Origin of Inconsistency in Experimentally Observed Transition Widths

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Origin of Inconsistency in Experimentally Observed Transition Widths and Critical Flow Rates in Ultrafiltration Studies of Flexible Linear Chains Tao Zheng, Jinxian Yang, Jing He, and Lianwei Li* Department of Chemical Physics, University of Science and Technology of China, Hefei, Anhui 230026, China

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ABSTRACT: Using special anisotropic membranes of bilayered cylindrical channels, we have experimentally clarified the origin of inconsistency in the transition widths and critical flow rates (qc) reported in related ultrafiltration studies of flexible linear chains. By studying the retention behavior of linear polystyrene chains passing through the same bilayered ultrafiltration membrane but in different translocation models, we reveal that the interaction among flow fields and the preconfinement effect are responsible for the previously observed inconsistency. In theory, only a single pore and a single chain are considered, but many pores and chains exist in experiment. The interaction among flow fields generated at different pore entrances could lead to the elongation/turbulent mixed flow fields, which will result in some unpredictable motions of polymer chains, such as the rotation, compression, reversed pulling, etc. Accordingly, much larger qc values and a broader transition were observed in experiments. On the basis of our previous free-draining model and present results, we propose a revised model by considering the partially draining nature of one confined blob, which provides a simple way for the rough estimation of the degree of draining for one confined blob in good solvents. Besides, the results also reveal that the preconfinement effect could lead to a reduced conformation entropy and an increased preconfinement energy for a polymer chain before its translocation through a small cylindrical pore, but such an influence seems to be chain length independent when the normalized confinement energy is compared. It satisfactorily explains why the critical flow rate is widely reported to be chain length independent, even though the preconfinement effect is not taken into consideration in related studies. Finally, our control experiment reconfirms that the asymmetry of bilayered membrane, instead of the variation of effective pore size, is the origin for the different ultrafiltration behavior of linear chains in different translocation models.

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the predicted first-order coil-to-stretch transition, to name but a few. Though most of studies support that qc is indeed independent of the chain length, significant inconsistency still exists in the transition widths and the critical flow rates (10−3 < qc/(kBT/(3πη)) < 101), experimentally determined by different groups.9−13,15−17 Though the explanations given in the literature seem reasonable, the origin of the inconsistency is rather difficult to clarify, which is mainly because of the complex factors in real experiments, such as the diversity of polymer sample, membrane structure, material, driven force, etc. We have been interested in the ultrafiltration study of topological polymers for the past few years.15−17,26−28 In the ultrafiltration studies of linear chains,15−17 we not only observed a sharp coil-to-stretch transition but also found that the measured qc values under various experimental conditions are 102−103 times smaller than those predicted by de

INTRODUCTION The translocation of (bio)macromolecules through nanoporous media not only is of academic interest but also exerts a profound influence on many real applications, such like gene delivery, protein transportation, and ultrafiltration separation of polymer chain mixture. Since the 1960s, Peterlin,1 Casassa,2 de Gennes,3,4 Pincus,5 Daoudi and Brochard,6 and Freed and Wu7,8 have been successively discussing theoretically how a flexible linear chain passes through a cylindrical pore. In particular, de Gennes3 and Pincus5 predicted that a flexible linear chain in a dilute solution can undergo a first-order coilto-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 diameter (d) much smaller than the chain size. On the basis of the blob model, de Gennes and Pincus further predicted that qc should be independent of both the chain length and the pore size, more specifically that qc ∼ kBT/(3πη), where kB, T, and η are the Boltzmann constant, the absolute temperature, and the solvent viscosity, respectively. In the past three decades, numerous experimental9−17 and computational18−25 studies have been performed to examine © XXXX American Chemical Society

Received: July 5, 2018 Revised: August 4, 2018

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DOI: 10.1021/acs.macromol.8b01436 Macromolecules XXXX, XXX, XXX−XXX

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transition and the interaction among flow fields generated at different pore entrances. Different from the continuously 3-dimensional ultrafiltration membranes with a poorly defined pore geometry (Scheme 1a,b), the isotropic/anisotropic membranes owing a precise cylindrical pore structure with nearly no lateral crossovers between individual pores are popular candidates for ultrafiltration study (Scheme 1c,d,f). In particular, Anderson et al.10,11 and Duval et al.13 used the track-etched mica and polycarbonate isotropic membranes in their ultrafiltration studies of PS and PEO linear chains, respectively; instead, a special bilayered AAO anisotropic membrane filter was used by us (inset in Scheme 2a).15,17 Statistically, each large pore

