Dependence of Solvent Diffusion on Hydrophobic Block Length within

Article pubs.acs.org/JPCB

Dependence of Solvent Diffusion on Hydrophobic Block Length within Amphiphilic−Hydrophobic Block Copolymer Membranes G. Dorenbos* T410-1118, 1107-2, sanno, Belle Crea 502, Susono, Japan ABSTRACT: Pore networks and water diffusion within model (amphiphilic−hydrophobic) diblock copolymer membranes in the presence of 16 vol % water is studied by dissipative particle dynamics in combination with Monte Carlo tracer diffusion calculations. The amphiphilic block (parent architecture (A[A3C])10) is composed of a backbone that contains 10 consecutively connected hydrophobic A beads; to each A bead, a side chain is grafted composed of three connected A beads and a pendant hydrophilic C bead. Hydrophobic blocks are constructed from x covalently bonded A beads, with x = 20, 30, or 50. Water diffusion through the pores is modeled by Monte Carlo tracer diffusion within more than 500 mapped morphologies. Long range water diffusion within the amphiphilic− hydrophobic ((A[A3C])10−Ax) diblock architectures increases with hydrophobic block length. Diffusion increases with Q = ⟨Nbond⟩|C||1 − C|−1, where C is the hydrophilic C bead fraction and ⟨Nbond⟩ the average number of bonds that A beads are separated from the nearest C bead. These trends are also anticipated for amphiphilic parent architectures (ACA3)10, (A2[C]A2)10, and (A2[AC]A)10. This is explained by the squeezing of water from the hydrophobic phase into the amphiphilic phase. Two characteristic distances are observed: The shorter distance corresponds to the interpore (or intercluster) separation within the “parent architecture-water” phase and obeys the earlier obtained linear relation between intercluster distance and ⟨Nbond⟩amphi of the amphiphilic parent architecture. The longer distance is governed by the phase separation between the amphiphilic-water phase and hydrophobic blocks. form ionic clusters in the dry state,6,7 can attract considerable amounts of water from moisture or liquid causing membranes to swell.7−14 Water and hydrophilic sulfonic sites then associate to shape a pore network. For Nafion1100 (for which the equivalent weight (EW) is 1100 g of polymer/mol of SO3H), experimental studies suggest that it requires about λ = 3 H2O molecules per sulfuric site in order for water to percolate through the membrane.7,14 Further increase in water content increases the diffusion of both protons and water. Dissipative particle dynamics (DPD) is a convenient tool to simulate microphase separation for polymeric systems. In DPD, groups of molecular fragments are coarse grained into beads. With DPD, much larger sizes and time scales can be accessed within reasonable CPU time than with molecular dynamics (MD) and equilibrium-like morphologies can be generated for system volumes of O(104−105) nm3 from unbiased (i.e., random) starting configurations. This is important, since the effect of phase separated pore morphology on the diffusive property of water and protons requires simulation box edge lengths to be larger than the characteristic distance between pores. These are within PFSA membranes of the order of 5 nm. Pore networks within hydrated Nafion were first modeled using DPD by Yamamoto and Hyodo.15 Their simulation studies

I. INTRODUCTION Microphase separation phenomena within polymeric systems are of relevance to technological application of, e.g., Li-ion batteries,1 bulk heterojunction organic (plastic) solar cells,2−5 dialysis, water purification, reverse osmosis, etc. Membranes composed of polymers that contain both hydrophobic and hydrophilic molecular fragments may swell by absorbing water from the environment. The absorbed water together with the hydrophilic sites may form a connected hydrophilic pore network within the hydrophobic matrix. The size of the pores and their extent of connection depend on water content, polymeric architecture, and the mutual compatibility between the constituent fragments. Pore sizes can be tuned down to the nm scale. Membranes composed of polymers that contain acidic (hydrophilic) molecular fragments and capable of conducting protons may serve as a polymer electrolyte membrane (PEM) in a proton electrolyte membrane fuel cell (PEMFC). This requires well connected pores that result in a low internal resistance for proton transport from anode to cathode. The most standard PEM is composed from the perfluorosulfonic acid (PFSA) ionomer Nafion. The backbone of a Nafion polymer contains several hundred to thousand connected −CF2−CF2− fragments. To the backbone amphiphilic (−O−CF2−CF(CF3)−O−CF2−CF2−SO3H) side chains are grafted. The ion exchange capacity (IEC) is therefore proportional to the grafting density. The hydrophilic sites, that © XXXX American Chemical Society

Received: October 30, 2016 Revised: November 29, 2016 Published: November 29, 2016 A

