(AmphiphilicâHydrophobic) Diblock Copolymer Membranes

Article pubs.acs.org/JPCB

Water Diffusion Dependence on Amphiphilic Block Design in (Amphiphilic−Hydrophobic) Diblock Copolymer Membranes Gert Dorenbos* T410-1118, sano 1107-2, Belle Crea 502, Susono, Japan ABSTRACT: Polyelectrolyte membranes (PEMs) are applied in polyelectrolyte fuel cells (PEFC). The proton conductive pathways within PEMs are provided by nanometer-sized water containing pores. Large-scale application of PEFC requires the production of low-cost membranes with high proton conductivity and therefore good connected pore networks. Pore network formation within four alternative model diblock (hydrophobic_amphiphilic) copolymers in the presence of water is studied by dissipative particle dynamics. Each hydrophobic block contains 50 consecutively connected hydrophobic (A) fragments, and amphiphilic blocks contain 40 hydrophobic A beads and 10 hydrophilic C beads. For one amphiphilic block the C beads are distributed uniformly along the backbone. For the other architectures C beads are located at the end of the side chains attached at regular intervals along the backbone. Water diffusion through the pores is modeled by Monte Carlo tracer diffusion through mapped morphologies. Diffusion is highest for the grafted architectures and increases with increase of length of the side chains. A consistent picture emerges in which diffusion strongly increases with the value of ⟨Nbond⟩ within the amphiphilic block, where ⟨Nbond⟩ is the average number of bonds between hydrophobic A beads and the nearest C bead. this reason Nafion of EW ∼ 1100 and Dow of EW ∼ 1000 are most optimal for fuel cell applications. Widespread application of fuel cells requires a reduction in production cost. Since PFSA membranes are very expensive (the prize of Nafion is ∼1000 $/m2) researchers are focusing on alternative PEMs such as sulfonated poly(ether−ether− ketone) (S-PEEK),10−12 styrene/ethylene−butylene styrene (SSEBS),13 polyarylene−ethersulfone (S-PAES),12,14,15 polybenzimidazole,16 and sulfonated polyimide.17 Among these alternatives also proton and water diffusion occurs within the water containing hydrophilic pore networks, and sulfonation degrees need to be restricted to prevent excessive swelling. Knowledge about the morphology of the phase separated pores is essential for designing new polymeric materials with optimized pore networks. Information about the membrane morphology can be obtained by application of expensive techniques such as small-angle X-ray (SAXS)1,2,7−9,12,18 and neutron scattering (SANS),18,19 atomic force microscopy AFM,12,15,20 and transmission electron microscopy (TEM).21 Theoretical methods on the other hand are much cheaper and various molecular dynamics (MD) studies on Nafion have been performed.22−36 The attractive point of MD is that the (sub) nanometer-scale can be studied. But CPU time restrictions limit the MD system size to simulation box edge lengths of ∼5

I. INTRODUCTION Nafion is a perfluorinated sulfonic acid (PFSA) polymer and is used as a polymer electrolyte membrane (PEM) in polyelectrolyte fuel cells (PEFC). The backbone of Nafion is composed of strongly hydrophobic connected −CF2−CF2− fragments. Amphiphilic side chains with chemical formula −O− CF2−CF(CF3)−O−CF2−CF2−SO3H are attached to the backbone. The ion exchange capacity (IEC) or equivalent weight (EW) is given by the density of the grafted side chains. In the dry state the hydrophilic sites form ionic clusters.1,2 When exposed to moisture or liquid water the membrane swells1−9 and takes up water that associates with the sulfuric sites. A Nafion1100 membrane, for which the EW is 1100 g of dry polymer per mole SO3H, requires a hydration level of around λ ∼ 3H2O molecules per sulfuric site for protons and water to percolate through the microphase separated water containing pore networks.3,9 Nafion membranes with EW significantly larger than 1200 hardly swell,2 and pores are poorly connected. Also for Dow membranes (with the shorter −O−CF2−CF2−SO3H side chains) when boiled in water, the water uptake decreases from ∼200 wt % for EW = 635 down to ∼20 wt % for EW = 1200.1 A poor pore connection and low proton density will result in low proton conductivities. For this reason high EW PFSA membranes are not suitable for fuel cell applications. On the other hand, too low EW (high IEC) PFSA membranes are neither attractive since excessive swelling decreases besides the proton density as does the membranes’ structural integrity. For © 2016 American Chemical Society