Gennes.15,17 Previously, we proposed a free-draining blob model to explain such a discrepancy, in which each confined blob is considered to be made of N Kuhn segments with a total effective subchain length (Le) along the flow direction.16,17 On the basis of the scaling relation d = kNα, and the balance between confinement force (fc = kBT/d) and hydrodynamic force (f h = 3πη(q/d2)Le), we have derived qc =

kBT 3πη

3 k1/ αd1 − 1/ α l

(1)

where kBT/(3πη) is the theoretical critical flow rate predicted in the hard-sphere model, l is the Kuhn length, Le = lN/ 3 , k is a constant for a given polymer solution, and α is the Flory exponent depending on the solvent quality. Such a scaling argument has further been supported by the theoretical work of Freed and Wu, in which qc was calculated by the analytical Green’s function/numerical inverse Laplace transform methods.8 Unfortunately, the free-draining model could still not explain the origin of inconsistency for the different qc values and transition widths reported in the literature.9−17 In principle, polymer samples could not cause any significant difference in the polymer rejection curves because the used samples are generally polymer standards such as polystyrene (PS) and poly(ethylene oxide) (PEO) from commercially available sources in related experimental studies. In contrast, the diversity of membrane material and pore geometry is more suspicious to affect the transition width and critical flow rate in a real experiment. Scheme 1 shows some typical pore

Scheme 2. (a) SEM and TEM Images of Top and Cross Section of Bilayered AAO Anisotropic Membranes from Whatman (d = 20 nm); (b) Schematic Illustration of the Bilayered Pore Structure of an AAO Anisotropic Membrane; (c) Schematic Illustration of the Two Models for Polymer Translocation through an AAO Anisotropic Membrane

Scheme 1. SEM (a−c) and TEM (d−f) Images of Commercially Available Ultrafiltration Membranes and the Schematic Pore Structures: (a) Cellulose Acetate Membrane from Whatman (800 nm); (b) Regenerated Cellulose Membrane from Whatman (450 nm); (c) Isotropic TrackEtched Polycarbonate Membrane from Whatman (50 nm); (d) Isotropic AAO Membrane from InRedox (50 nm); (e) Anisotropic AAO Membrane from InRedox (5 nm); (f) Anisotropic AAO Membrane from Whatman (20 nm) (From Web Sites of Whatman and InRedox)

contains one small pore underneath. Unlike the smooth transition reported by the group of Anderson and the group of Duval, we observed a sharp transition with much smaller qc. We initially attributed the sharp transition to the special bilayered structure of AAO membranes. Logically, the chains are supposed to first pass through a large pore in support layer and then through a small pore in active layer. Thus, we previously hypothesized that the large pores could function as effective barriers to isolate the small pores and to block the interaction among flow fields generated at small pore entrances.15 Objectively, the proof based solely on the comparison of our and other groups’ results is not direct or solid enough to support such a hypothesis. To reach an unambiguous conclusion, one has to use two membranes with exactly the same active layer structure (small pore structure) in experiments, where only one of the membranes should have a support layer (large pore barrier). The realization of this requirement, however, is rather difficult, if not impossible, due

structures of ultrafiltration membranes including polymer (Scheme 1a−c) and inorganic anodic aluminum oxide (AAO) (Scheme 1d−f) membranes. Obviously, the pore geometry will play a profound role in determining both the confinement state of a polymer chain during the coil-to-stretch B

DOI: 10.1021/acs.macromol.8b01436 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules to the complex membrane manufacturing process. In addition, the polymer samples, solvents, and driving forces are needed to be identical in experiments to preclude their potential influence. In this work, we report our recently discovered experimental evidence supporting the above-mentioned hypothesis. The new evidence was found by accident in one experiment, in which we mistakenly placed an anisotropic AAO membrane in holder by reversing the two faces of membrane (insets in Scheme 2c). Consequently, we actually measured the rejection curves of polymer chains directly passing through the small cylindrical pores without first passing through the large cylindrical pores in the support layer. After that we realized that the study of the reversed translocation process would be exactly an ideal experiment to clarify the effect of pore structure on the polymer ultrafiltration. Scheme 2c shows the two models for the polymer translocation through an anisotropic membrane (forward and reversed models). Experimentally, we observed more smooth transition and much larger qc values (5−20 times) in the reversed translocation model, markedly different from the result in the forward model. The related experimental evidence has for the f irst time clarified that the interaction among flow fields and the preconfinement by large pores are responsible for the observed inconsistency of critical flow rates and transition widths reported in different studies.