DOI: 10.1021/acs.jpcb.6b10913 J. Phys. Chem. B XXXX, XXX, XXX−XXX

Article

The Journal of Physical Chemistry B

diffusion increases as a function of ⟨Nbond⟩. (2) DMC(W) increases approximately linearly with ⟨Nbond⟩|C||A|−1, where |C| and |A| (=|1 − C|) are the hydrophilic C and hydrophobic A bead fractions within the architecture. (3) Percolation thresholds expressed in water volume fraction decrease inversely proportional with ⟨Nbond⟩ + 4 − (1.612|C|−1). Besides exceptions,34 these trends occur when C beads are distributed uniformly on the scale of the polymer backbone, but experimental validation is difficult to obtain. At least for PFSA polymers, some of the trends are confirmed. For instance, the longer side chain Nafion membranes seem to reveal higher water diffusion constants as compared to the shorter side chain (composed of −O−CF2− CF2−SO3H) Dow membrane of similar IEC at similar water contents, as predicted by DPD.16 This agrees with a coarse grained MD study by Allahyarov and Taylor.36 and strongly suggests that maximizing ⟨Nbond⟩ might be a route for designing uniform architectures that favor large pores and high water diffusion constants. Whether an increase of ⟨Nbond⟩ also results in higher diffusion for model architectures for which the hydrophilic fragments are not uniformly distributed on the length scale of the whole backbone length is one focus of study in this work. The aggregation of polymers for which the hydrophilic fragments are nonuniformly distributed on the scale of the whole polymer backbone length was addressed recently.35 For that purpose, block copolymers were considered that are composed of one amphiphilic block covalently bonded with a hydrophobic block. Each block contained 50 DPD beads while the architecture of the amphiphilic block was systematically varied while keeping its composition (C/A bead ratio) the same. Interestingly, as for the studies on uniform amphiphilic architectures, water transport also increased with the value of ⟨Nbond⟩amphi of the amphiphilic block. The aim here is to predict microphase separation and water diffusion within membranes composed of polymers for which also the hydrophilic fragments are nonuniformly distributed on the length scale of the polymer backbone. Similar as in ref 35, amphiphilic−hydrophobic copolymer architectures are modeled, but here the hydrophobic block length is systematically varied while the architecture of the amphiphilic block architecture is kept the same. At the same water content, pore morphology and water diffusion will turn out to depend drastically on the length of the hydrophobic block. These results are expected to be of interest for experimentalists working on the synthesis of amphiphilic−hydrophilic block copolymers with the aim to produce new PEMs with high proton conductivities. The outline of the paper is as follows. In the Computational Details section, the DPD parametrization is outlined and the diblock architectures are defined. In the Results and Analysis section, the distribution of water (W) beads within the DPD pore morphologies is studied and calculated MC tracer diffusion constants are presented. In the Discussion, the results are interpreted, explained, and placed in the context of the trends that were found recently for the uniform amphiphilic polymer architectures.

confirm experimental observations7,14 that for Nafion1200 the water containing pores or clusters increase in size with hydration level (varying from 10 to 30 vol %) with calculated Bragg spacings that compare well with experimental values. Various DPD studies on membranes with PFSA-like architecture16−26 and alternative chain architectures27−34 followed. In particular, in order to gain more insight into the dependence of pore network and water diffusion on polymeric architecture, DPD simulations were combined with Monte Carlo trajectory calculations.16,21−34 In these studies, water diffusion is mimicked by performing lattice MC walks of tracer particles on a 3D grid through copies of the morphologies obtained from DPD. Diffusion constants derived from the combined DPD-MC studies16,25 on Nafion membranes resemble experimental values when the local water mobility within the pores is assumed to be similar to that of pure water. Also, at fixed water volume fractions, a decrease of long-range water diffusion with increase of EW for fixed water contents was predicted.16,25 Recently, Vishnyakov and Neimark21 and Lee et al.22 also applied such a kind of combined DPD-MC simulations to study water diffusion in Nafion21 and sulfonated polystyrene membranes.22 By analyzing results obtained from more than 100 model architectures,23−34 various trends were predicted. All polymer architectures in refs 23−34 were composed of a minority of hydrophilic C beads and a majority of hydrophobic A beads, with C beads uniformly distributed on the length scale of the whole polymer backbone (each polymer contained several repeat units). The mutual Flory−Huggins interaction parameters between A, C, and water (W) beads were intentionally kept the same. For architectures with hydrophilic fragments end-grafted to side chains (each side chain equidistantly distributed along the hydrophobic backbones), long-range water diffusion increases with side chain length (for fixed IEC), and when side chain lengths are kept the same, diffusion increases with IEC,23,24 provided that the water contents are the same. From simulations performed at various water contents, the percolation threshold λp was found to depend on the fraction of C beads and side chain length.24 In searching and proposing alternative polymers that promote good connected pores that facilitate long-range diffusion, architectures have been considered such as those where each repeat unit contains (i) a branched (Y-shaped) side chain,28,33 (ii) two side chains but each of them of different length,31,32 and (iii) similar side chains but separated by alternating distances along the backbone.25,26,29,30 At a fixed water volume fraction of 16% (ϕw = 0.16), the average distance between water containing pores or clusters, DCl−Cl, turned out to depend considerably on the architecture. Among the architectures with side chains alternatingly distributed (type iii), the differences in DCl−Cl between architectures25,29,30 increase quadratically with Dtopol,2−Dtopol,1, where Dtopol,1 and Dtopol,2 are the number of DPD springs (covalent bonds) between a C bead and its nearest and next nearest C bead, respectively. Reanalysis31−33 augmented by additional studies revealed that in general DCl−Cl increases linearly with the parameter of merit ⟨Nbond⟩, which is the average number of bonds between hydrophobic A beads and the nearest hydrophilic C bead within the architecture. Moreover, ⟨Nbond⟩ is also useful when interpreting MC derived diffusion coefficients and percolation thresholds,31−33 resulting in the following trends: (1) For a fixed IEC (or |C|) and water content (ϕw = 0.16), water

