical flow field-flow fractionation. Anal. Bioanal. Chem.
**2015**, 407, 4327–4338. [Google Scholar] [CrossRef] [PubMed] - Mudalige, T.K.; Qu, H.; Sánchez-Pomales, G.; Sisco, P.N.; Linder, S.W. Simple Functionalization Strategies for Enhancing Nanoparticle Separation and Recovery with Asymmetric Flow Field Flow Fractionation. Anal. Chem.
**2015**, 87, 1764–1772. [Google Scholar] [CrossRef] [PubMed] - Perez-Rea, D.; Bergenståhl, B.; Nilsson, L. Development and evaluation of methods for starch dissolution using asymmetrical flow field-flow fractionation. Part I: Dissolution of amylopectin. Anal. Bioanal. Chem.
**2015**, 407, 4315–4326. [Google Scholar] [PubMed] - Ngaza, N.; Brand, M.; Pasch, H. Multidetector-ThF3 as a Novel Tool for the Investigation of Solution Properties of Amphiphilic Block Copolymers. Macromol. Chem. Phys.
**2015**, 216, 1355–1364. [Google Scholar] [CrossRef] - Greyling, G.; Pasch, H. Tacticity Separation of Poly(methyl methacrylate) by Multidetector Thermal Field-Flow Fractionation. Anal. Chem.
**2015**, 87, 3011–3018. [Google Scholar] [CrossRef] [PubMed] - Tasci, T.O.; Johnson, W.P.; Fernandez, D.P.; Manangon, E.; Gale, B.K. Biased Cyclical Electrical Field Flow Fractionation for Separation of Sub 50 nm Particles. Anal. Chem.
**2013**, 85, 11225–11232. [Google Scholar] [CrossRef] [PubMed] - Tasci, T.O.; Johnson, W.P.; Fernandez, D.P.; Manangon, E.; Gale, B.K. Circuit modification in electrical field flow fractionation systems generating higher resolution separation of nanoparticles. J. Chromatogr. A
**2014**, 1365, 164–172. [Google Scholar] [CrossRef] [PubMed] - Choi, J.; Kwen, H.D.; Kim, Y.S.; Choi, S.H.; Lee, S. γ-ray synthesis and size characterization of CdS quantum dot (QD) particles using flow and sedimentation field-flow fractionation (FFF). Microchem. J.
**2014**, 117, 34–39. [Google Scholar] [CrossRef] - Woo, I.S.; Jung, E.C.; Lee, S. Retention behavior of microparticles in gravitational field-flow fractionation (GrFFF): Effect of ionic strength. Talanta
**2015**, 132, 945–953. [Google Scholar] [CrossRef] [PubMed] - Ponyik, C.; Wu, D.; Ratanathanawongs Williams, S. Separation and composition distribution determination of triblock copolymers by thermal field-flow fractionation. Anal. Bioanal. Chem.
**2013**, 1, 1–8. [Google Scholar] [CrossRef] [PubMed] - Gale, B.; Sant, H. Nanoparticle analysis using microscale field flow fractionation. In Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series (San Jose, USA); SPIE: Bellingham, WA, USA, 2007; Volume 6465, p. 18. [Google Scholar]

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Shendruk, T.N.; Slater, G.W. Adverse-Mode FFF: Multi-Force Ideal Retention Theory. *Chromatography* **2015**, *2*, 392-409.
https://doi.org/10.3390/chromatography2030392

**AMA Style**

Shendruk TN, Slater GW. Adverse-Mode FFF: Multi-Force Ideal Retention Theory. *Chromatography*. 2015; 2(3):392-409.
https://doi.org/10.3390/chromatography2030392

**Chicago/Turabian Style**

Shendruk, Tyler N., and Gary W. Slater. 2015. "Adverse-Mode FFF: Multi-Force Ideal Retention Theory" *Chromatography* 2, no. 3: 392-409.
https://doi.org/10.3390/chromatography2030392