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EXPERIMENTAL SECTION

In ultrafiltration study, four narrowly distributed PS samples with different peak molar masses (Mp) were used (Table 1), and their SEC

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,29 where the data of PS samples used in experiments were also plotted for comparison.

Table 1. Molecular Parameters of Four Linear Polystyrene Samples Used in Experiments sample

Mpa/(g/mol)

Mw/Mna

⟨Rg⟩b/nm

2⟨Rg⟩/d

PS9k PS70k PS2200k PS8000k

× × × ×

1.05 1.06 1.09 1.15

∼2.8 9.0 72 155

∼0.28 (d = 20 nm) 0.9 (d = 20 nm) 7.2 (d = 20 nm) 3.1 (d = 100 nm)

9.0 7.0 2.2 8.0

3

10 104 106 106

respectively, which are at least ∼1 order of magnitude smaller than their critical overlap concentrations (C*). Experimentally, the macroscopic flow rate (Q) was controlled by a syringe pump (Harvard Apparatus, PHD 2000) and the temperature by an incubator (Stuart Scientific, S160D). A mechanical stirring was applied at a stirring rate of ∼100 rpm during the extrusion to suppress the concentration polarization. The rejection probabilities (Pr) of PS2200k and PS8000k long chains were determined by measuring the normalized relative retention concentrations Pr = [(C0 − C)/C0] by size exclusion chromatography (SEC) in THF, where C0 and C are the polymer concentrations normalized by short chain reference in the retentate and permeate solutions, respectively. The measurement uncertainty for Pr is typically within ±3%. At a given flow rate, 0.15 mL of solution was extruded into a small vial (sealed) for SEC measurement, and 0.05 mL of solution was discarded before collection. Both the forward and reversed translocation models were studied by using the same membrane. The typical SEC curves of PS2200k solutions collected under different experimental conditions are shown in Figure S2. The pressure drop across the membrane was measured with two high-accuracy pressure sensors installed before and after the membrane, which were connected to a multimeter for voltage output (Figure S3a).

a

Mp and Mw/Mn data were from suppliers, which show satisfactory consistency with the results obtained by our SEC characterization, where Mw and Mn are the weight- and number-average molar mass, respectively. b⟨Rg⟩ data were estimated based on the Rg−M scaling relation in toluene: Rg/nm = 1.23 × 10−2M0.594.29 characterization results are summarized in Figure S1. The two short PS chains (PS9k, Mp = 9.0 × 103 g/mol; PS70k, Mp = 7.0 × 104 g/mol) can act as internal references in solutions to increase the measuring accuracy of rejection curves of large PS chains (PS2200k, Mp = 2.2 × 106 g/mol; PS8000k, Mp = 8.0 × 106 g/mol). Two types of AAO membranes with the same large pore diameter (D = 200 nm) but different small pore diameters (d = 20 nm; d = 100 nm) were used to explore the influence of asymmetric feature of cylindrical channels on the polymer translocation. The effective filtration area of membrane is ∼0.8 cm2. The asymmetric coefficient (D/d) varies from 10.0 to 2.0 as d increases from 20 to 100 nm. Specifically, by comparing the relative size (2⟨Rg⟩/d) of the small pore radius (d/2) and the average radius of gyration (⟨Rg⟩) of each PS sample (Figure 1a), we decided to use PS2200k in the study of 20 nm membrane and PS8000k in the study of 100 nm membrane. Quantitatively, 2⟨Rg⟩/d is equal to 7.2 and 3.1 for PS2200k (d = 20 nm) and PS8000k (d = 100 nm), respectively, ensuring that the chains are large enough to undergo a coil-to-stretch transition. The polymer concentration (C) was fixed at 0.3 and 0.1 g/L for PS2200k and PS8000k in 15 mL of stock solution (Figure 1b),

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RESULTS AND DISCUSSION Figures 2a and 2b show how the translocation model quantitatively affects the rejection behavior of PS long chains passing through nanoporous membranes with different diameters. Clearly, for the forward model, the narrow transition was observed, while the reverse model exhibited a much broader transition. This observation unambiguously C

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Figure 3. (a) Normalized microscopic flow rate (qc/[kBT/(3πη)]) dependence of the rejection probability (Pr) of PS2200k and PS8000k chains passing through AAO membranes in toluene at T = 20 °C, where Anderson’s10,11 and Duval’s13 data were plotted for comparison (2⟨Rg⟩/d > 1.0). (b) Comparison of the microscopic flow rates (qc) at Pr = 0.5 determined by the group of Anderson (d = 40−160 nm), by the group of Duval (d = 30 and 50 nm), and by us under different experimental conditions.

