Revisiting the Flow-Driven Translocation of Flexible Linear Chains

Nov 13, 2018 - In the strong confinement regime (R/r > λ*), qc is found to be independent of the chain length, consistent with the prediction by clas...
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Revisiting the Flow-Driven Translocation of Flexible Linear Chains through Cylindrical Nanopores: Is the Critical Flow Rate Really Independent of the Chain Length? Tao Zheng,† Mo Zhu,† Jinxian Yang, Jing He, Muhammad Waqas, and Lianwei Li* Department of Chemical Physics, University of Science and Technology of China, Hefei, Anhui 230026, China

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S Supporting Information *

ABSTRACT: We aim to clarify how the chain length influences the critical flow rate (qc) for flexible linear chains translocating through cylindrical nanopores in the whole length range (9.5 > R/r > 1.0), where R and r represent the chain size and the pore size, respectively. By studying the translocation behavior of both mono- and polydisperse polystyrenes through 20 nm nanopores, we have, for the first time, experimentally revealed that there exist two different translocation regimes, i.e., strong and moderate confinement regimes. In the strong confinement regime (R/r > λ*), qc is found to be independent of the chain length, consistent with the prediction by classical theories, while in the moderate confinement regime (R/r < λ*), qc increases with the chain length significantly, where λ* represents the critical relative chain length. Theoretically, λ* is determined by the critical penetration length (l*), at which the free energy change (ΔE) of a translocating chain reaches its maximum value (ΔE ∼ kBT). For longer chains (R/r > λ*), they always reach the position l* of the barrier (Ebar*) before being completely confined in the channel; but for shorter chains (R/r < λ*), they could overcome the energy barrier before reaching l*. By considering the variation of both position (L) and height (Ebar) of the energy barrier in the moderate confinement regime, a normalization equation has been proposed to correlate the normalized critical flow rate (qc/qc*) with the normalized confined length (L/l*) and energy barrier (Ebar/Ebar*), i.e., qc/qc* = (2L/l* − Ebar/ Ebar*)/(L/l*)2, where qc*, l*, and Ebar* are the corresponding physical quantities in the strong confinement regime. The theoretical equation describes our experimental data well. In addition, we discuss the possibility of utilizing single membranebased ultrafiltration for linear polymer fractionation. The experimental result demonstrates that one single membrane is enough to simultaneously tailor the boundaries of low and high molar masses of polymer fractions in ultrafiltration fractionation. As an example, we experimentally show how one can “dynamically” regulate the effective pore size of a given membrane, by properly choosing the flow rates, to fractionate a polydisperse sample (Mw/Mn ∼ 3.25) into a series of monodispersed samples (1.08 ≤ Mw/Mn ≤ 1.30) with different boundary molar masses. The current work not only strengthens our understanding of the lengthdependent translocation behavior of linear polymers but also provides a versatile method for linear chain fractionation.

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INTRODUCTION How flexible linear chains translocate through cylindrical nanochannels with the channel size smaller than the chain size (macromolecule ultrafiltration) is a fundamental and important question in polymer physics. Since the 1960s, Peterlin,1 Casassa,2 de Gennes,3,4 Pincus,5 Daoudi and Brochard,6 and Freed and Wu7,8 have successively theoretically discussed how a flexible linear chain passes through a cylindrical pore. In particular, de Gennes3 and Pincus5 predicted that a linear chain can undergo a coil-to-stretch transition in an elongational flow field at a critical/minimum flow rate (qc), at which the chain can pass through a cylindrical pore with its pore radius (r) much smaller than the chain size (R). In de Gennes’ blob picture, the flow-driven polymer translocation through a nanopore can be regarded as a tunneling process, in which the continuously increased confinement energy (entropic loss due to the reduced chain conformation) has to be overcome by the gained hydrodynamic energy provided by the shearing flow for a successful © XXXX American Chemical Society

translocation event. In the theory, the total confinement energy (Ec,t) is expressed as Ec,t ∼ nbkBT ∼

D2 l kBT ξ3

(1)

where ξ, nb, and l represent the blob size, the blob number (nb = D2l/ξ3), and the penetration length of chain segments into the confining tube, respectively, and kB, T, and D are the Boltzmann’s constant, the absolute temperature, and the pore diameter, respectively. On the other hand, the total hydrodynamic energy (Eh,t), i.e., the total work done by the shearing fluid, can be expressed as Received: September 27, 2018 Revised: November 3, 2018

A

DOI: 10.1021/acs.macromol.8b02082 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules E h,t ∼ −

∫0

l

fh (l) dl ∼ −

ηqLe ξ

3

l2

position l* of the barrier before being completely confined in the channel; but for shorter chains (L < l*), they could overcome the energy barrier before reaching l*, and both the height and the position of energy barrier might be determined by the fully confined length L of a given chain. Physically, both L and l* should be pore size dependent, so the size ratio (λ = R/r) of the chain and nanopore should be very critical. Actually, we anticipate the existence of a critical size ratio λ*, at which the translocation model changes, but such a critical ratio has never been tested experimentally. It is worth mentioning the recent theoretical work by Ledesma-Aguilar et al.,21 in which they provided a thorough discussion about the chain length dependent translocation of linear chains through nanochannels. (2) In experimental aspect, some of previous results seem to support the chain length independency argument, but the λ range is generally not appropriately chosen in related studies. Specifically, the λ values are either too large (10.0 > λ > 5.0)15,22 or unknown23,24 in related reports. Consequently, the obtained results are not convincing, and the dependency relation of the critical flow rate and the chain length still needs to be further clarified experimentally. In the current work, we revisited the flow-driven ultrafiltration of flexible linear chains through nanopores. We aim to experimentally clarify how the chain length influences the critical flow rate qc in the whole length range (9.5 > λ > 1.0). Our previous study of the ultrafiltration of topological polymers lays a solid foundation for present work.17 By measuring the translocation curves of both narrowly and broadly distributed polystyrene (PS) standards passing through a narrowly distributed 20 nm membrane in toluene, we systematically investigated how the translocation probability (Pt) changes with the fluid flow rate and the chain length. The experimental result has shown, for the first time, that there exist two distinct regimes for linear chain translocation, i.e., moderate and strong confinement regimes. Namely, in the strong confinement regime (9.5 > λ > 4.5), qc was found to be indeed chain length independent, but in the moderate confinement regime (4.0 > λ > 1.0), qc was found to gradually decrease as the chain length decreases, due to the decreased energy barrier. Scheme 1 illustrates our new findings in this work. By considering the variation of both position and height of energy barrier in the moderate confinement regime, a modified theoretical model was proposed to explain the discrepancy between our experimental result and the classical theories. Finally, we showed how one can take advantage of the chain length dependent translocation behavior of linear chains (moderate confinement regime), by using only one ultrafiltration membrane with a fixed pore size, to fractionate a polydisperse sample into a series of monodispersed samples.