Received: March 29, 2016 Revised: June 6, 2016 Published: June 6, 2016 5634

DOI: 10.1021/acs.jpcb.6b03171 J. Phys. Chem. B 2016, 120, 5634−5645

Article

The Journal of Physical Chemistry B nm.22−34 This length is about the size where for hydrated Nafion1200 the Bragg spacing is observed,2,9 and representative for the interpore distance. Information on pore connectivity is thus difficult to obtain from (classical) MD, which would require a system containing O(106) atoms. Efforts to study the Nafion-water system with millions of atoms have been performed but requires preassumed starting morphologies.35 A recently reported MD study starting from random geometries for very large system size required repetitive cycles of adding periodic images that are equilibrated.36 An attractive route to predict pore morphology is by application of methodologies in which molecular fragments are coarse grained (CG). CG methods allow a much longer time scale and system size to be accessed. CG approaches applied to hydrated PFSA membranes are, for example, coarse grained MD (CGMD),37,38 bond fluctuation model (BFM),39 combined Monte Carlo and reference interaction site modeling (MC/RISM),40 and the self-consistent MESODYN approach.41 Especially DPD is attractive to study microphase separation within PFSA membranes42−49 within reasonable computer time. Previous studies42,43 on the Nafion−water system reproduced the experimentally observed9 increased distance between water clusters with water uptake,42 and trends regarding dependency of pore morphology and water diffusion on EW could be predicted.43 In refs 43 and 45 Monte Carlo (MC) tracer diffusion calculations were performed through the pores within discretized copies of DPD morphologies mapped on a 3D lattice. Diffusion constants derived from the DPD-MC studies resembled experimental values when the local water mobility within the pores is assumed equal to that of pure water.43,45 This is in line with quasi elastic neutron scattering (QENS) experiments on Nafion that revealed at λ = 5 that local H2O mobility within the confined pores approaches the pure water value.50,51 Vishnyakov and Neimark49 and Lee et al.52 applied a similar “DPD-MC” strategy to study water diffusion in Nafion49 and within sulfonated polystyrene membranes of various degree of sulfonation.52 From a series of studies53−62 an exhaustive database containing more than one hundred amphiphilic grafted- or block-type architectures has been established. The architectures were composed of hydrophobic (A) and hydrophilic (C) beads (in DPD a bead represents a molecular fragment). For grafted architectures the dependency of pore morphology and diffusion on the side chain length53−55 and on side chain distribution was studied.55−57 Combinations in which side chains of different lengths (i.e., bimodal length distribution) are attached to the polymer backbone and the distributions along the backbones are varied were also studied,58,59 as well as the dependency of water diffusion on side chain architecture.60,61 The database also contains several amphiphilic block polymers with varying hydrophilic site distribution along the backbones53,55,62 and amphiphilic star polymer architectures.63 In all these studies the differences in compatibility of A and C beads toward water was kept the same. From the database53−63 it was found that at fixed water volume fraction the characteristic distance between pores increases linearly with ⟨Nbond⟩,58−60 where ⟨Nbond⟩ is the average number of bonds that each A bead in an architecture is separated from the nearest C bead. For fixed IEC membranes water diffusion mostly also increases with ⟨Nbond⟩ when the water content is kept fixed. Consequently, in general lower percolation thresholds for diffusion are predicted when ⟨Nbond⟩ is increased for model membranes of the same IEC.58−60

In the DPD-MC studies53−62 the polymers contain various repeat units and the hydrophilic C beads are therefore uniformly distributed on the length scale whole polymer length. The interest here is to study the aggregation of polymers for which the distribution of the hydrophilic fragments on the length scale of the whole polymer is highly nonuniform. For this purpose diblock copolymers are considered that are composed of a fully hydrophobic block that is connected with an amphiphilic block (of same size) that contains both hydrophobic (80%) and hydrophilic (20%) beads. Deliberate design of the amphiphilic block architecture might be an interesting route to optimize the pore networks for proton and water transport. From the database an optimizing strategy for obtaining good connected pore networks could be deduced in that architectures with high ⟨Nbond⟩ values favor the formation of good connected pores. Whether and to what extent this trend is valid for (amphiphilic−hydrophobic) diblock copolymers is a question addressed in this study. Phase separated morphologies are calculated by DPD for several (hydrophobic−amphiphilic) diblock copolymer membranes that contain a water volume fraction of ϕw = 0.16, and pore connectivity/water diffusion will be studied by MC tracer diffusion through 400 mapped morphologies (frozen lattices). For each copolymer the hydrophobic block is the same while the amphiphilic blocks are of identical composition but differ in the way the hydrophilic molecular fragments are distributed within the amphiphilic block. There are two main advantages for splitting the simulation of water diffusion and morphology in the DPD-MC studies. First during a DPD simulation, polymer beads actually diffuse much too fast: Since bond crossings are allowed entanglements are not taken into account. Although the amount of bond crossings can be drastically reduced by additional programming efforts64 it will require much more CPU time to reach (eventually the same) equilibrium-like morphologies. As in conventional applications of DPD in this work, bond crossings are allowed and equilibrium like morphologies are obtained within an acceptable CPU time of ∼102 hours on a 1 GHz PC. Second by studying MC tracer diffusion through frozen morphologies, the polymer motion is set at zero. In the Computational Details section the DPD parametrization is explained, and the model diblock copolymer architectures are presented. In the Results and Analysis section, the distribution of water within the morphologies is analyzed and water diffusion is studied by MC tracer particle trajectory calculations and by tracking the motion of water during the DPD simulations. In Discussion the obtained relations between amphiphilic block architecture pore morphology and water diffusion are discussed.

II. COMPUTATIONAL DETAILS Dissipative Particle Dynamics. After its introduction in 199265 DPD has become a suitable tool to model microphase separation related phenomena. Since molecular fragments are coarse grained into beads, much larger time scales and system size can be accessed by DPD than by MD. The framework as described by Groot and Warren is adapted here.66 The motions of the beads are calculated as dri/dt = vi, midvi/dt = f i, where mi, ri and vi, are the mass, position, and velocity of bead i. Beads interact by conservative forces FijC, dissipative forces FijD, and random forces FijR. The force that bead j exerts on bead i is given by eq 1, with the sum taken over all particles j within cutoff distance rc. 5635


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The Journal of Physical Chemistry B fi =