Figure 2. (a) Macroscopic flow rate (Q) dependence of the rejection probability (Pr) of PS2200k chains passing through an AAO membrane with d = 20 nm in toluene at T = 20 °C. (b) Macroscopic flow rate (Q) dependence of the rejection probability (Pr) of PS8000k chains passing through an AAO membrane with d = 100 nm in toluene at T = 20 °C. The measurement uncertainty for Pr is typically within ±3%.

evidence the importance of pore isolation in blocking the interaction among flow fields generated at different small pore entrances. Physically, the rejection probability Pr represents the mass percentage of long chains retained by the membrane at a given flow rate. To better interpret the rejection curves, we define Qc,Pr=1.0 and Qc,Pr=0.5 as two macroscopic critical flow rates at which 100% and 50% of initial chains are retained by nanopores, respectively. Without the pore isolation, Qc,Pr=1.0 increases from ∼2.0 × 10−2 to ∼3.0 × 10−1 mL/h and Qc,Pr=0.5 increases from ∼5.0 × 10−2 to ∼1.1 mL/h for d = 20 nm and Qc,Pr=1.0 increases from ∼5 × 10−3 to ∼5 × 10−2 mL/h and Qc,Pr=0.5 increases from ∼2.6 × 10−2 to ∼1.7 × 10−1 mL/h for d = 100 nm. Overall, 10−20 times larger Qc values were observed for the reversed model. This phenomenon indicates that the interaction among flow fields actually leads to the generation of elongation/turbulent mixed flow fields, which could result in some unpredictable motions of polymer chains, such as the rotation, compression, and reversed pulling, instead of elongated deformation. Clearly, a much larger apparent hydrodynamic force (f h) is needed to stretch and drag the long linear chains into cylindrical nanochannels when the interaction among flow fields is not screened. In addition, we converted the macroscopic flow rate Q into the microscopic flow rate q (q = Q/N, N ≃ 5 × 108 pores/ membrane) and plotted the normalized-q (q/[kBT/(3πη)]) dependent rejection curves in Figure 3a. It is clear that the rejection curve (hollow squares) of PS2200k chains (d = 20 nm, reversed model) is not far from those determined by the groups of Anderson and Duval using single-layer membranes; these curves show smooth transition and similar thresholds on the order of ∼kBT/(3πη). Considering the existence of

transition width and the measurement accuracy of qc in different studies, we prefer to use Qc,Pr=0.5 and qc,Pr=0.5 to represent the experimentally determined Qc and qc in the following comparison and discussion. Table 2 and Figure 3b show a summary of experimental values for Qc, qc, and qc/ [kBT/(3πη)] at Pr = 0.5. Numerically, the values for qc/[kBT/ (3πη)] are 0.02 (d = 100 nm, forward), 0.13 (d = 100 nm, reversed), 0.04 (d = 20 nm, forward), and 0.84 (d = 20 nm, reversed) in our study, while ∼1.5 and ∼1.7 were reported by the groups of Anderson and Duval, respectively.10,13 Figure 4a shows how the asymmetric coefficient (D/d) of membrane affects the ratio (qc,reversed/qc,forward) of microscopic critical flow rates determined in different translocation models. Clearly, qc,reversed/qc,forward increases from 6.5 to 22.0 as the asymmetric coefficient D/d increases from 2.0 to 10.0, supporting that the difference between qc,reversed and qc,forward is indeed originated from the asymmetry of membrane, which is consistent with our hypothesis. Figure 4b further shows how qc depends on the small pore diameter d in different translocation models. Experimentally, qc,20 nm/qc,100 nm = 6.5 in the reversed model, which is 3−4 times larger than qc,20 nm/ qc,100 nm = 1.9 in the forward model. It is obvious that the much larger qc,20 nm/qc,100 nm for the reversed translocation model is because of the nonideal elongational flow fields. It is worth noting that the qc,20 nm/qc,100 nm ratio is associated with the draining property of a given polymer in solution; namely, qc ∼ d0 for the nondraining limit (hard-sphere model)3,4 and qc ∼ d1−1/α for the free-draining limit (eq 1).16,26 Specifically, eq 1 predicts that qc,20 nm/qc,100 nm ≈ 3 and 5, respectively, for α = 0.6 (athermal) and α = 0.5 (theta). In reality, the degree of D