(2)

where f h(l), q, Le, and η represent the hydrodynamic force (f h = 3πηqLe/D2), the microscopic flow rate, the effective shear length of chain segments along the flow direction, and the solvent viscosity, respectively. Accordingly, the free energy change (ΔE) can be found: ΔE = Ec,t + E h,t ∼ ∼

D2kBT

ηqL yz D2kBT ijj jl − 2 e l 2zzz 3 j j ξ k D kBT z{ ξ

3

l−

ηqLe ξ3

l2

(3)

Mathematically, at a given microscopic flow rate q, the corresponding optimal penetration length (l*) where ΔE has its maximum can be found when dΔE /dl = 0: l* ∼

kBTD2 ηqLe

(4)

Thus, by combining eqs 3 and 4, assuming that the critical energy barrier (Ebar*) is on the order of ∼kBT, de Gennes derived the final expression of the critical flow (qc*) at l = l*: qc* ∼

D4 kBT ξ 3Leη

(5)

Specifically, by considering ξ = Le = D in eq 5, qc* ≃ kBT/η. More importantly, eq 5 predicts that the critical flow rate qc is independent of the chain length. Such a prediction has been supported by numerous theoretical work,3,4,6,8,9 computational simulation,10−12 and experimental observation.13−17 Unfortunately, this prediction simultaneously means that the flowdriven ultrafiltration is not a feasible method for the fractionation/separation of linear polymers with different chain lengths. In contrast, a number of novel nanopore-based characterization methods have been proposed for branched polymers. The core ideas are generally based on the fact that the polymer translocation process strongly depends on the degree of branching. Namely, by exploring the passing/clogging transition across the nanochannel, Sakaue and BrochardWyart proposed a novel method for the characterization of randomly branched polymers using the finite length channel;18 by establishing the relation between the chain topology and the critical flow rate, Wu and Li proposed that the ultrafiltrationbased method is a versatile method for the separation of polymer chains with different branching structures.19,20 However, none of these methods are applicable to linear polymers. The development of feasible nanopore-based separation/characterization methods for linear polymers is still challenging. Note that whether the critical flow rate of linear chains is really independent of the chain length is still an open question. This argument is based on the following facts: (1) In theoretical aspect, one of the most important hypotheses in de Gennes’ scaling argument is that the confined chain always needs to overcome the energy barrier at l = l*, which corresponds to an assumption that the chain size has to be large enough to make its equilibrium/fully confined length (L) longer than the theoretically optimal penetration length l*. Obviously, for longer chains (L > l*), they always reach the

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EXPERIMENTAL SECTION

In an ultrafiltration study, four narrowly distributed linear polystyrene samples (PSN) with different peak molar masses (Mp) and one broadly distributed sample (PSB) were used. The SEC characterization results are summarized in Figure S1 and Table 1. The shortest PS chains (PSN‑9K, Mp = 9.00 × 103 g/mol) can act as internal references in solutions to increase the measuring accuracy of translocation curves of large PS chains (PSN‑400K, Mp = 4.00 × 105 g/mol; PSN‑1030K, Mp = 1.03 × 106 g/mol; PSN‑3500K, Mp = 3.50 × 106 g/mol). To minimize the influence of nanoporous membrane, the state-of-art anodic aluminum oxide membranes (AAO, from Whatman, D = 20 nm and r = 10 nm) owning a precise bilayered cylindrical pore structure with nearly no lateral crossovers between individual pores were used as ultrafiltration membranes. The detailed B

DOI: 10.1021/acs.macromol.8b02082 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules Scheme 1. Schematic Illustration of the Flow-Driven Translocation of a Flexible Linear Chain through a Cylindrical Channel in the Strong Confinement and Moderate Confinement Regimes

Table 1. Molecular Parameters of Polystyrene Samples Used in Experiments sample PSN‑9K PSN‑400K PSN‑1030K PSN‑3500K PSB‑285K

Mpa (g/mol) 9.00 4.00 1.03 3.50

× × × ×

103 105 106 106

Mwa (g/mol)

Mw/Mna

× × × × ×

1.02 1.05 1.05 1.08 2.48

9.30 4.30 1.06 3.71 2.85

103 105 106 106 105

⟨Rg⟩b (nm)

λ

∼2.7 26.0 45.0 95.0

∼0.27 2.6 4.5 9.5

Figure 1. Molar mass dependence of (a) the average radius of gyration (⟨Rg⟩) and (b) the overlap concentration (Coverlap) of linear polystyrene chains in toluene at T = 20 °C,25 where the data of PS samples used in experiments were also plotted for comparison.