∑ FijC+FijR + FijD (1)

j≠i

FijC

For the other three architectures the C beads within the amphiphilic blocks are end-grafted to side chains that spring off at regular intervals along the backbone. The side chain length is varied systematically. The fraction of hydrophilic C beads (|C|) in each of the four architectures is 0.1. Water is represented by W beads. To allow for a conversion to physical units the volume of each (A, C, and W) bead is assumed to be V = 0.12 nm3. This reproduces the specific mass density of liquid water when each W bead represents four H2O molecules. The acidic site density for the architectures in Figure 1 is then 1.38 mmol cm−3. For supposed mass density of 1.0 (2.0) g cm−3, the EW is then 722 (1444) g mol−1, or IEC = 1.38 (0.69) mmol g−1. Repulsion Parameters. Repulsions (eq 2a) between similar beads are aii = 104 (i = W, A, or C) which reproduces the water compressibility.42,44 Hydrophobic A and hydrophilic W or C beads are incompatible with each other. This is reflected by the repulsions aAW and aAC which are set at 124.4. For this value the proportionality between χ and excess repulsion Δaij is χij ≈ (4.16)−1Δaij,62 and χAW=χAC ≈ 4.9. The χ parameter between C and W beads is set at −2.6 and the repulsion aCW = 93.2.62 The χ_values are comparable to those calculated for Nafion in ref 42 where for example χAW and χAC was, respectively, 5.79 and 3.11. A or C beads might contain for instance ether linkages, (fluoro) aromatic or (hydro)fluoro-carbon molecular fragments and C beads might also contain different acidic groups. For instance, molecular fragments like −O−CF2−CF(CF3)−O− and −O−CH2−CF2−CF2 are hydrophobic. The χ parameter between −O−CF2−CF(CF3)−O− and four water molecules, was estimated to be ∼4.942 which is the same as between A and W beads here. Wu et al.46,47 assumed a χ parameter approximately equal to 7 between hydrophobic and hydrophilic beads (CF3SO3H·3H2O complex) of volume ∼0.18 nm3. These correspond to χ ≈ 4.7 when bead volumes are rescaled to V = 0.12 nm3. In a DPD study by Vishnyakov and Neimark49 the excess repulsion between CF2−CF2−CF2−CF2 (A) and water (W) was deduced from fitting with MD calculations. They obtained aAW = 65 when the diagonal terms are set at aAA = aWW = 50. Since χij = 0.306Δaij for aii = 2566 and 0.231Δaij for aii = 78,68 a conservative estimate for χAW is then 4.0 which compares with the value adapted here. The simulation conditions are summarized in Table 1. The system is cubic with edge length L = 30rc and contains 680 polymers and 81000 (=ρL 3 ) beads. One length unit corresponds to rc = ρV1/3 = (3 × 0.12 nm3)1/3 = 0.71 nm and the system volume is ρL3V = 9.7 × 103 nm3. The water volume fraction ϕw is equal to 0.16 = NW(NA + NW + NC)−1, where NA, NW, and NC are the number of A, W, and C beads. Periodic boundaries are included and simulations are performed until 2 × 105Δt with bead positions stored on file every 2000Δt.

decreases with distance:

⎧ ⎛ rij ⎞ ⎪ aij⎜1 − ⎟riĵ (rij < rc) ⎪ rc ⎠ FijC = ⎨ ⎝ ⎪ ⎪0 (rij ≥ rc) ⎩ rij = ri − rj,

rij = |rij| ,

FijR

(2a)

riĵ = rij/|rij|

(2b)

FijD

introduces noise and is proportional to the relative velocity. Together they act as a thermostat. FijR = σωR (rij)ζij(Δt )−0.5 (kBT )−1riĵ

(3)

FijD = −γωD(rij)(riĵ ·vij)riĵ

(4)

where vij = vi − vj. ω and ω are weight functions. Randomness is introduced by a randomly fluctuating variable with Gaussian statistics with unit variance and zero mean: ⟨ζij(t)⟩ = 0, and ⟨ζij(t)ζkl(t′)⟩ = (δikδjl + δilδjk)δ(t − t′). ωD and ωR are given by eq 5. D

R

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

(5)

σ and γ are related by σ2 = 2γkBT, where σ = 3, γ = 4.5, kB is the Boltzmann constant, and T is the temperature. FijC, FijD, and FijR conserve linear and angular momentum. Adjacent polymer beads are joined by a spring force FSij = −C(rij − R0)r̂ij with C = 50 and R0 = 0.85rc. The unit of time τ = rc(m/kBT)0. 5 and bead masses are scaled to 1, and kBT is kept at 1.0 by a modified Verlet algorithm with empirical factor 0.65 with time step Δt = 0.05τ.66 The bead density ρ is 3.66 Diblock Architecture. DPD representations of the model diblock copolymers are displayed in Figure 1. The hydrophobic block is composed of 50 consecutively connected hydrophobic A beads. Each amphiphilic block contains 40 A beads and 10 hydrophilic C beads. For architecture I the C beads are distributed uniformly within the linear (i.e., (ACA3)10) block. 67

III. RESULTS AND ANALYSIS Phase-Separated Morphologies. Figure 2 displays the morphologies generated at t = 2 × 105Δt together with their periodic images. To highlight the pore network the A beads are removed in the lower half of each image. The C and W beads are clustered forming a pore network within the majority A phase. For architecture I the pores appear most poorly connected and blobby. For grafted-type architectures III and IV the water channels seem to be more elongated and better connected than for architecture I. The C beads are located near

Figure 1. DPD representations of diblock (amphiphilic−hydrophobic) architectures. Hydrophobic A beads are red (dark gray) and orange (gray), hydrophilic C beads are yellow (light gray). Values within A beads in the amphiphilic blocks indicate the number of bonds, Nbond, toward the nearest C bead. Italic numbers in parentheses give the range of Nbond for A beads located in the hydrophobic block. 5636