II. COMPUTATIONAL DETAILS Dissipative Particle Dynamics. In 1992, Hoogerbrugge and Koelman37 introduced DPD which has become widely applied in modeling microphase separation related phenomena. Most DPD studies adopt the mathematical formulation B


Article

The Journal of Physical Chemistry B published by Groot and Warren.38 Molecular fragments are coarse grained into beads that move according to Newtons equations of motion, i.e., dri/dt = vi, midvi/dt = f i, with mi, ri, and vi being, respectively, the mass, position, and velocity of bead i. The forces that a bead j acts on bead i are conservative forces FijC, dissipative forces FijD, and random forces FijR. The total force acting on bead i is calculated from eq 1 with the sum over all particles j separated less than the cutoff distance rc. fi =

∑ FijC + FijR + FijD

Figure 1. Amphiphilic−hydrophobic polymer architectures II (x = 20), III (x = 30), and IV (x = 50). Hydrophobic A beads within the amphiphilic (hydrophobic) block are red/dark-gray (orange/gray), and hydrophilic C beads are yellow (light gray). Values within A beads in the amphiphilic block are the number of bonds, Nbond, toward the nearest C bead. Values within parentheses give the range of Nbond for A beads within the hydrophobic block.

(1)

j≠i

FijC decreases with distance: ⎧ ⎛ rij ⎞ ⎪ aij⎜1 − ⎟riĵ (rij < rc) ⎪ rc ⎠ FijC = ⎨ ⎝ ⎪ ⎪0 (rij ≥ rc) ⎩

rij = |rij| ,

rij = ri − rj,

(2a)

riĵ = rij/|rij|

total number of terms in the numerator of eq 6 is thus equal to the amount of A beads in the architecture, and the value of each term is the total number, Nbond, of (A−A and A−C) bonds of that A bead toward the nearest C bead. For each architecture, ⟨Nbond⟩ is obtained by dividing the value of the numerator by the total number of A beads which is equal to 4 × Nrep + x. Thus, obtained ⟨Nbond⟩ values are listed in Table 1 and increase approximately quadratic with x. To make a conversion to physical units, the volume of each DPD bead is set at V = 0.12 nm3. This value is of the same order17,18,21,22 or exactly the same15,16,19,20,24−35 as in recent DPD work15−35 on hydrated membranes. The actual choice for V = 0.12 nm3 has its origin located in the parametrization of Nafion in the DPD studies by Yamamoto and Hyodo.15 Since water is represented by W beads, this volume of 0.12 nm3 then contains four water molecules which reproduces the specific mass density of liquid water of 1 g cm−3 or 0.03 nm3/H2O molecule. The acidic site density is then between 1.38 mmol cm−3 (architecture VI) and 2.76 mmol cm−3 (architecture I). DPD Repulsions. By setting the DPD repulsions between similar beads at aii = 104, the water compressibility is reproduced.15,16 All other repulsions are given by the assumed Flory−Huggins χ parameters. The proportionality between χ and Δaij was calculated as χij ∼ (4.16)−1Δaij in ref 27. The χ values between incompatible A and W and between A and C beads are set at χAW = χAC = 4.9. These correspond with an excess repulsion of ΔaAW = ΔaAC = 20.4 and therefore aAW = aAC = 124.4. The repulsion between C and W beads is set at aCW = 93.2. The χ parameter is then χCW ∼ −2.6. The assumed χ values are exactly the same, or comparable, to those between noncompatible beads in other DPD studies.15−35 For instance, for Nafion, χAW = 5.79 and χAC = 3.11 was estimated from MD calculations.15 Simulations are performed with cubic box edge lengths of L = 30rc, and periodic boundaries are applied in each orthogonal direction. All simulations contain 81 000 (=ρL3) beads. The number of polymers Np is between 1360 (architecture I) and 680 (architecture IV) (Table 1). The length unit is rc = ρV1/3 or (3 × 0.12 nm3)1/3 = 0.71 nm. The box edge is ∼21.3 nm and corresponds with a system volume of ∼9.7 × 103 nm3. The box edges are much smaller than the thickness of a typical PEM (∼25−200 μm). Instead of a realistic (sheet or planar) membrane with two interfaces, a bulk system is modeled and interfaces (e.g., gas−membrane, liquid−membrane, or solid−membrane) are not considered. The membrane morphology near an interface may be different than that of the