a

Mp, Mw, and Mw/Mn were obtained based on PS standard-calibration method, where Mp, Mw, and Mn represents the peak molar mass, weight-average molar mass, and number-average molar mass, respectively. b⟨Rg⟩ was estimated based on the Rg−M scaling for linear PS in toluene: Rg/nm =1.23 × 10−2M0.594.25

extruded and collected for SEC measurement. Note that the ultrafiltration membranes composed of a single-layered cylindrical channel are widely used in industrial applications and related studies;13,14,22−24 therefore, the polymer solution was directly extruded through the small cylindrical pores (active layer, D = 20 nm) without first passing through the large cylindrical pores (support layer, D = 200 nm) of AAO membrane. In this way, the bilayered AAO membrane can be simplified as a special single-layered membrane, which makes our results more comparable with the results obtained by other groups.13,14,22−24

specification and characterization of AAO membrane can refer to Whatman and our previous work.17 The precise pore structure and narrow pore size distribution of AAO membrane ensure the sharp cutoff molar mass/size, which lays a solid foundation for this study. The effective filtration area of membranes used in this study is ∼3.9 cm2. Specifically, the average radius of gyration (⟨Rg⟩) of each PS sample was used for the calculation of the size ratio (λ = ⟨Rg⟩/r) of the chain and nanopore (Figure 1a). The λ values for different PS samples are summarized in Table 1. In experiments, the polymer concentrations (C) were fixed at ∼0.2 g/L for PSN samples in ∼20 mL of stock solution (Figure 1b), which are at least ∼1 order of magnitude smaller than their critical overlap concentrations (C*). The macroscopic flow rate (Q) was controlled by a syringe pump (Harvard Apparatus, PHD 2000), and the temperature was controlled to be 20 ± 1 °C. A mechanical stirring was applied at a stirring rate of 100 rpm during the extrusion to suppress the concentration polarization. The experimental setup for ultrafiltration study is shown in Figure S2. The translocation probabilities (Pt) of long PS chains were determined by measuring the normalized relative concentrations Pt = C/C0 by size exclusion chromatography (SEC) in THF, where C0 and C are the polymer concentrations normalized by short polystyrene chains in the retentate and permeate solutions, respectively. The measurement accuracy for Pt is ±3%. At a given flow rate, ∼0.2 mL of solution was

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RESULTS AND DISCUSSION Experimental Observation of the Chain Length Dependent Translocation Behavior for Linear Chains in the Moderate Confinement Regime. First, to test whether the chain length independent translocation behavior for linear chains is valid or not in the whole chain length range, we directly measured the translocation curves of a solution mixture containing three narrowly distributed PS samples with different λ values under different experimental flow rates. As shown in Figure 2a, when the macroscopic flow rate Q is < 0.50 mL/h, only PSN‑9K chains (λ = 0.27) could pass through the nanopores; as Q further increases to 1.00 mL/h, nearly half of PSN‑400K (λ = 2.6) chains in stock solution could pass through nanopores, but longer PSN‑1030K (λ = 4.5) and PSN‑3500K chains (λ = 9.8) are still completely retained, which clearly indicates that the chain length independency is not C

DOI: 10.1021/acs.macromol.8b02082 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules

Figure 2. (a) SEC curves of polymer solutions of linear PS mixture after passing through an AAO membrane with r = 10 nm in toluene at T = 20 °C under different macroscopic flow rates (Q), where PSN‑9K was used as internal reference. (b) Macroscopic flow rate (Q) dependence of the translocation probabilities (Pt) of PSN‑400K, PSN‑1030K, and PSN‑3500K chains passing through an AAO membrane with r = 10 nm in toluene at T = 20 °C.

Figure 3. (a, b) SEC curves of polymer solutions of one broadly distributed PS sample (PSB‑285K) after passing through an AAO membrane with r = 10 nm in toluene at T = 20 °C under different macroscopic flow rates (Q), where the molar mass information can be converted in to the size information based on the equation Rg = 1.23 × 10−2M0.594 (in toluene),25 and the dashed lines in (b) represent the tangent lines for each curves.

always valid in the whole chain length range, different than the theoretical prediction. Figure 2b further shows how the translocation probability (Pt) quantitatively increases as Q increases for linear chains with different lengths. Obviously, there is no big difference for PSN‑1030K and PSN‑3500K even though PSN‑1030K chains are more than 2 times larger than PSN‑1030K chains. This is reasonable because both the position and height of the energy barrier are actually irrelevant with the chain length in the strong confinement regime. In contrast, Figure 2b shows that PSN‑400K chains are much easier to pass through nanopores at Q = 1.00 mL/h, and a big gap exists between the translocation curves of PSN‑400K and PSN‑1030K/ PSN‑3500K. The result strongly implies that PSN‑400K (λ = 2.6) might not be long enough to ensure its fully confined length L larger than the optimal penetration length l* assumed in theory,3,4,6,8,9 which leads to a much smaller qc required to drag PSN‑400K chains into nanopores. The preliminary result directly confirms the chain length dependent translocation behavior for linear chains in the moderate confinement regime. To fully confirm the existence of the moderate confinement regime experimentally, we further studied the translocation behavior of one polydisperse sample (PSB‑285K) with the molar mass from ∼3.00 × 104 to ∼1.00 × 106 g/mol and λ from 0.56 (Rg = 5.6 nm) to 4.5 (Rg = 45.0 nm). In principle, by using one polydisperse sample, it is much easier to monitor the chain length dependent translocation behavior in a continuous mode. Figure 3a clearly shows that the SEC curve of permeate

solution of PSB‑285K gradual shifts to the higher molar mass range as Q increases. Note that under extremely weak fluid shearing (Q ≤ 0.50 mL/h), the polymer translocation should be a diffusion-controlled process, so the observed narrow transition from full transmission (λ ∼ 1.1) to full retention (λ ∼ 2.2) actually signifies a narrow pore size distribution of used AAO membrane. It is also worth emphasizing that even if the pore size is monodispersed, we would still anticipate the existence of a transition width for the diffusion-limited translocation due to the diversity of chain conformation.26 By further converting the molar mass into the chain size, Figure 3b shows that for a given flow rate (0.75 mL/h < Q < 4.00 mL/h) Pt significantly decreases as the chain length increases when ∼1.0 < λ < ∼4.0. Considering the unavoidable peak broadening effect in SEC measurements, the SEC data in the tail range (4.0 < λ < 6.0) with poor signal-to-noise ratio are not reliable and will not be adopted for analysis. Clearly, the results from mono- and polydisperse samples cross-validate the chain length dependency for the polymer translocation in the moderate confinement regime. Interestingly, we noticed that the estimated macroscopic critical flow rates (Qc) for linear chains with λ = 2.6 are different in Figures 2b and 3b. Namely, Qc at Pt = 0.5 is ∼1.00 mL/h in Figure 2b, while Qc at Pt = 0.5 is ∼1.80 mL/h in Figure 3b. Note that the only difference lies in the polymer sample used in ultrafiltration study. A solution mixture of three D