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The Journal of Physical Chemistry B

Table 1. DPD Formulae of Di-block Polymer Architectures, ⟨Nbond⟩, ⟨Nbond⟩amphi, Total Number Np of Polymers, NA of A Beads, NC of C Beads and NW of W Beads, Water and Amphiphilic Bead Volume Fractions ϕW and ϕW+C, and Total Duration of Each Simulation Run architecture

⟨Nbond⟩

⟨Nbond⟩amphi

Np

NA

NC

NW

ϕW

ϕW+C

I: (ACA3)10−A50 II: (A2[C]A2)10−A50 III:(A2[AC]A)10−A50 IV: (A[A3C])10−A50

16.5 16.72 16.83 17.5

1.5 2 2.25 2.5

680 680 680 680

61200 61200 61200 61200

6800 6800 6800 6800

13000 13000 13000 13000

0.1605 0.1605 0.1605 0.1605

0.2444 0.2444 0.2444 0.2444

no. of time steps 2 2 2 2

× × × ×

105 105 105 105

Pair correlation functions obtained at t = 2 × 105Δt are shown in more detail in Figure 3e (g(r)W−W) and Figure 3f (g(r)C−C). For each diblock copolymer an upturn occurs beyond 6 nm and a maximum is observed between ∼10 nm and ∼14 nm. Distances where these maxima occur for g(r)W−W (Dinter(W−W)) and g(r)C−C (Dinter(C−C)) increase in the order IV < III < II < I (insets Figure 3e−f). Notable differences in g(r)W−W are observed in the 3−5 nm region where a shoulder or maximum appears whose approximate locations (Dintra) decrease in the order IV > III > II > I (inset Figure 3e). These are due to differences in pore organization within the amphiphilic block−water aggregates. The time required to obtain the structures in Figure 2 is up to 1 order of magnitude longer than in previous DPD studies.43−45,53−63 This is due to the high anisotropic (nonuniform) distribution of hydrophilic C beads for the present architectures. In refs 43−45 and 53−60 the average number, ⟨Nbond⟩, of bonds (or springs) between an A bead and a nearest C bead was in the range 2−6. Here they are a least ∼3 times larger (Table 1). Due to the larger ⟨Nbond⟩ also large hydrophobic regions are expected which will slow down the time for obtaining equilibrium like morphologies. Indeed the characteristic distance between amphiphilic aggregates is ∼10− 14 nm, significantly larger than in previous work where the average distance between water containing pores was less than ∼6.5 nm. Largest Water Cluster. The distribution of W beads within the stored morphologies was calculated for various values of Rcrit. Rcrit is defined as the distance for which for every W bead that belongs to a cluster there is at least one W bead in that cluster located within less than Rcrit. For a system that contains only W beads, due the DPD repulsions, most W beads will not have another W bead located within a very small distance Rcrit (of say 0.5rc ≈ 0.35 nm). But for Rcrit beyond the value for which the first peak in the pair correlation of g(r)W−W occurs (at ∼0.83rc = 0.59 nm), nearly each bead is contained within the same cluster. Figure 4a and 4b display the largest cluster size versus DPD time for Rcrit = 0.75rc and 0.85rc, respectively. Beyond ∼1 × 105Δt these sizes increase in the order I < II < III < IV. The fraction of W beads, Fr(W), contained within the largest cluster sampled over the DPD time window of 1.8 × 105Δt to 2 × 105Δt is plotted against the architecture label in Figure 4c. The pore connectivity is thus lowest for architecture I that contains amphiphilic blocks, and for the grafted amphiphilic blocks (II, III, IV) increase with side chain length. Monte Carlo Tracer Diffusion through the Pore Networks. A profound estimation of the effect of pore connectivity on diffusion is obtained by performing lattice MC walks onto mapped replicas of the DPD morphologies. Tracer particles are restricted to move through the pore networks.43 From the DPD bead positions a “W” and “W+C” phase (or pore network) is constructed on a cubic grid that contains 10L3 = 2.7 × 107 nodes. Each node for which the nearest bead is a W

Figure 2. DPD morphologies and their periodic images sampled at 2 × 105Δt. For the sake of visualization all beads in the top front boxes are removed. In the lower half of each image only C and W beads are displayed. A beads within amphiphilic blocks (red, dark gray), A beads within hydrophobic block (orange, gray), W beads (blue, dark gray) C, beads (yellow, light gray). Visualization was generated using the VMDvisual molecular dynamics package.69

the pore boundaries or hydrophobic−hydrophilic interface. This is because they are connected via springs to hydrophobic polymer A beads, and the only way to prevent A beads to enter the hydrophilic W−C phase is to place the C beads near the pore boundaries. In the upper part of the images hydrophobic A beads that belong to the amphiphilic block are colored differently from those located within the hydrophobic block. For architecture I A beads within the hydrophobic blocks are segregated from those located within the amphiphilic blocks. Within the amphiphilic phase the water containing pores are located. The extent of segregation between the amphiphilic and hydrophobic blocks becomes progressively less for architecture II, III and IV. An estimate for the distance between water containing aggregates, or amphiphilic aggregates, can be obtained from the pair correlation functions g(r)W−W of the W beads, and g(r)C−C of the C beads. Their time dependencies are displayed in Figure 3a−d. For all four copolymers the positions of the maxima and minima in g(r), indicated by vertical bars in Figure 3a, shift toward larger distances with time. This is an indication that amphiphilic aggregates become separated further apart with simulation time. 5637