(2b)

FijR (eq 3) together with FijD (eq 4) act as a thermostat FijR = σωR (rij)ζij(Δt )

−0.5

(kBT )−1riĵ

(3)

FijD = −γωD(rij)(riĵ ·vij)riĵ

(4)

with vij = vi − vj and weight functions ω and ω . Randomness is implemented by the fluctuating variable with Gaussian statistics, zero mean, and unit variance, ⟨ζij(t)⟩ = 0, and ⟨ζij(t)ζkl(t′)⟩ = (δikδjl + δilδjk)δ(t − t′). Equation 5 defines ωR and ωD. D

R

2 ⎧⎛ rij ⎞ ⎪ 1 (rij < rc) − ⎜ ⎟ ⎪ rc ⎠ ωD(rij) = [ωR (rij)]2 = ⎨ ⎝ ⎪ ⎪0 (rij ≥ rc) ⎩

(5)

Furthermore, σ = 2γkBT, with σ = 3, γ = 4.5, and Boltzmann constant kB and temperature T. FijC, FijD, and FijR conserve linear and angular momentum. A spring force connects adjacent polymer beads: FSij = −C(rij − R0)r̂ij, where R0 = 0.85rc and C = 50. The unit of time τ = rc(m/kBT)0.5 and bead masses are scaled to 1. kBT is set at 1.0 by a (modified) Verlet algorithm with empirical factor 0.65 and time step Δt = 0.05τ.38 The bead density is ρ = 3.38 Block Polymer Design. The polymers are shown in Figure 1. Architecture I is composed of a single amphiphilic block that contains 10 hydrophobic backbone beads with side chains springing off from each backbone bead. Each side chain contains three A beads and a pendant hydrophilic C bead. For architectures II, III, and IV, a hydrophobic block is covalently bonded to this amphiphilic block. The hydrophobic blocks differ in length and contain, respectively, 20, 30, and 50 consecutively connected A beads. The C bead fractions (|C|) thus decrease in the order IV < III < II < I. For the architectures in Figure 1, ⟨Nbond⟩ is calculated according to eq 6 2

39

i=4

⟨Nbond⟩ =

i=4+x

(N rep × ∑i = 1 i) + ∑i = 5 (4 × N

rep

+ x)

i (6)

rep

where N (=10) is the number of repeat units within the amphiphilic block and x is the number of hydrophobic A beads located within the backbone of the hydrophobic block. The C


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Table 1. Model Polymer Architectures A[A3C]10−Ax; Hydrophobic Block Length x; ⟨Nbond⟩; C Bead Density |C|; λ; Respective Number of Polymers (Np), A Beads (NA), C Beads (NC), and W Beads (NW); Water Bead Volume Fraction ϕw, and the Duration of Each Simulation Run architecture