DOI: 10.1021/acs.macromol.8b02082 Macromolecules XXXX, XXX, XXX−XXX

Macromolecules monodispersed samples (PSN‑400K, PSN‑1030K, and PSN‑3500K) was used for Figure 2b, but a solution of polydisperse sample (PSB‑285K) was used for Figure 3b. In fact, both of the two tested samples are a mixture of chains with different chain lengths. The repeated experiments reconfirmed that the use of monodispersed PSN‑400K always led to a higher experimental Qc value. How does one explain such a discrepancy? Note that only a single pore and a single chain are considered in theory in which there are only two opposite situations: passing or not passing. In a real ultrafiltration experiment, there are many pores on the membrane and many chains in the solution. For the solution mixture of monodispersed samples, when the macroscopic flow rate is on the order of magnitude of the threshold for PSN‑400K, the flow rate is still not high enough to stretch/deform the extremely long chains (PSN‑1030K and PSN‑3500K) at the entrance of nanopores so that they start to temporarily block/clog them. This will lead to an increase of microscopic flow rate in those unblocked pores because the macroscopic flow rate is a constant in each of our experiments. Therefore, even though the macroscopic flow rate is lower than the real threshold for PSN‑400K, the microscopic flow rate in those unblocked pores can still reach the threshold to flush the PSN‑400K chains through nanopores. In contrast, the extremely long chains are nonexistent in the PSB‑285K solution, and the probability for nanopore clogging should be greatly reduced. Therefore, it is not difficult to understand the observed difference for Qc values in Figures 2b and 3b. Unfortunately, the possible pore blocking is an inevitable question when we are dealing with extremely long chains in the ultrafiltration study. Nevertheless, the result from polydisperse PSB‑285K, obviously, could reflect the translocation behavior of linear chains in the moderate confinement regime in real situation more objectively, which is preferred to be used in the following discussion. Overall, the observation for polydisperse PSB‑285K unambiguously demonstrates that the critical flow rate for linear chains indeed strongly depends on the chain length in the moderate confinement regime, qualitatively consistent with our observation for monodispersed samples. On the basis of the combined results for mono- and polydisperse samples, it is safe to estimate that the value for λ* separating the strong and moderate confinement regimes is around 3.5−4.0. Theoretical Modeling of the Chain Length Dependent Critical Flow Rate for Linear Chains in the Moderate Confinement Regime. Before further analyzing our experimental data, we need to discuss theoretically in advance the origin of why Qc increases with the chain length in the moderate confinement regime. Here, we present a modified theoretical description on the basis of the previous discussion by Ledesma-Aguilar et al.21 To drop the ∼ symbols used in de Gennes’ scaling argument and have a more quantitative discussion, we first introduce two numerical prefactors A and B in the expressions of confinement energy (eq 1) and hydrodynamic energy (eq 2), respectively: Ec,t = A

E h,t = B

D2kBT ξ3 ηqLe ξ

3

l2

l

ΔE = A

l* =

Article

D2kBT jij B ηqLe 2yzz jl − l zz 3 j j A D2kBT z{ ξ k

(8)

2 A kBTD 2B ηqLe

(9)

By further assuming that the critical energy barrier Ebar* is on the order of ∼kBT, i.e., Ebar* = xkBT, where x is the prefactor, qc* can be extracted by putting l = l* into eq 8: qc* =

4 A2 D kBT 4xB ξ 3Leη

(10)

A combination of eqs 9 and 10 leads to a simple expression of l*: 2 4 A kBTD ijj A2 D kBT yzz jj zz lc* = 2B ηL B jk 4Bx ξ 3Leη z{

−1

=

2x ξ 3 A D2

(11)

On the other hand, by defining the normalized penetration length l/l* and the normalized microscopic flow rate q/qc*, ΔE can be further represented by quadratic function of l/l* in a more concise manner: É ÄÅ 2Ñ ÑÑ ÅÅÅ l i y q l 2 yzz q 2xkBT ijj 1 l j z jj zz ÑÑÑÑ jjl − zz = E bar*ÅÅÅ2 ΔE = − j z ÅÅ l * 2 qc* lc* zz lc* jj qc* jk lc* z{ ÑÑÑ ÅÅÇ c k { ÑÖ (12)

Theoretically, both l* and qc* are constant parameters when the type of polymer solution and the size of nanopore are fixed for a given system. Figure 4 schematically shows how ΔE

Figure 4. Normalized penetration length (l/l*) dependence of the free energy change (ΔE) at different normalized microscopic flow rates (q/qc*), where the curves for q/qc* = 0.4, 0.8, 1.0, and 1.4 were plotted for comparison and Ebar* was assumed to be constant.

changes with l/l* at different q/qc* values based on eq 12. It is not difficult to understand why the translocation can always occur when q/qc* ≥ 1.0 (two square symbols in Figure 4) because the maximum value of the quadratic function for ΔE is always smaller than Ebar* (xkBT) in the whole penetration length range (purple rectangular region), irrelevant to the chain length, which corresponds to the strong confinement regime. However, when q/qc* < 1.0 (triangle and circle symbols in Figure 4), the relative values of ΔE and Ebar* depend closely on the penetration length l. Mathematically, if we still assume the energy barrier to be constant (xkBT) when q/qc* < 1.0, the normalized penetration length l/l* has to be

(6)

(7)

Following a similar scaling derivation, eqs 3 and 4 can be accordingly rewritten as E

DOI: 10.1021/acs.macromol.8b02082 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules smaller than (1 − 1 − q/qc* )/(q/qc*). Thus, for a given chain, if its fully confined length L is smaller than [(1 − 1 − q/q * )/(q/qc*)]l*, the chain can still pass through the c

nanopore under a much weaker fluid shearing (qc < qc*). Accordingly, the energy barrier can be overcome at l = L, much smaller than the strong confinement limit at l = l*. In principle, both the position and height of the energy barrier will vary with the chain length in the moderate confinement regime. By assuming a smaller energy barrier Ebar (Ebar ≤ Ebar*), and letting ΔE = Ebar in eq 12, the scaling relation between qc/qc* and L/l* can be derived as qc 2L /lc* − E bar /E bar* = q* (L /lc*)2 c