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Table 2. Determined Macroscopic (Qc), Microscopic (qc), and Normalized Microscopic (qc/[kBT/(3πη)]) Critical Flow Rates in the Forward and Reversed Translocation Models at Pr = 0.5 in Toluene at T = 20 °C Qca/(10−2 mL/h) forward reversed qc,reversed/qc,forward

qca/(10−10 mL/h)

qc/[kBT/(3πη)]a,b

20 nm

100 nm

20 nm

100 nm

20 nm

100 nm

qc,20 nm/qc,100 nm

5.0 110 22.0

2.6 17 6.5

1.0 22 22.0

0.5 3.4 6.5

0.04 0.84 22.0

0.02 0.13 6.5

1.9 6.5

The uncertainties in the calculation of Qc, qc, and qc/[kBT/(3πη)] at Pr = 0.5 are within ±5%. bT = 20 °C and η = 0.59 mPa·s were used for the calculation.

a

model (eq 1). On the basis of the scaling relation d ≃ kNα, we can rewrite the expression of Le as idy Le ≃ jjj zzz kk {

β/α

l

α ≤ β ≤ 1.0

(2)

By considering the balance of the confinement force fc = kBT/d and the hydrodynamic force f h = 3πη(q/d2)Le, we will have a new expression of critical flow rate by considering the partially draining nature of one confined blob, i.e. qc ≃

kBT k β / αd1 − β / α 3πη l

(3) 1−β/α

Thus, we have qc,20 nm/qc,100 nm = (20/100) . If we consider the athermal condition (α = 0.6) and input qc,20 nm/qc,100 nm = 1.9 measured in the forward model (without the flow−field interaction), we will get β = 0.85, semiquantitatively reflecting the partially draining nature. Note that Freed and Wu previously pointed out that the quantitative evaluation of the hydrodynamic force and confinement energy in good solvents are rather challenging, and the analytical calculations with partially screened hydrodynamics are not possible for the cylindrical geometry.8 Therefore, our revised scaling model by considering the partially draining nature provides a simple method for the estimation of the degree of draining of confined segments. Scheme 3 further visually illustrates one of the probable reasons explaining why the apparent qc would be much higher if the interaction among flow fields is not blocked in the reversed translocation model. Note that a polymer chain is always undergoing the Brownian motion under the thermal agitation. Thus, the chain will not statically stay in front of one specific pore entrance; instead, the chain will be simultaneously pulled by the dragging flows generated at different pore entrances in the reversed translocation model. In other words, the chain might already undergo a certain degree of predeformation before entering a specific pore. It is not difficult to realize that the probability for one polymer chain to find/enter a specific pore entrance will also decrease accordingly. As shown in Scheme 3, at some point, even the chain is lucky to enter one pore entrance (pore 1); the hydrodynamic force f h has to be large enough to overcome the sum of the confinement force fc and the reversed pulling force (f rp) for a successful translocation, where f rp is originated from the hydrodynamic shearing of chain segments by the fluid flows generated at other pore entrances (pore 2 in Scheme 3). Actually, the AAO membranes used in this work have been previously well characterized in related studies. For example, the study by Fisch et al. has shown that the interpore distances of AAO membrane conform to Gaussian distribution and the average interpore distance is ∼320 nm, which is on the order of sizes of polymer samples (⟨2Rg⟩ = 144 nm for PS2200k and

Figure 4. (a) Asymmetric coefficient (D/d) dependence of the ratio (qc,reversed/qc,forward) of microscopic critical flow rates determined under different translocation models. (b) Small pore diameter (d) dependence of the microscopic flow rate (qc) at Pr = 0.5 determined under different translocation models.

draining is supposed to be between the two limits. Note that a partially permeable sphere model has been recently proposed by An and Lu to describe the hydrodynamic property of topological polymers by combining Einstein’s theory for hard spheres with Debye’s theory for free draining polymer chains.30 This partially draining model was successfully applied to the description of the intrinsic viscosity of flexible polymers of arbitrary architecture.31 Considering the association between the qc,20 nm/qc,100 nm ratio and the draining property, one can actually reversibly use the determined qc,20 nm/qc,100 nm to roughly estimate the degree of draining of confined segments in good solvents. As discussed earlier, the effective subchain length Le of a confined blob is related to the Kuhn segment number N and the Kuhn length l, i.e., Le ≃ Nl for the free-draining limit and Le ≃ Nαl ≃ d for the nondraining limit. Theoretically, one unified equation is enough to cover the two limits, i.e., Le ≃ Nβl, where β varies from α to 1.0, and could reflect the degree of draining; β = α corresponds to the nondraining limit in de Gennes’ model, and β = 1.0 corresponds to the free-draining limit in our previous E