x

⟨Nbond⟩

|C|

λ

Np

NA

NC

NW

ϕw

I: A[A3C]10 II: (A[A3C])10−A20 III: (A[A3C])10−A30 IV: (A[A3C])10−A50

0 20 30 50

2.5 6.5 9.786 17.5

0.2 0.143 0.125 0.1

3.82 5.33 6.11 7.65

1360 972 850 680

54400 58320 59500 61200

13600 9720 8500 6800

13000 12960 13000 13000

0.1605 0.160 0.1605 0.1605

duration (Δt) 1.84 2.8 2.72 3

× × × ×

105 105 105 105

bulk system and be affected by substrate type or gas phase humidity, which is outside the scope of this study. For instance, for very thin (∼5−10 nm) Nafion layers on a catalyst support, DPD studies predict that morphology and water diffusion depend strongly on the hydrophobicity of the support.40 Also, experimental characterization41−43 of thin membranes (thickness III > II > I. This means that an increase in hydrophobic block length facilitates long-range diffusion. The cause for this counterintuitive result is addressed in the Discussion. The increase of DMC(W) with hydrophobic block length is not observed for relatively short DPD runs. This is illustrated in Figure 5a where DMC(W) is plotted against DPD time. For times shorter than 104 time steps, diffusion is actually highest for the uniform (A[A3C])10 architecture I and decreases with hydrophobic block length. Diffusion initially increases with DPD time, but the required times to approach equilibrium appear to increase with hydrophobic block length. This necessitates taking care that simulation runs should be long enough to ensure that equilibrium-like morphologies are generated. When plotting DMC(W) against ⟨Nbond⟩, a linear increase is obtained, as shown in Figure 5b. It is notable that the C bead fraction decreases by a factor of 2 when going from architecture I (|C| = 0.2) to architecture IV (|C| = 0.1), yet diffusion more than doubles.

N

∑ |R⃗ i(t′) − R⃗ i(0)|2 i=1

(7a)

N

∑ |R⃗ i(t′) − R⃗ i(0)|2 i=1

(7b)

DMC(W) = MSD(t ′ = 2 × 106MCS) − MSD(t ′ = 1 × 106MCS) 106

IV. DISCUSSION In recent years, various DPD-MC studies23−26,28−33 have been published on model architectures composed of hydrophobic backbones to which amphiphilic side chains are attached with hydrophilic C beads located at the end of linear or branched (Y-shaped) side chains. Also, star polymers composed of linear or Y-shaped arms34 and block polymers with C beads distributed along various positions within the polymer backbones27 were modeled. From those works, a persistent trend was deduced in which DMC(W)ϕw=0.16 increases with Q = ⟨Nbond⟩|C||A|−1 (=⟨Nbond⟩|C||1 − C|−1). This suggests the possibility of a design strategy toward optimally connected pore networks by searching for architectures with high Q values. In Figure 6, the dashed line is the trend line obtained from fitting DMC(W)ϕw=0.16 against Q for various architectures31−33 with C beads uniformly distributed on the length scale of the polymer backbone. The vertical line drawn at Q = 1 is the highest possible Q value for block polymers and for

(7c)

Ntracer is the number of tracer particles and R⃗ i(t′) is the position of particle i at time t′. The water mobility within the pores is thus assumed to be the same as that of pure water. Some justification in favor of this at first sight rough assumption is retrieved from quasi elastic neutron scattering (QENS) measurements45,46 on Nafion that revealed that the local water diffusion coefficient increases with hydration from 1.2 × 10−5 cm2/s (λ = 1) to 2.14 × 10−5 cm2/s (λ = 16),45 close to the pure water coefficient of 2.3 × 10−5 cm2/s. In particular, at ∼14 vol % (λ = 5), the local water diffusion coefficient within Nafion1100 was 1.6 × 10−5 cm2/s or ∼0.7 times that of pure water. Since the actual longrange diffusion coefficient at this water content is ∼2 × 10−6 cm2/s,14 the decrease in long-range diffusion is thus mainly caused by restrictions in diffusive pathway within the topological pore network. F


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Figure 5. (a) Dependence of DMC(W) on DPD time. (b) DMC(W) plotted against ⟨Nbond⟩ (values are averages over the time interval 1.66 × 105Δt− 1.84 × 105Δt (I), 2.62 × 105Δt−2.8 × 105Δt (II), 2.54 × 105Δt−2.72 × 105Δt (III), and 2.82 × 105Δt−3.0 × 105Δt (IV).