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Figure 3. W bead (solid curves) and C bead (dashed curves) pair correlation functions at various DPD time for the membranes (a) I, (ACA3)10−A50, (b) II, (A2[C]A2)10−A50, (c) III, (A2[AC]A)10−A50, and (d) IV, (A[A3C])10−A50. Vertical bars in panel a are located near minima and maxima. (e) g(r)W−W at 2 × 105Δt. Inset: Dinter(W−W) and Dintra(W−W) vs ⟨Nbond⟩amphi; (f) g(r)C−C at DPD time t = 2 × 105Δt. Inset: Dinter(C−C) and Dmin(C−C) vs ⟨Nbond⟩amphi.

bead belongs to the “W” phase, and nodes for which a nearest bead is a W or a C bead belong to the “W+C” phase. The fraction of nodes that joins the W (W+C) phase is close to the respective W and W+C bead fraction of ∼0.1605 (=NW(NW + NC + NA)−1) and ∼0.244 (=(NW + NC) (NW + NC + NA)−1). Diffusion coefficients are determined by tracking the random movement of two thousand tracer particles (Ntracer = 2000) that are put at MC time t′ = 0 on randomly selected nodes that belong to the pore phase. At each Monte Carlo step (MCS) for each particle a jump trial toward a neighboring node in a random direction (±x, ±y, ±z) is selected. A trial is accepted when that site belongs to the pore phase. Diffusion constants, D, are derived from the particles’ mean squared displacements

(MSD) (eq 6a). By plotting MSD against MC time, D is derived from the slope d(MSD)/dt′ (eq 6b). MSD =

D=

1

lim

Ntracer t ′→∞

d dt ′

1 d lim 6Ntracer t ′→∞ dt ′

1 = d(MSD)/dt ′ 6

N

∑ |R⃗ i(t′) − R⃗ i(0)|2 i=1

(6a)

N

∑ |R⃗ i(t′) − R⃗ i(0)|2 i=1

(6b)

R⃗ i(t′) is the position of particle i at time t′. From the slopes the value 6D (eq 6b) is obtained. MSD curves are shown in Figure 5638


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Figure 4. Fraction (Fr(W)) of total number of water beads contained within largest connected W cluster versus DPD time for architecture I, (ACA3)10−A50; II, (A2[C]A2)10−A50; III, (A2[AC]A)10−A50and IV, (A[A3C])10−A50. (a) Rcrit = 0.75rc (0.54 nm), (b) Rcrit = 0.85rc (0.62 nm). (c) Fr(W) vs architecture with averages taken over the time window 1.8 × 105Δt to 2 × 105Δt.

Figure 5. Examples of MSD versus MC time (at DPD time 1.4 × 105Δt). (a) Diffusion through the W pore networks. (b) Diffusion through the W +C networks. Data points indicated by crosses represent pure water.

Figure 6. MC diffusion constants (a) DMC(W) and (b) DMC(W+C) against DPD time obtained for architecture I−IV. Solid lines are averages obtained over the time window t ± 6 × 103Δt.

5. Diffusion is always fastest through the W+C network as compared to diffusion through the W network. For both pore

network definitions diffusion increases in the order I < II < III < IV. 5639


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in the order I < II < III < IV, similar to the dependence of DMC(W) and DMC(W+C) on architecture (Figure 5a,b). Figure 7 includes MSD calculated for a pure water system. The time evolutions of DDPD, sampled over consecutive time intervals of 20.000Δt are displayed in the inset of Figure 7. The values seem to converge beyond ∼1.5 × 105Δt. The DDPD values obtained for the time window t = 1.8 × 105Δt to 2 × 105Δt are listed in Table 2. The dependence of DDPD on architecture is the same as that obtained from the MC-grid calculations, with DDPD being larger or close to DMC(W+C).

For pure water every jump trial is accepted. The MSD is expressed in units of (internode distance)2 and time in MCS. The slope, d(MSD/dt′, obtained for pure water is equal to 1 (internode distance)2/MCS. The diffusion constant DMC relative to that of pure water is thus given by the slopes. The slopes, Δ(MSD)/Δ(MCS), are determined over the time interval 1 × 106 to 2 × 106 MCS. DMC is then given by eq 7. DMC =

MSD(t ′ = 2 × 106MCS) − MSD(t ′ = 1 × 106MCS) 106 (7)