DOI: 10.1021/acs.macromol.8b01436 Macromolecules XXXX, XXX, XXX−XXX

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Duval are also partially because of the dissipation of hydrodynamic energy. Indeed, the pore isolation is one of the most important prerequisites for the generation of an ideal elongational field as required by de Gennes’ theory, which is also the prerequisite for an experimental study.14,15 From an experimental point of view, the use of bilayered membrane is always preferred for the screening of flow field interaction. Ideally, the size of large pores in the support layer should be infinite, so that all the chain conformations in equilibrium state could be accessible (without preconfinement). For a real membrane, however, the pore size is always finite, and may be on the order of chain size, due to the limitation of manufacturing technique as well as the requirement of high flux for real applications. It is reasonable to assume that large pores with finite size in support layer might influence the polymer translocation in certain ways, but whether the large pore barrier will exert preconfinement on polymer chains, and how such a phenomenon will affect the polymer translocation, has never been discussed in a real experiment, although the theoretical study of the preconfinement on the polymer translocation has been receiving increasing attention in recent years.18,22,24,25 In this study, ⟨Rg⟩ is respectively 72 and 155 nm for PS2200k and PS8000k, which are very close to large pore radius (D/2 = 100 nm), implying that the preconfinement might already start to play a role in the polymer translocation. To better understand the preconfinement effect, the chain conformation distribution and the preconfinement free energy need to be discussed. Note that the quantitative description of the chain conformation distribution and the confinement energy in a good solvent is rather difficult. Thus, we can first analyze the radius of gyration distribution function [P(Rg)] for ideal chains to get some insight, which could be expressed as33

Scheme 3. Schematic Comparison of the Force Balance for One Confined Blob in the Forward Translocation Model (Single Pore) and in the Reversed Translocation Model (Multipores)a

a

f rp represents the reversed pulling force originated from the hydrodynamic shearing of chain segments by the fluid flow at the entrance of pore 2.

ji P(R g,max ) zyz 4 jij R g zyz zz = jj zz logjjjj j P(R g) zz 5 jj R g,max zz k { k {

⟨2Rg⟩ = 310 nm for PS8000k) used in this study, indicating that a polymer chain is very likely to be in the flow fields generated by different pores simultaneously.32 Scheme 3 only shows the simplest multipore case with only two pores coexisting. In principle, to overcome the large f rp, the segments of one confined blob, inevitably, have to be stretched completely to reach the maximum Le and f h. This situation significantly deviates from the classic blob model, the discussion of which is beyond the scope of this work. Obviously, as the pore density increases, more segments will be under the shearing of interference flow fields, which further results in significant increase of both f rp and apparent qc, satisfactorily explaining our experimental observation. In addition to the pore structures, the used experimental setups are also different in the literature. For example, a deadend ultrafiltration setup was used by us (Figures S3b) and the group of Anderson (Figure S3c), while a cross-flow ultrafiltration setup was used by the group of Duval (Figure S3d). Note that the cross-flow direction is parallel to membrane surface, which inevitably leads to a disturbance of the elongational fields generated at pore entrances, but how such an interaction affects the chain deformation and translocation kinetics is not clear and not predictable. In Anderson’s work, a mechanical stirring (240−360 rpm) was applied to suppress the concentration polarization, but such a strong stirring may also generate interference fields. In our modified design, a gentle stirring (100 rpm) was applied at the position far away from the membrane surface to suppress the generation of interference flow fields. Hence, we may expect that the larger critical flow rates observed by the groups of Anderson and

−15/4

6 ji R g zyzz + jjjj z 5 j R g,max zz k {

5/2

−2 (4)