10), a high DMC(W) value is obtained. For architectures with such low C bead fraction, previous DPD-MC simulations on uniform architectures resulted in much lower diffusion constants. The increased diffusion with hydrophobic block length is caused by the phase separation between the hydrophobic blocks and the amphiphilic blocks. As illustrated in Figure 2, those A beads (colored orange) contained within the hydrophobic Ax block are clearly phase separated from those A beads located within the amphiphilic blocks. Since water beads are expelled from this hydrophobic phase, they are forced to associate with the amphiphilic blocks (that contain both hydrophobic A and hydrophilic C beads), as clearly visible in Figure 2. This also explains that DCl−Cl(1) values (Table 2; Figure 3a) tend to be near DCl−Cl calculated for the amphiphilic “parent” architecture I (A[A3C])10 for which the hydrophobic block length is zero. Since water is squeezed out from the phase that contains the hydrophobic (−Ax) blocks, the effective water content within the remaining “amphiphilic block-water” phase increases with x. Therefore, with increasing length of the hydrophobic block, the pores become also better connected facilitating diffusion. An increase of DMC(W) with Q (Figure 6) is also retrieved when the hydrophobic block length is kept constant while the architecture of the amphiphilic block is changed. This is demonstrated in Figure 6 for three diblock (amphiphilic− hydrophobic) architectures modeled in ref 35. Each of them contains a hydrophobic A50 block but with the amphiphilic block being, respectively, (ACA3)10, (A2[C]A2)10, and (A2[AC]A)10. The corresponding DMC(W) values (empty triangles in Figure 6) also increase with Q (=0.111⟨Nbond⟩). When the hydrophobic block lengths are reduced to zero, the DMC(W) values for the amphiphilic parent architectures (ACA3)10, (A2[C]A2)10, and (A2[AC]A)10 (filled triangles in Figure 6) follow the dashed trend line deduced earlier31−33 for the uniform amphiphilic architectures. The dotted red lines in Figure 6 illustrate that an overall increase in hydrophobic block length is anticipated to result in increased diffusion (provided that the water volume fraction is kept fixed). The occurrence of long-range order structure for amphiphilic-W phases was not verified. Depending on the sulfonation degree within the amphiphilic block and the length of both blocks, phase separation toward, e.g., lamellar, hexagonally perforated lamellae, hexagonally packed cylinder, and gyroid morphologies might occur. Such phases have been observed, e.g., by Park et al.47−49 for a series of polystyrenesulfonateblock-polymethylbutylene (PSS-b-PMB) copolymers, and by Komarov et al.50 using DPD simulations for amphiphilic diblock (AnB24−n) copolymers in simulation cells with cubic

Figure 6. DMC(W) for the diblock copolymers (filled circles) against Q = ⟨Nbond⟩|C||A|−1 (=⟨Nbond⟩|C||1 − C|−1). The dashed line is the trend31−33 obtained for ∼100 architectures (refs 23−33) with C beads uniformly distributed on the length scale of the whole polymer backbone. The line drawn at Q = 1 is the maximal Q value possible (asymptotic limit, i.e., infinite number of repeat units) for architecture types studied in refs 23−34 (C2 dimers, C3 trimers, etc., within repeat units are left out of consideration). Open triangles: DMC(W) for diblock copolymers (ACA3)10−A50, (A2[C]A2)10−A50, and (A2[AC]A)10−A50 (each sampled over time interval 2.02 × 105−2.20 × 105Δt). Filled triangles: DMC(W) for parent architectures (ACA3)10, (A2[C]A2)10, and (A2[AC]A)10. Dotted lines are anticipated trend lines when the hydrophobic block lengths are increased from 0 to 50. The water contents are ϕw = 0.16.

architectures in which C beads are distributed within linear or Y-shaped side chains. For the block architectures II, III, and IV, the C beads are nonuniformly distributed over the length scale of the whole polymer backbone. By increasing the hydrophobic block length from x = 0 to x = 50, ⟨Nbond⟩ increases by a factor of 7 (Table 1), while the C bead fraction is reduced from |C| = 0.2 to |C| = 0.1. As a result, Q increases from 0.625 (I), 1.083 (II), 1.398 (III), to 1.944 (IV) (Table 2). In this same order, diffusion increases also, as shown in Figure 6 (filled circles). Although the increase of DMC(W) with Q for the block architectures (filled circles/thin solid trend line) is around 1/3 of that for the uniform architectures (dashed line), higher diffusion constants can be obtained for the block architectures. For architecture IV, DMC(W) = 0.26, which is higher than that for the Y-shaped architectures (A3[A8[C][C]])633 and (A3[A6[AC][AC]])633 (| C| = 1/6.5) for which DMC(W) ∼ 0.19 and branched star polymers (A[A10[AC][AC]])4 (DMC(W) ∼ 0.2)34 and (A[A12[C][C]])4 (DMC(W) ∼ 0.18)34 with |C| = 1/7.125. Despite the relative low C bead density within architecture IV (|C| = 1/ G