IV. DISCUSSION For each architecture the hydrophobic block was the same (A50) and within the amphiphilic block the composition was fixed (C bead ratio |C| = 5−1) but its architecture was systematically varied. The characteristic distances (∼10 nm to 14 nm) between hydrophilic aggregates are much larger than in a previous work.43−45,53−63 This is due to the, on average, larger number of bonds (DPD springs) between A beads and C beads for the current architectures. Here ⟨Nbond⟩ is between 16.5 and 17.5 (Table 1), while in prior studies58−60 ⟨Nbond⟩ was between 2.5 and 6. The Flory−Huggins χ parameters between A, W, and C beads cause the system to evolve toward morphologies with maximized (minimized) number of hydrophobic−hydrophobic (hydrophobic−hydrophilic) interactions. The A beads then agglomerate into a hydrophobic phase from which hydrophilic W and C beads are expelled. The W and C beads form the hydrophilic pore network. Any A bead belonging to the amphiphilic block (label nr. 1−40 in Figure 8a) is always in the proximity of a C bead. This is because ⟨Nbond⟩amphi (listed in Table 1) is between 1.5 (architecture I) and 2.5 (architecture IV). These A beads are thus forced to be located in the proximity of the pore boundaries. Only A beads that belong to the hydrophobic block and for which Nbond is large are capable to be separated far away from the hydrophobic−hydrophilic boundaries. As a result A beads within the hydrophobic blocks are segregated from A beads located within the amphiphilic blocks. This is demonstrated in Figure 8a in which the average distance that each A bead is separated from the nearest W bead is plotted against label number. The A beads located within the amphiphilic blocks (label numbers 1 to 40) are, on average, located within 0.6 and 1 nm from the nearest W bead. The A beads that belong to the hydrophobic blocks (label nr. 41−90) are much further away from a nearest W bead. The actual (average) distance that these A beads are separated from a nearest W bead increases with label nr. up to label nr. ∼60 (Figure 8a). When plotted against Nbond (Figure 8b) these averages increase until Nbond approaches ∼30. Near this value the nearest W bead is located on average at 1.6 nm (architecture I) and 1.75 nm (architectures III and IV). Note that these are averages of a distribution of 680 distances for each label number and the actual (characteristic) size of the hydrophobic phase is much larger. In Figure 8c the largest distance between A beads and the nearest W bead is plotted. At least for one A bead located within the hydrophobic block (label nr. 41−90) is separated 3.8 nm (architecture II) to 4.2 nm (architecture III) from the nearest W bead. The diameter of the hydrophobic aggregates is thus ∼8 nm and the average distance between amphiphilic aggregates must be larger than 8 nm. This agrees with the observed ∼10−14 nm locations of the maxima in g(r)W−W (Figure 3e,f). In Figure 13 of ref 58 linear fitting for the data

The evolution of DMC(W) and DMC(W+C) with DPD time is plotted in Figure 6a,b. Initially DMC increases sharply with DPD time. This means that the water clusters become better connected. The averages sampled over the time window 1.8 × 105Δt to 2 × 105Δt are listed in Table 2. Table 2. Locations of Maxima in g(r)W−W within the 10−14 nm Region (2nd Column), Approximate Position of Shoulder in the 3−5 nm Region at t = 2 × 105Δt (3rd Column)a amphiphilic block

Dinter (nm)

Dintra (nm) or DCl−Cl

DMC (W)

DMC (W +C)

Ddpd

I: (ACA3)10 II: (A2[C] A2)10 III: (A2[AC] A)10 IV: (A[A3C])10

14.4 12.1

3.5 3.75

0.052 0.120

0.156 0.208

0.249 0.269

11.95

4.3

0.190

0.311

0.331

10.8

4.5

0.245

0.377

0.402

a DMC(W), DMC(W+C), and Ddpd are listed in 4th, 5th, and 6th column, respectively (averages are sampled over the time window 1.8 × 105Δt to 2 × 105Δt).

DPD W Bead Diffusion. Within the membrane the water diffusion coefficient, DDPD, relative to the pure water diffusion coefficient of 2.3 × 10−5 cm2/s is calculated by comparing the MSD of W beads through the morphologies during the DPD simulation with the MSD calculated for a system that contains only W beads. DDPD is then simply calculated from the ratio of both slopes when MSD is plotted against DPD time. The DPD morphologies are continuously subjected to rearrangements.57 Therefore, DDPD might be larger than DMC(W) and DMC(W +C).55,57 Figure 7 gives examples of the W bead MSD with respect to their positions at t = 160.000Δt. The slopes increase

Figure 7. Examples of W bead MSD during a DPD simulation for architectures I−IV over time window 1.6 × 106Δt to 1.8 × 106Δt. The MSD for a system composed of only W beads is indicated by crosses. Inset: DDPD versus DPD time. 5640


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Figure 8. Average separation between A beads and a nearest W bead plotted against (a) A bead label number and (b) Nbond. (c) The maximum distance that these A beads of certain label number are separated from a nearest W bead. The data are sampled for morphologies generated at DPD time 2 × 105Δt.

I), ∼3.75 nm (II), ∼4.3 nm (III) and ∼4.5 nm (IV) (inset Figure 3e). For the current architectures a DCl−Cl increase with ⟨Nbond⟩amphi seems therefore to be retained here. For C beads uniformly distributed on the scale of the polymer length it was found that water diffusion generally increases with ⟨Nbond⟩ when the IEC (or |C|−1) is kept fixed (see for example Figure 15 in ref 59 or Figure 14 in ref 58). For the diblock copolymer architectures a similar increase of diffusion with ⟨Nbond⟩amphi is obtained in Figure 9. For the amphiphilic block with the longest side chains (architecture IV) the DMC(W) value of ∼0.25 (Table 2) is actually the highest that is obtained at ϕw = 0.16 among all DPD-MC studies thus far. Even more intriguing is that this high relative diffusion was