where Rg,max is the radius of gyration for which the maximum probability occurs, at which P(Rg) = P(Rg,max). By assuming P(Rg,max) = 1.0, we could obtain the Rg,max-normalized radius of gyration distribution function (Figure S4a,b). Note that Rg,max is very close to ⟨Rg⟩ for monodispersed samples under theta condition; namely, the ⟨Rg⟩/Rg,max ratio calculated according to eq 4 is ∼1.04. Figure 5a further shows the absolute radius of gyration distribution function for PS2200k and PS8000k by taking Rg,max = 69 nm and Rg,max = 149 nm into eq 4. Under thermal equilibrium, it is clear that most of the equilibrium conformations for PS8000k is not accessible due to the preconfinement by large pores. For PS2200k, though the confinement is not as strong as PS8000k, the increase of confinement energy (Ac) might be large enough to influence the translocation behavior. In real conditions, more extended chain conformations and a stronger preconfinement are expected in good solvents due to the excluded-volume interaction. Second, we could roughly estimate Ac via the equation Ac ≃ kBT[1.05 + 5.78(2Rg/D)2] deduced under theta condition by Freed and Wu.8 Quantitatively, Ac is estimated to be ∼4.04kBT and ∼ 14.93kBT for PS2200k and PS8000k chains preconfined in a 200 nm cylindrical pore (Figure 5b), respectively. Obviously, the preconfinement effect leads to a reduced conformation entropy and an increased confinement free energy. Physically, F

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conformations under preconfinement, and accordingly, more segments will be exposed to the effective shearing by solvent molecules (larger Le). Accordingly, a smaller qc is enough to reach the balance between confinement force fc = kBT/d and hydrodynamic force f h = 3πη(q/d2)Le. More recently, TabardCossa and co-workers have revealed that the preconfinement effect can lead to narrower passage time distributions of DNA molecules translocating through a nanopore by changing the conformational entropy of each molecule prior to translocation,37 which strongly supports our argument. More challenging theoretical derivation and experimental observation should be performed in the future to estimate the accurate confinement energy in good solvents and to seriously answer the thermodynamics and kinetics related issues. Finally, one polydispersed sample (PSbroad, Mw = 2.85 × 105 g/mol and Mw/Mn = 2.48) with molar mass from 3.0 × 104 to 1.0 × 106 g/mol was used to test whether the asymmetry of membrane will influence the effective pore size during the polymer translocation. Similarly, one short PS sample (Mp = 4.0 × 103 g/mol) was used as internal reference in experiments. In different translocation models, a flow rate much smaller than Qc was used to ensure that the chains could pass through the membrane by only diffusion without undergoing any significant deformation. Figure 6a clearly shows that two similar SEC retention curves with cutoff molar masses around 3.0 × 105 g/mol were observed, unambiguously

Figure 5. (a) Radius of gyration distribution functions [P(Rg)] of PS2200k and PS8000k under theta condition, where Rg,max = 69 nm and ⟨Rg⟩ = 72 nm for PS2200k and Rg,max = 149 nm and ⟨Rg⟩ = 155 nm for PS8000k. (b) Relative chain size (2Rg/D) dependence of the confinement energy (Ac) of a polymer chain preconfined in a cylindrical pore with a diameter of D estimated by Freed and Wu under theta condition, where Ac ≃ kBT[1.05 + 5.78(2Rg/D).28

it is more important to estimate the confinement energy per Kuhn segment (Ac,K) or per blob (Ac,b) confined in a large pore (inset in Figure 5b), because we only need to consider the segments in the first subchain in front of a small pore entrance for a successful translocation event. Theoretically, Ac,K could be expressed as Ac,K = Ac/(M/Mk), where M represents the molar mass of one polymer chain and Mk represents the molar mass of one Kuhn segment (702 g/mol).29 Thus, we have Ac,K ∼ 1.29 × 10−3kBT for PS2200k and ∼1.31 × 10−3kBT for PS8000k. Clearly, the result demonstrates that the energy state of each preconfined segment/blob shows almost no dependence on the chain length, which implies that from a viewpoint of thermodynamics the influence of preconfinement effect on the critical flow rate of flexible linear chains might also be chain length independent. This finding satisfactorily explains the widely reported independence of qc from the chain length in related theoretical 3,4,6,8,18,34−36 and experimental studies,10,11,15,17 even though the preconfinement effect was never taken into consideration. In addition, we expect that the preconfinement may influence the initial chain conformation, the capture process, and the translocation kinetics, which are all essential in the flow-induced polymer translocation. It is not difficult to realize, before entering a small pore, one polymer chain is supposed to exhibit more stretched

Figure 6. (a) SEC curves of PSbroad stock solution before ultrafiltration and permeate solution after passing through an AAO membrane with d = 20 nm by different translocation models at small flow rates. (b) Relative chain size (2Rg/d) dependence of the rejection probability (Pr) of PSbroad passing through an AAO membrane with d = 20 nm by different translocation models at small flow rates. G