(13)

Equation 13 quantitatively describes how the normalized critical flow rate changes with the normalized fully confined length in the moderate confinement regime. Theoretically, it is more important to correlate the normalized critical flow rate qc/qc* and the relative chain size λ because they are the measurable characteristic quantities in a real experiment. For a chain confined in a cylindrical tube, its fully confined length L is essentially determined by the chain size R in good solvents and the pore size D, irrelevant with the chain topology:4 L i 2R y ≃ jjj zzz D kD{

5/3

≃ λ 5/3

Figure 5. (a) Relative chain size (λ = Rg/r) dependence of the translocation probability (Pt) of PS linear chains (PSB‑285K) under different macroscopic flow rates. (b) Macroscopic flow rate (Q) dependence of the translocation probability (Pt) of PS linear chains (PSB‑285K) with different λ values.

(14)

Mathematically, L = l* corresponds to λ = λ*, which further gives L/l* = (λ/λ*)5/3. Thus, eqs 12 and 13 can be rewritten in terms of λ/λ*: É ÅÄÅ 10/3Ñ ÅÅ ij λ yz5/3 ij λ yz ÑÑÑÑ q Å j z j z z − jj zz ÑÑ ΔE = E bar*ÅÅÅ2jj ÅÅ j λc* zz qc* jk λc* z{ ÑÑÑÑ ÅÅÇ k { (15) ÑÖ qc 2(λ /λ*)5/3 − E bar /E bar* = q* (λ /λ*)10/3 c

polymer fraction with a specific λ in permeate solution and stock solution, respectively. By considering that the polymer concentrations of stock solutions before and after extrusion show only a minor change (5%−7%) due to the solute accumulation effect (Figure S4), the normalization calculation is directly based on the signal of stock solution before extrusion. Figure 5a shows how the macroscopic critical flow rate Qc at a given Pt increases with the relative chain size λ. Figure 5b further shows the evolution of translocation curve as a function of λ. Obviously, as λ increases from 1.25 to 3.50, the translocation curve become less dependent on λ, indicating that λ gradually approaches the limit λ*. In theory, the translocation flow rate determined at an arbitrary translocation probability Pt can be defined as the critical flow rate. Figure S5 shows how the determined Qc correlates with λ at different Pt values (Pt = 0.8, 0.5, and 0.2), which demonstrates that the dependency between Qc and λ is a universal relation, regardless of the selection of Qc at an arbitrary Pt. In the fitting, λ* = 3.5 and qc determined at λ = 3.5 were used as λ* and qc*, respectively. Based on our experimental results of mono- and polydisperse samples, the above estimation for λ* and qc* would be accurate enough for the data fitting. We first compared the experimental data and the theoretical fitting by assuming a constant energy barrier (Ebar/Ebar* = 1.0) in the moderate confinement regime. Figure S6 clearly shows that by normalizing Qc, the dispersed curves in Figure S5 collapse into a narrow region, indicating that the normalized critical flow rate Qc/Qc* (or qc/qc*) is a characteristic quantity for the description of the polymer translocation in the moderate confinement regime. The fitted curve (dashed line

(16)

Therefore, we have, for the f irst time, established the quantitative correlation between the normalized characteristic quantities qc/qc* and λ/λ* for the flow-driven polymer translocation in the moderate confinement regime. Clearly, qc/qc* is determined by the relative chain size λ/λ* and the relative height of energy barrier Ebar/Ebar*. In the limit when λ = λ* and Ebar = Ebar* = xkBT, qc = qc*; when λ < λ*, qc increases monotonously with λ (Figure S3). The correlation equation not only is very concise but also owns clear physical meaning. Note that for a given system qc* is a constant and can be determined in the strong confinement regime, λ is known for a given polymer solution, and qc can be determined for a given polymer solution in ultrafiltration experiments. Therefore, we can quantitatively fit our experimental data according to eqs 15 and 16 to test whether the theoretical model captures the core issues of linear chain translocation in the moderate confinement regime. Comparison between Experimental Result and Theoretical Derivation. To better analyze the experimental data in the moderate confinement regime, we converted the original SEC curves (Figure 3b) into the corresponding translocation probability curves (Figure 5a) by Pt = Ip/Is, where Ip and Is represent the signal intensities for a given F

DOI: 10.1021/acs.macromol.8b02082 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules

unambiguously confirms the chain length dependent translocation behavior for the ultrafiltration of flexible linear chains in the moderate confinement regime. The results also reveal that both the penetration length and the energy barrier are functions of the chain length. To further test whether the interaction among flow fields and the preconfinement effect, discussed in our recent work,27 play decisive roles in dominating the chain length dependency for linear chain translocation, control experiments were carried out. By reversing the two faces of the anisotropic AAO membrane in holder, we measured the translocation curves of PSB‑285K chains successively passing through the large cylindrical pores (D = 200 nm) and small cylindrical pores (D = 20 nm), as shown in Figure S7. The analysis results clearly show that the translocation probability of linear chains depends on both the flow rate and the chain length in the reversed translocation mode. The control experiment demonstrates that the chain length dependent translocation behavior in the moderate confinement regime is a universal phenomenon for linear chains, no matter whether the flow-field interaction/preconfinement effect was taken into consideration. Combined with the observation in our previous work,27 we conclude that only the transition width and the magnitude of critical flow rate, rather than the chain length dependency, could be significantly influenced by the flow-field interaction/ preconfinement effect in the whole chain length range. Single Membrane-Based Fractionation/Separation of Polydisperse Samples. Based on the above result, one potential and attractive application is to utilize the chain length dependent translocation for the fractionation/separation of linear synthetic polymers. Actually, the possibility of using the flow-driven ultrafiltration process for linear chain fractionation/separation has been considered in some recent computational simulation work. For example, using a hybrid simulation method, Ding et al. found that different than linear polymers, the critical flow rate for ring polymers decreases slowly with the increase of the chain length, indicating that the separation of linear and ring polymers is feasible;12 their follow-up work further demonstrated that the critical flow rate for linear chains increases with the enhancement in the polymer insolubility, which establishes a principle for the separation of linear chains with different lengths.28 Unfortunately, none of these have been tested experimentally so far. Literature search shows that the applicability of ultrafiltration technique to the fractionation of linear synthetic polymers is very limited.29−31 Generally, a combination of ultrafiltration membranes with different cutoff sizes is necessary to produce/retain a medium molar mass polymer fraction. Moreover, the 3-dimensional ultrafiltration membranes with poorly defined pore geometries and broad pore size distributions are generally adopted in real ultrafiltration industry, which makes the separation of polymer fractions extremely difficult if their molar masses are within ∼1 order of magnitude. More importantly, the separation process is mainly based on the size separation principle, i.e., the static relative size of polymer chains and nanopores, which means that only the polymer fractions with their chain sizes smaller than the pore size can be removed by using a single membrane with a fixed pore size. It is not difficult to realize that the confirmation of the chain length dependent translocation behavior in the moderate confinement regime endows the nanopores with “dynamic” features; namely, we are actually able to vary the effective