points obtained for architectures with ⟨Nbond⟩ between 2 and 6 resulted in the maxima in g(r)W−W to be located at 0.7 ⟨Nbond⟩ + 2.5 nm. For ⟨Nbond⟩ = 16.5 this indeed extrapolates to 14 nm. However, for the current architectures the g(r)W−W maxima actually decreases with ⟨Nbond⟩ which is opposed to the trends found in refs 58−60. This may be indicative for the onset of formation of hierarchical morphologies. Hierarchical morphologies can be easily imagined for very long polymer architectures, viz. A5000-(A2CA2)1000. Then the A5000 blocks, for which a vast majority of A beads have a very large Nbond value, fully segregate from the amphiphilic ((A2CA2)1000) blocks that associate with water. For ϕw = 0.16 the hydrophobic blocks then fill up a volume fraction of ϕHfobic‑block ≈ 0.84/2−0.42. Within the remaining volume (fraction = 0.58) a pore network then establishes with properties (pore size, diffusive property) similar as for a fully amphiphilic-block/water system (A2CA2/W) but with water contents ϕw′ = 2 × ϕw. Previous DPD work on amphiphilic polymers revealed that the distance between pores (DCl−Cl) increases with ⟨Nbond⟩amphi. Since ⟨Nbond⟩amphi increases as I < II < III < IV the distance between pores is then also expected to increase in that order for the current architectures. For architectures I−IV such an increase with ⟨Nbond⟩amphi is not easily deduced from g(r)W−W. At ϕw = 0.16 the expected average distance (assigned as DCl−Cl in refs43−45 and 53−63) between pores (or water clusters) estimated from DCl−Cl ≈ 0.7⟨Nbond⟩ + 2.5 nm is between ∼3.55 nm (⟨Nbond⟩amphi = 1.5) nm and ∼4.25 nm (⟨Nbond⟩amphi = 2.5). But, as mentioned in the Results and Analysis section, near 4 nm a shoulder or maximum occurs which is located near ∼3.5 nm (architecture

Figure 9. Tracer diffusion coefficients DMC(W), DMC(W+C), and DDPD plotted against ⟨Nbond⟩amphi. All values are averages over time window 1.8 × 105Δt to 2 × 105Δt. 5641


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The Journal of Physical Chemistry B obtained for an architecture for which |C|−1 = 10. In the DPDMC modeling studies so far for that C bead density the (A[A8C]-5) architecture revealed the highest diffusion constant with DMC(W) = 0.0459 at ϕw = 0.16. The ∼5 times higher diffusion constants obtained for architectures III and IV is thus related with the segregation between the amphiphilic blocks and hydrophobic blocks. As a result all the water is associated with the amphiphilic block phase. Since the C bead density within the amphiphilic block is |C|−1 = 5, the effective hydration level is then twice the nominal value of 0.16, which strongly enhances long-range water transport. The sulfonation degree (cq. IEC) was the same within each amphiphilic block. In this way solely the impact of amphiphilic block design on pore connectivity and diffusion could be studied, resulting in clear relations with ⟨Nbond⟩amphi. Experimental confirmation of the dependence of water and proton diffusion on ⟨Nbond⟩amphi is difficult to obtain from the literature. Noteworthy Park et al.70−72 synthesized a series of polystyrenesulfonate-block-polymethylbutylene (PSS-b-PMB) copolymers. The amphiphilic PSS blocks contain partially sulfonated styrene rings and the PMB blocks are hydrophobic. Park et al.70−72 varied the length of the hydrophobic and amphiphilic blocks while keeping their volume fractions nearly the same. These architectures are thus somehow alike those displayed in Figure 1. Park et al.70−72 also varied the sulfonation degree within the amphiphilic block. Depending on sulfonation degree their diblock copolymers phase separate toward lamellar, hexagonally perforated lamellae, hexagonally packed cylinder, and gyroid morphologies.70−72 When hydrated the sulfonated styrene within the (amphiphilic) PSS phase associate with water leading to the formation of hydrophilic domains with water containing pores. This is actually what is also observed for the morphologies in Figure 2, in which the hydrophobic and amphiphilic blocks segregate while simultaneously within the amphiphilic phase the acidic sites (C beads) associate with water. Differences in pore morphology (Figure 2) among architectures are observed when ⟨Nbond⟩amphi is varied, with distances between pores (and thus their size) increasing with ⟨Nbond⟩amphi. Park et al.70−72 noticed also a difference in pore size within the amphiphilic phase when the architecture of the amphiphilic block is varied. They mentioned that when the sulfonation degree (ie. IEC) is increased from 20% to 40% the size of the water-rich domain spacing within the amphiphilic phase decreases from 4.2 to 3.0 nm.72 An increase in sulfonation degree automatically implies a decrease in ⟨Nbond⟩amphi, therefore their result seems to confirm that an increase of ⟨Nbond⟩amphi leads to larger distances between pores within the amphiphilic phase. The interesting prediction in the present DPD study is that the water rich domain spacing can also be controlled by control of ⟨Nbond⟩amphi while keeping the sulfonation degree (or IEC) within the amphiphilic block the same. Model diblock copolymer architectures were considered without taking into account bending constraints. The phase separated morphology within real membranes may depend on the stiffness of the polymers’ backbone and/or side chains. Polymer stiffness can be implemented in a DPD simulation in a rather straightforward way. Such an implementation was done very recently in a DPD study on Nafion by Vishnyakov and Neimark49 for the same system size as in this work with box edge length ∼21 nm. For this system size morphologies cannot be obtained by conventional MD from a random mixture of