DOI: 10.1021/acs.macromol.8b01436 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules

by introducing a draining parameter (β) into the expression of the effective subchain length (Le) of a confined blob along the flow direction, which enables us to semiquantitatively estimate the degree of draining of confined segments in a good solvent. In addition, the analyses of the radius of gyration distribution function and the confinement energy further confirm that the preconfinement by large pores already starts to play a role in polymer translocation, but the calculation of confinement energy per Kuhn segment/per blob indicates a chain length independent energy state. It satisfactorily explains why the critical flow rate is always reported to be chain length independent no matter whether the preconfinement effect is taken into consideration or not in related studies. Overall, this study has clarified that the interaction among flow fields and the preconfinement effect are responsible for the inconsistency of critical flow rates and transition widths reported in different studies, but the influence of preconfinement on the initial chain conformation, the capture process, and the translocation kinetics is still not clear and needs to be further addressed in the future. We hope this work not only could be helpful for the understanding of the translocation behavior of polymer chains in porous medium of more complex microscopic geometries but also could provide theoretical guidance for the design of novel nanodevice to regulate the polymer transportation.

indicating similar effective pore sizes for the two translocation models. Moreover, the transition from full transmission to full retention for the diffusion-limited translocation corresponds to two critical molar masses, i.e., Mc,lower ∼ 8.5 × 104 g/mol and Mc,upper ∼ 3.0 × 105 g/mol, corresponding to a Mc,upper/Mc,lower ratio of ∼3.5. It is worth noting that for most of commercially available ultrafiltration membranes the separation of polymer fractions with molar mass difference within ∼1 order of magnitude is very difficult. Therefore, the small Mc,upper/Mc,lower ratio reflects an extremely narrow pore size distribution of AAO membrane used in this study. In addition, compared with our previous result,15 we found no difference of the transmembrane pressure drops for the two translocation models. All this evidence clearly demonstrates that the geometry of small pores is symmetric and the effective pore size is constant in different translocation models. Based on the scaling relation Rg = 1.23 × 10−2M0.594 for linear polystyrene in toluene,29,38 the molar mass information can be further converted into the size information (Figure S5a), which shows similar cutoff sizes (R ≃ 9.5 nm at Pr = 1.0), irrelevant with the translocation model. Accordingly, the transition from full transmission to full retention for the diffusion-limited translocation actually corresponds to the relative chain size (2Rg/d) from ∼0.9 to ∼1.9. Figure 6b quantitatively shows how the rejection probability Pr depends on the 2Rg/d ratio. Experimentally, Pr increases almost linearly with 2Rg/d when 0.9 < 2Rg/d < 1.9. By considering the radius of gyration distribution functions for polymer chains with 2Rg/ d ≃ 0.9 and 2Rg/d ≃ 1.9 (Figure S5b), it is not difficult to understand why the full retention starts to occur when 2Rg/d ≥ 1.9, at which all accessible conformations correspond to Rg values much larger than the pore radius. On the other hand, though a small portion of chain conformations (∼30%) are not thermodynamically favorable for the translocation when 2Rg/d ≃ 0.9, a full transmission is still possible. This is because a polymer chain is not a hard sphere and all the accessible conformations are energetically equal, which leads to an easy achievement of conformation−conformation transition under the excitation of thermal energy. Overall, the above evidence clearly demonstrates that the observed different ultrafiltration behavior of polymer chains in different translocation models should be attributed to the asymmetric bilayered membrane structure, instead of the variation of effective pore size in different translocation models.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.macromol.8b01436. Figures S1−S5; the SEC curves for permeate solutions of monodispersed and polydispersed samples, the experimental setups and the radius of gyration distribution functions (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail [email protected] (L.L.). ORCID

Lianwei Li: 0000-0002-1996-6046 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. We thank Prof. Chi Wu at The Chinese University of Hong Kong for helpful discussions.

CONCLUSION In summary, using special bilayered anisotropic membranes, we experimentally studied the influence of pore isolation and chain preconfinement on the (minimum) critical flow rate (qc) and transition width in ultrafiltration of flexible linear chains through small cylindrical tubes (“nanopores”). The results revealed that the interaction among flow fields generated at small pore entrances could lead to much larger apparent qc values and a much broader transition observed in experiments due to the existence of elongation/turbulent mixed flow fields. Quantitatively, without the screening of interaction among flow fields, the critical flow rate of linear chains measured in the reversed translocation model (qc,reversed) through 20 nm nanopores is ∼0.84kBT/(3πη), which is ∼20 times larger than the critical flow rate measured in the forward translocation mode (qc,forward), but not far from 1.5−1.7kBT/(3πη) reported by other groups. After slightly modifying our previous freedraining model, we further proposed a partially draining model

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REFERENCES

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