in Figure S6) seems to agree well with the experimental data in the range 0.80 < λ/λ* < 1.00; however, a strong deviation exists when 0.40 < λ/λ* < 0.80, which indicates that the assumption of a constant energy barrier Ebar = Ebar* is not accurate enough. According to eq 15, both the position and height of Ebar are supposed to vary with λ/λ* in the moderate confinement regime. Thus, we further calculated the true values for Ebar based on our experimental data. As shown in Figure 6a, Ebar/Ebar* is almost constant in the range 0.80 < λ/

Figure 6. (a) Normalized relative chain size (λ/λ*) dependence of the calculated energy barrier (Ebar) based on eq 15. (b) Comparison of experimental and fitted normalized relative chain size (λ/λ*) dependence of the normalized macroscopic critical flow rate (Qc/ Qc*), where Qc at λ = 3.5 was used as Qc* and λ* = 3.5.

λ* < 1.00 but significantly decreases from ∼1.0 to ∼0.5 as λ/λ* decreases from ∼0.80 to ∼0.40, which explains the origin of deviation in the fitting in Figure S6. Mathematically, the calculated Ebar can be approximately expressed as Ebar = Ebar* = xkBT when 0.80 < λ/λ* < 1.00 and Ebar = Ebar*[11/8(λ/λ*) − 1/10] when 0.40 < λ/λ* ≤ 0.80. Thus, eq 16 can be rewritten as qc 2(λ /λ*)5/3 − 1 = qc* (λ /λ*)10/3

0.80 < λ /λ* < 1.00 (17)

qc 2(λ /λ*)5/3 − [11/8(λ /λ*) − 1/10] = qc* (λ /λ*)10/3 0.40 < λ /λ* < 0.80

(18)

Clearly, by considering the variation of energy barrier, the theoretical model satisfactorily describes our experimental data, as shown in Figure 6b. The above cross-validation based on the comparison of experimental and theoretical results G

DOI: 10.1021/acs.macromol.8b02082 Macromolecules XXXX, XXX, XXX−XXX

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Scheme 2. Schematic Illustration of the Principle of Semicontinuous Ultrafiltration Fractionation for a Polydisperse Sample under a Fixed Flow Rate, Where Three Repeated Ultrafiltration Cycles Are Necessary in One Experiment for the Complete Removal of Low Molar Mass Polymer Fractions

Figure 7. (a) SEC curve of the stock solution of the tested polydisperse sample with Mw/Mn ∼ 3.25. (b) SEC curves of the retentate solutions during the repeated ultrafiltration separation processes. (c) SEC curves of the permeate solutions during the repeated ultrafiltration separation processes. (d) Ultrafiltration cycle number (Nu) dependence of the molecular parameters for polymer fractions in retentate solutions.

cutoff size for a given membrane during fractionation process, by adjusting the operation flow rate. Thus, it is possible to realize the fractionation of a medium molar mass polymer fraction by using only a single membrane with fixed pore size. In principle, one just needs to properly choose the flow rates to successively remove the low and high molar mass polymer fractions from a polydisperse sample in real experiments. To test such an idea, the single membrane-based ultrafiltration

fractionation for a highly polydisperse sample was experimentally carried out. In experiments, a lab-scale semicontinuous ultrafiltration separation protocol was adopted, and the operation principle is illustrated in Scheme 2. Briefly, in each cycle of ultrafiltration separation, four-fifths of the stock solution (20 mL) is extruded through the ultrafiltration membrane, and both the retentate (∼4 mL) and permeate (∼16 mL) solutions are collected. The collected retentate H