water and Nafion49 or for the diblock copolymers considered here. Komarov et al.73 noted in their DPD study on AnB24‑n block copolymers there might be some biasing of the morphology with system size and the importance to match the size of the simulation box with the periodicity of the (supposed) equilibrium morphology. Therefore, they varied the box edge length between L = 20 and L = 36. Although here such a systematic variation was not performed, it is noted that in prior trial studies (not published) for L = 24 a similar dependence of diffusion on architecture was obtained as in Figure 9. It would be interesting to verify whether the results obtained here are also representative for much larger system size in future. It might also be worthwhile to consider other side chain architectures (e.g., branched or Y-shaped60,61) and/or to have differences in Flory−Huggins parameters between water beads and the hydrophobic beads located within the amphiphilic and hydrophobic blocks. The main reason to select ϕw = 0.16 is that trends could be obtained and the results could be compared with previous DPD-MC calculations that were performed at this same water contents. Percolation thresholds for various architectures have been estimated.54,56,59,60 Among those with C bead density |C| = 0.1 ϕp was between 0.13 (A[AC]A15[AC])-6)56 and 0.27 (A8[AC]-12).56 When selecting for the current diblock polymers ϕw much smaller than 0.16 then diffusion might be absent making comparison between architectures meaningless. Selecting ϕw much larger than 0.16 might be unphysical for |C| = 0.1 since for real membranes the water uptake increases with IEC. For instance, for Nafion when boiled in 0.2% NaOH, the weight % water uptake depends strongly on EW and is between 6.3 wt % (EW = 1790) and 42 wt % (EW = 944).2 The equilibrium water contents, ϕw(eq ), decrease as ϕw(eq ) ≈ 4 × 105 × EW−2. For Nafion (assumed dry mass density ∼2.2 gr/ cm3) the volume of a EW1600 repeat unit (assuming side chains equidistantly distributed along the backbones) is then ∼1.2 nm3, which is equivalent to 10 DPD bead volumes. For Naf1600 a 8.1 wt % mass gain was reported,2 this corresponds with ϕw(eq) ≈ 0.15, close to the ϕw = 0.16 assumed for the diblock architectures I−IV for which also |C| = 10−1. It is noted that for realistic (e.g., PFSA) membranes the amount of absorbed water can depend, besides on IEC, also on the preparation method2,4 and the extent of crystallite formation within the hydrophobic phase. DPD does not take into account crystallization effects. For architectures I−IV ϕw(eq) is not known. Since for ϕw = 0.16 pore networks are different among architectures I−IV, the number of unfavorable (e.g., W-A) contacts might also vary. Therefore, despite the same composition among membranes I−IV, differences in architecture might well result in different ϕw(eq). Within real polymeric membranes also interchain cross-links and entanglements occur. These and the possible formation of small crystals are not dealt with in this DPD study but might affect the swelling capacity. When these would be taken properly into account in future DPD studies then it might be worthwhile to model swelling in an asymmetric box (e.g., Lx × Ly × Lz, Lx ≫ Ly = Lz, e.g., 200 × 30 × 30), with initially the (preequilibrated) dry or modestly hydrated polymer membrane placed in the center of the box (e.g., 75 < x < 125) and remaining volume occupied with water. The membrane will then absorb water and expand until an equilibrium volume fraction of water (ϕw(eq) is established accross the membrane. ϕw(eq) can then be estimated from the water density profile 5642



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along the x direction and/or from the locations of the membrane−water interfaces. As noted in section II, χ values resemble those calculated for Nafion, but A and C beads might represent different molecular fragments than in DPD parametrizations of PFSA polymers,42,46,47,49 when χ parameters are of comparable value. For SPEEK two solubility parameters δ1 and δ2 are observed74 with δ1 = 26.4 MPa0.5 (corrsponding with unsulfonated hydrophobic A fragments) and δ2 = 35.7 MPa0.5 (partly sulfonated hydrophilic C fragments). For bead sizes of V = 0.12 nm3 these correspond with χAC ≈ 2.5 and χAW ≈ 3.9,59 being lower than χAC = χAW = 4.9 in this work. When these values would be chosen then diffusion constants are expected to differ from those in Table 2, but the trends are anticipated to be the same, with DMC(W) decreasing with architecture in the order I < II < III < IV. Also in a DPD-MC study53 on block polymer architectures with C beads distributed at various positions along the backbones, a reduction of χAW = χAC from ∼5.3 to ∼2.5 resulted in the same dependency of DMC(W) on architecture, but with the trends becoming less pronounced (Figure 3a of ref 53). Finally, in this and other DPD-MC studies43−45,53−63 χ parameters were assumed not to be affected by the architecture, which might be an oversimplification for real polymers. The compatibility, or χ parameter, between molecular A and C fragments within an, for example, ACACACACACAC architecture might be different from that within AAAAAACCCCCC or A4[A2C]C4 due to cooperative effects (e.g., polarization) or steric constraints. Such issues were beyond the scope of this work, although effective χ parameters might depend on molecular architecture.75

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V. CONCLUSION Within fuel cell membranes the water containing pores act as ion channels for protons and a good connection between them is essential for fuel cell applications. The pore formation was studied here for alternative model diblock copolymer architectures. Here microphase separation within four types of (hydrophobic_amphiphilic) copolymer membranes that contain 16 vol % water was studied by DPD. The composition of the hydrophobic and amphiphilic blocks were kept the same but the architecture of the amphiphilic blocks was systematically varied: For one architecture the hydrophilic fragments were distributed uniformly within a linear block, in three other architectures the hydrophilic fragments were end-grafted at side chains that spring off at regular intervals along the backbone. Monte Carlo derived tracer diffusion constants revealed that water diffusion is slowest for the block-type architecture. For the grafted architectures diffusion was found to increase with side chain length. Solvent diffusion is predicted to increases with increase of the parameter ⟨Nbond⟩ within the amphiphilic blocks where ⟨Nbond⟩ is the average number of bonds between hydrophobic A beads and the nearest C bead.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] Tel.: +81 80 3396 4286. Notes

The authors declare no competing financial interest. 5643


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