Article

The Journal of Physical Chemistry B edge lengths between L = 21rc and 36rc. A DPD study of (amphiphilic−hydrophobic) diblock copolymers that involve much larger blocks such as (A[A3C])100−A1000 might be interesting but would demand a very large DPD simulation box. Figure 7 displays the highest possible Q value, Qmax, as a function of C bead fraction for four types of uniform

thin lines do not cross each other in Figure 6. Since amphiphilic blocks can be designed that contain Y- or Ψ-shaped side chains with high Q values, then when these are used as parent architectures the corresponding amphiphilic−hydrophobic diblock derivatives are anticipated to also provide good connected pore networks. In this and related DPD work,15,16,24−34 electrostatic interactions between the acidic (deprotonated) sites and the counter charges (water−proton complex) were not taken explicitly into account. The mutual interactions between beads were given by χ parameters. These χ parameters however do depend on the partial atomic charges within each molecular fragment and can be estimated from atomistic simulations on PFSA polymers. It would be worthwhile to verify whether the predicted trends in refs 16 and 24−34 and in particular that of Figure 6 are also retainable from a full atomistic simulation. Conventional MD takes the electrostatic interactions between acidic sites and counterions explicitly into account, as well as those between the (fixed) partially charged atoms that are contained within the coarse grained (overall neutral) A and W beads. In DPD, these electrostatic interactions were implicitly mapped via the assumed χ parameters toward the corresponding DPD repulsions aij (eq 2a). For this reason, the possible impact of long-range Coulomb interactions on the final DPD morphology could not be studied. Interestingly, atomistic MD studies that include electrostatic interactions also tend to reproduce trends or PFSA membranes that are predicted by DPD studies that involve only neutral beads. As mentioned, DPD simulation results are in line with CGMD studies36 that predict that for similar IEC architecture those PFSA ionomers that contain the longer side chains also perform better with respect to diffusion than those containing the shorter side chains, and that pore sizes increase with side chain length. The consistency is perhaps partly caused by the highly incompatible interactions between hydrophobic and hydrophilic beads that were assumed and therefore govern the phase separated morphology. If the χ parameters between incompatible beads would be significantly reduced, then longrange electrostatic interactions due to the presence of deprotonated acid sites and their counter charges might eventually determine the final morphology. This issue might be interesting to study in the future. The effect of pore connectivity on water diffusion was studied by forbidding MC tracer particles to cross pore boundaries that were fixed in time (frozen), and water mobility within the pore space was assumed to be equal to that of pure water. This allowed simple acceptance criteria for tracer particle jumping. When one aims to study gas permeation through the frozen morphologies, then the gas solubility and diffusion coefficients within the respective water and polymer phases should be reproduced. This requires that in a MC calculation gas species are allowed to cross the pore boundaries. For this purpose, a kinetic Monte Carlo (KMC) based algorithm has already been developed51 and applied25,51 to model N2, O2, and H2 gas permeation through Nafion. In real membranes, water and ion diffusion will depend on the electrostatic interactions with the acidic groups and ions. The polarization of water molecules near acidic sites can reduce their mobility, with water molecules located further apart from the acidic sites being more mobile. More sophisticated jump acceptance rules are then required for hopping toward a randomly selected nearest node in such a way that locally the mobility of water molecules is reproduced. Counter ion

Figure 7. Qmax vs C bead fraction (|C|) for architectures of family (i) (AxC) (uniform C bead distribution block polymer); (ii) (AxCAyC)x>0,y=1 (nonuniform C bead distribution in block polymer); (iii) (Ax[AyC])x=1,y>0 (grafted side chains uniformly distributed along the backbone); and (iv) (Ax[Ay][AzC][AzC]])x=1,z=1 (Y-shaped side chains uniformly distributed along the backbone).

architectures with a hypothetically infinite number of repeat units. Those for which these maxima are obtained for |C| = 0.2 are illustrated with arrows. For fixed IEC (or C bead fraction | C|), Qmax increases in the order: “equidistantly distributed C beads within block polymers (AxC)” < “highly non-uniformly distributed C beads (AxCAyC) within block polymers” < “linear side chains for graf ted type polymers” < “Y-shaped side chains for graf ted type polymers”. In general, diffusion for these “parent architectures” is expected to follow this same order which is indeed in line with ref 28 where for architectures containing Yshaped side chains at ϕw = 0.16 higher water diffusion constants were obtained than those containing linear side chains. Also, the studies in refs 27 and 30 revealed that for block polymers with C beads equidistantly distributed along the backbone (AxC) water diffusion is less than that for a nonequidistant distribution. Nevertheless, block polymer architectures can compete with grafted (linear side chain) architectures of the same IEC with regard to water diffusion. For instance, at ϕw = 0.25, DMC(W) = 0.18 for the (ACA9C)5 block architecture27 similar as for the grafted (A4[AC]A4[AC])3 (DMC(W) = 0.17) and (A5[A2C]A[A2C])3 (DMC(W) = 0.19) architectures.30 However, that block polymer eventually could not compete anymore with the longer side chain architecture (A3[A3C]A[A3C])3 (DMC(W) = 0.24).30 Actually, Q values greater than 1 can be designed for alternative parent architectures. For example, the architecture composed of (A[A5[AC][AC][AC]]) repeat units contains trifurcated (Ψ-shaped) side chains and Q = 1.111. DMC(W) for parent amphiphilic architectures increases with Q (dashed line in Figure 6). For the amphiphilic−hydrophobic block copolymers derived from these parent architectures, the order of increase of DMC(W) with Q is conserved, since dotted and H