DOI: 10.1021/acs.macromol.8b02082 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules solution is further diluted to a new stock solution (20 mL) for the second cycle of ultrafiltration separation. Typically, at least three repeated ultrafiltration cycles are necessary under a fixed flow rate to completely remove the low and high molar mass polymer fractions (Scheme 2). The three permeate and three retentate solutions are quantitatively characterized by SEC. Figure 7a shows the SEC curve for the tested polydisperse sample (Mn ∼ 7.38 × 104 g/mol and Mw/Mn ∼ 3.25). Figure 7b shows the evolution of SEC curve for retentate solution as a function of the cycle number of ultrafiltration separation (Nu). First, a small flow rate Q ∼ 0.75 mL/h was used to ensure that the separation follows the static size separation principle; i.e., the chain deformation is not considered. Obviously, the mass fraction of low molar mass polymer fractions (Vretention > 23.0 mL) significantly decreases as Nu increases. Compared with the stock solution, the SEC peak for retentate solution becomes much narrower as Nu increases. A clear cutoff boundary located at Vretention ≃ 24.0 mL emerges in the SEC curve for retentate solution after the third ultrafiltration cycle (purple curve in Figure 7b), corresponding to an effective cutoff molar mass ∼1.10 × 105 g/mol and cutoff size Rg ∼ 12.0 nm. On the other hand, Figure 7c qualitatively shows that the polymer concentration in permeate solution decreases dramatically as Nu increases, indicating that the repeated ultrafiltration separation is indeed, necessary to completely remove the low molar mass fractions. It is worth noting that the high molar mass boundaries in the SEC curves for three permeate polymer solutions are highly consistent with each other (Vretention ∼ 23.0 mL), indicating a constant effective cutoff size of membrane during multiple separation processes. Figure S8 quantitatively shows how the relative concentration (Cr/Cs) of each polymer fraction in retentate solution varies with the ultrafiltration cycle number Nu, where Cr and Cs are the absolute concentration for a specific polymer fraction in retentate solution and stock solution, respectively. By taking Vretention = 24.4 mL as a reference point, Cr/Cs decreases from 100% to ∼28%, ∼12%, and ∼3% after 1, 2, and 3 cycles of ultrafiltration separation, respectively. Figure 7d summarizes how the molecular parameters of polymer chains in retentate solution change with Nu. Experimentally, Mn dramatically increases from ∼7.38 × 104 to ∼2.96 × 105 g/mol and Mw/Mn decreases from ∼3.25 to ∼1.30, demonstrating the complete removal of low molar mass polymer fractions during the repeated separation process. In this experiment, the polymer fraction in the retentate solution is the target fractionated product, while the polymer fraction in the permeate solution is the side product. Clearly, by using the “static” separation principle, we can tailor the low molar mass boundary of target polymer fraction. To further tailor the boundaries of low and high molar masses of the fractionated product (purple curve in Figure 7a), we could “dynamically” change the effective cutoff size of the membrane by operating the ultrafiltration separation under higher flow rates. As shown in Figure 8, at Q2 ∼ 1.50 mL/h, some polymer chains with their sizes larger than the static cutoff size of nanopores can deform themselves to squeeze through the nanopores under enhanced flow shearing, making the low molar mass boundary of retentate solution gradually move to the high molar mass direction. After three cycles of ultrafiltration purification at Q2 = 1.50 mL/h, the low molar mass boundary of polymer fraction varies from ∼1.08 × 105 to ∼1.80 × 105 g/mol (Figure 8, green curves), which is consistent with our earlier observation in Figure 3. It is worth

Figure 8. SEC curves for retentate and permeate solutions during the repeated ultrafiltration separation processes under different macroscopic flow rates, where Q1 = 0.75 mL/h, Q2 = 1.50 mL/h, and Q3 = 2.00 mL/h.

noting that either the polymer fraction in the permeate solution or retentate solution could be the target product. Therefore, we further increased the flow rate to Q3 = 2.00 mL/ h in the purification process; in this way, the high molar mass boundary of polymer fraction in permeate solution could be tailored (Figure 8, red curve). Eventually, via single membranebased repeated ultrafiltration fractionation, an extremely narrowly distributed fraction (Mn ∼ 3.85 × 105 g/mol) could be obtained, with molar mass from ∼1.80 × 105 to ∼7.20 × 105 g/mol and Mw/Mn ∼ 1.08. Overall, the above example clearly demonstrates, for the f irst time, that a single membrane is enough for linear polymer fractionation. Practically, by properly choosing the flow rate, both the low and high molar mass boundaries of target polymer fractions can be arbitrarily tailored in ultrafiltration fractionation process as long as the relative chain size λ is smaller than λ* to ensure that the translocation behavior is in the moderate confinement regime. Compared with the liquid chromatography fractionation techniques, the membrane-based ultrafiltration technique shows numerous advantages for industrial applications such as economy, simplicity, speed, and so forth. The pioneering work by Daoudi and Brochard theoretically predicted that the critical flow rate qc for linear chains should be scaled to the polymer concentration C as qc ∼ C0 in dilute solution and qc ∼ C15/4 in semidilute solution.6 This prediction implies that the maximum polymer concentration in ultrafiltration fractionation could reach the overlap concentration limit (C*). By using PSN‑1030K as a tested sample, we further examined whether the translocation probability is independent of the polymer concentration in dilute solution region. As shown in Figure 9, 95%−100% of PSN‑1030K chains are retained when C ≤ 5.0 g/ L, but only ∼30% of PSN‑1030K chains are retained when C = 8.0 g/L under the same flow rate. Figure 9 shows the plot of the translocation probability Pt versus the normalized concentration (C/C*). Clearly, the transition occurs between C* and 2C*, qualitatively consistent with the theoretical prediction by Daoudi and Brochard.6 Consequently, by assuming the maximum limit of polymer concentration to be C* in ultrafiltration fractionation, the processing capacity (Cp) can be estimated as Cp = C* × Q, where C*/(g/L) = 2.11 × 10−5M−0.782 for polystyrene in toluene and Q is 1.00−3.00 mL/ I

DOI: 10.1021/acs.macromol.8b02082 Macromolecules XXXX, XXX, XXX−XXX

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Figure 9. Normalized polymer concentration (C/C*) dependence of the translocation probability (Pt) for PSN‑1030K solution at a fixed macroscopic flow rate Q = 1.00 mL/h, where C* = 4.20 g/L for PSN‑1030K in toluene, and the inset shows the original SEC curves of permeate solutions.

h in this study (the filtration area of membrane is ∼3.9 cm2). Numerically, Cp is estimated to be 4.2−12.6 mg/h for M ≃ 106 g/mol (C* ≃ 4.2 g/L) and 26−78 mg/h for M ≃ 105 g/mol (C* ≃ 26 g/L). The result indicates that the processing capacity of single-membrane ultrafiltration fractionation is 2−3 orders of magnitude higher than the standard liquid chromatography methods, which is especially attractive for the production of standard polymers in large scale. By increasing the filtration area of ultrafiltration membrane, the processing capacity can be further enhanced. Figure 10a shows how the concentration limit of polymer solution used in ultrafiltration fractionation relies on the fractionation model. Figure 10b schematically illustrates the basic principles of how we can set up semicontinuous and continuous singlemembrane fractionation apparatuses for the fractionation of linear polymer samples in the laboratory.

Figure 10. (a) Molar mass dependence of the concentration limit (Climit) of polymer solution used in single-membrane fractionation, where the dashed and solid lines represent the semicontinuous fractionation model and continuous fractionation model, respectively. (b, c) Schematic diagrams of semicontinuous fractionation system and continuous fractionation system. For the semicontinuous fractionation system, the volume of filtration cell decreases as the fractionation process proceeds, and for continuous fractionation system, the volume of filtration cell is constant.

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CONCLUSION In summary, by a combination of experimental observation and theoretical derivation, we have clarified the existence of two regimes for the flow-driven ultrafiltration of flexible linear chains through cylindrical nanopores. In the strong confinement regime, the critical flow rate qc is indeed independent of the chain length, which agrees well with the prediction by classical theories; in the moderate confinement regime, the critical flow rate qc significantly increases with the chain length. Our result reveals that both the position (penetration length) and height of the energy barrier are chain length dependent in the moderate confinement regime, which is the origin for the chain length dependent translocation behavior. By considering the variation of energy barrier, our proposed theoretical model could satisfactorily describe the experimental data, further cross-validating our experimental findings. Quantitatively, the measured critical size ratio λ* (λ = R/r) separating the two regimes is around 3.5−4.0, indicating that it is possible to fractionate linear samples (λ < 3.5−4.0) by using the single membrane-based ultrafiltration method in a real experiment. In addition, we demonstrate that one can effectively fractionate a poly dispersed sample (Mw/Mn = 3.25) into a series of monodispersed samples (Mw/Mn = 1.08−1.30) by properly choosing the flow rates. Experimentally, both the boundaries of low and high molar masses of the fractionated products can be

tailored during the fractionation process. In principle, the use of a single membrane with “dynamic pore” size is enough for linear chain fractionation as long as 1.0 < λ < λ*. The result also demonstrates that the maximum limit of polymer concentration used in ultrafiltration fractionation could reach the critical overlap concentration for a given system. From a practical point of view, the merit of high processing capacity for ultrafiltration fractionation shows greater advantage compared with the widely used liquid chromatography fractionation techniques such as SEC and LCCC. We hope this work could not only be helpful for the theoretical understanding of the chain length dependent translocation behavior of linear chains but also motivate to design and manufacture novel instruments for polymer characterization/ separation. J

DOI: 10.1021/acs.macromol.8b02082 Macromolecules XXXX, XXX, XXX−XXX

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(15) Jin, F.; Wu, C. Observation of the First-Order Transition in Ultrafiltration of Flexible Linear Polymer Chains. Phys. Rev. Lett. 2006, 96, 237801. (16) Ge, H.; Jin, F.; Li, J.; Wu, C. How Much Force Is Needed To Stretch a Coiled Chain in Solution? Macromolecules 2009, 42, 4400− 4402. (17) Li, L.; Chen, Q.; Jin, F.; Wu, C. How does a polymer chain pass through a cylindrical pore under an elongational flow field? Polymer 2015, 67, A1−A13. (18) Sakaue, T.; Brochard-Wyart, F. Nanopore-Based Characterization of Branched Polymers. ACS Macro Lett. 2014, 3, 194−197. (19) Ge, H.; Wu, C. Separation of Linear and Star Chains by a Nanopore. Macromolecules 2010, 43, 8711−8713. (20) Li, L.; He, C.; He, W.; Wu, C. How Does a Hyperbranched Chain Pass through a Nanopore? Macromolecules 2012, 45, 7583− 7589. (21) Ledesma-Aguilar, R.; Sakaue, T.; Yeomans, J. M. Lengthdependent translocation of polymers through nanochannels. Soft Matter 2012, 8, 1884−1892. (22) Beguin, L.; Grassl, B.; Brochard-Wyart, F.; Rakib, M.; Duval, H. Suction of hydrosoluble polymers into nanopores. Soft Matter 2011, 7, 96−103. (23) Nguyen, Q. T.; Neel, J. Characterization of ultrafiltration membranes.: Part IV. Influence of the deformation of macromolecular solutes on the transport through ultrafiltration membranes. J. Membr. Sci. 1983, 14, 111−127. (24) Beerlage, M. A. M.; Heijnen, M. L.; Mulder, M. H. V.; Smolders, C. A.; Strathmann, H. Non-aqueous retention measurements: ultrafiltration behaviour of polystyrene solutions and colloidal silver particles. J. Membr. Sci. 1996, 113, 259−273. (25) Teraoka, I. Polymer Solutions: An Introduction to Physical Properties.; John Wiley & Sons: New York, 2002. (26) Bishop, M.; Saltiel, C. J. The distribution function of the radius of gyration of linear polymers in two and three dimensions. J. Chem. Phys. 1991, 95, 606−607. (27) Zheng, T.; Yang, J.; He, J.; Li, L. Origin of Inconsistency in Experimentally Observed Transition Widths and Critical Flow Rates in Ultrafiltration Studies of Flexible Linear Chains. Macromolecules 2018, 51, 6504−6512. (28) Ding, M.; Duan, X.; Shi, T. Flow-induced polymer separation through a nanopore: effects of solvent quality. Soft Matter 2017, 13, 7239−7243. (29) Barker, P. E.; Alsop, R. M.; Vlachogiannis, G. J. Fractionation, purification and concentration of dextran solutions by ultrafiltration. J. Membr. Sci. 1984, 20, 79−91. (30) Blatt, W. F. Membrane partition chromatography: A tool for fractionation of protein mixtures. J. Agric. Food Chem. 1971, 19, 589− 594. (31) Baker, R. W. Methods of fractionating polymers by ultrafiltration. J. Appl. Polym. Sci. 1969, 13, 369−376.

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.macromol.8b02082.

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Figures S1−S8; details of the SEC curves for used PS samples and permeate solutions of broadly distributed samples, the experimental setups, the normalized relative chain size functions (PDF)

AUTHOR INFORMATION

Corresponding Author

*E-mail [email protected]. ORCID

Lianwei Li: 0000-0002-1996-6046 Author Contributions †

T.Z. and M.Z. contributed equally to this work.

Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The National Natural Scientific Foundation of China Projects 21774116 and 51703216 are gratefully acknowledged.

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REFERENCES

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