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results augment conclusions obtained for architectures with hydrophilic fragments uniformly distributed on the length scale of the polymer backbone, for which overall DMC(W) increases linearly with Q = ⟨Nbond⟩|C||A|−1. For the current diblock architectures, this trend is retrieved, i.e., DMC(W) is proportional to ⟨Nbond⟩|C||A|−1 but with a proportionality constant being around 1/3 of that observed for the uniform architectures. Because of the quadratic increase of ⟨Nbond⟩ with hydrophobic block length, larger Q values and higher diffusion constants can be obtained for the diblock copolymers as compared with parent amphiphilic architectures for which hydrophilic fragments are uniformly distributed. These trends are also expected to occur for the amphiphilic parent architectures (ACA3)10, (A2[C]A2)10, and (A2[AC]A)10.

diffusion (not modeled here) would require a Metropolis based MC algorithm in which the jump probability of the selected ion depends on the difference in total electrostatic energy of the system with the ion in its present position with that where the ion is at the aimed position. Since frozen morphologies were used in the MC tracer particle calculations, polymer motion was ignored, while in a real system continual polymer rearrangements occur. To realize morphologies and subsequent tracing of water molecules by MD would require exhaustive computer effort. Although DPD polymer beads diffuse too fast due to unphysical bond crossings, the advantage is that morphologies were obtained within acceptable CPU time (O(102) hours per architecture). The subsequent modeling of water through the frozen pore networks sets the polymer motion back to zero. For architectures I−IV, the pendant C beads are located near the pore boundaries. In DPD simulations, the C bead motion (and that of A beads) indeed continually rearranges the pores and W beads might occupy a position that was earlier occupied by a polymer C (or A) bead.29 This increases the volume available for W bead (or water) diffusion. DPD studies on architectures with C beads located within the side chains indeed reveal that the W bead diffusivities (D D P D ) are larger than DMC(W)22,28,29,35 but differ only slightly from DMC(W+C) values for diffusion through the W+C pore networks.28,29,35 For architectures I−IV, DDPD and DMC(W+C) were also calculated over the time intervals given in Table 2. DDPD was 0.33 (architecture I), 0.30 (II), 0.34 (II), and 0.41 (IV), and DMC(W +C) increases in the same order, i.e., II (DMC(W+C) = 0.27) < I (0.31) < III (0.33) < IV (0.39). The shift in order for architectures I and II as compared with DMC(W) (Table 2, Figure 5b) is partly explained by the higher W+C bead fraction for architecture I (ϕW+C = 0.1605 × ϕW + 0.8395 × |C| = 0.3284) compared to architecture II (ϕW+C = 0.28). Further decrease of W+C bead fraction increases water diffusion for architectures III and IV. It is emphasized that for application as a PEM in a fuel cell the best candidate membrane is not necessarily the one for which water diffusion (and likely proton diffusion as well) is the highest. Under working conditions, different membranes might absorb different amounts of water. In the case that the water contents would hypothetically be the same as for the architectures in Table 1, proton diffusion coefficients may not scale linearly with the MC derived diffusion coefficients. Even if such a scaling would exist, then it must be realized that the proton concentration (which does scale with IEC, see the fourth column in Table 1) within the membrane decreases with hydrophobic block length. This will negatively affect the membrane’s proton conductivity, and suggests that for each amphiphilic parent architecture an optimal hydrophobic block length might exist that favors proton conductivity.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone: +81 80 3396 4286. ORCID

G. Dorenbos: 0000-0002-1664-9432 Notes

The author declares no competing financial interest.

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ACKNOWLEDGMENTS I gratefully thank the anonymous reviewers for suggested improvements and corrections of this manuscript.

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REFERENCES

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V. CONCLUSION The pore networks within model diblock copolymer membranes (each of them composed of one amphiphilic and a hydrophobic block) and that of a reference membrane which contains only amphiphilic blocks were simulated. For each polymer, the amphiphilic block (“parent architecture”) was exactly the same (i.e., (A[A3C])10) but the length of the hydrophobic block was varied (A20, A30, and A50). Each membrane contained 16 vol % water. Monte Carlo tracer diffusion calculations through the morphologies predict that water diffusion increases with hydrophobic block length. These I


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Dependence of Solvent Diffusion on Hydrophobic Block Length within

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