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Water diffusion within hydrated model grafted polymeric membranes with bimodal side chain length distributions G. Dorenbos* The effect of bimodal side chain length distributions on pore morphology and solvent diffusion within hydrated amphiphilic polymeric membranes is predicted. Seven polymeric architectures are constructed from hydrophobic backbones from which at regular intervals side chains branch off that are alternatingly short (composed of p hydrophobic A fragments or beads) and long (q A fragments, q > p). The side chains are end-linked with a hydrophilic C fragment. Pore morphologies at a water volume fraction of 0.16 are calculated by dissipative particle dynamics (DPD). Water diffusion through the water containing pores is calculated by tracer diffusion calculations through 140 selected snapshots and from the water bead motions. Diffusion constants decrease with difference in side chain lengths, q p. Overall, the distance between pores also decreases with q p. The results are explained by counting for every
Received 5th January 2015 Accepted 11th February 2015
architecture the average number of bonds hNbondi between an A and the nearest C fragment. These results are in line with a database that contains more than 60 architectures. Diffusion constants tend to
DOI: 10.1039/c5sm00016e
increase linearly with hNbondi|C|1|A|, where |C| and |A| are the C and A bead fractions within the architecture. hNbondi is therefore expected to be an interesting design parameter for obtaining low
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percolation thresholds for solvent and/or proton diffusion.
Introduction Proton exchange membranes (PEMs) are used in fuel cells where their purpose is to separate the anode from the cathode and to allow protons to ow from the anode to the cathode. Naon® is mostly used as a PEM. Naon polymers are constructed from hydrophobic (–CF2–CF2–) backbones to which side chains are attached with pendant hydrophilic SO3H groups, see Fig. 1. When exposed to humid environments water is absorbed1–9 and microphase separation occurs with the acidic sites and water forming a hydrophilic pore network. The pores are a few nm in diameter and provide the diffusive pathway for protons.6–8 One route to increase proton transport in a PEM is therefore by optimizing the pore morphology. Molecular dynamics (MD)10–24 has been frequently applied to model microphase separation within Naon. Due to the computational demand MD simulations are mostly performed for small system size. In order to gain insight in pore connectivity simulations for length scales longer than that of corresponding Bragg spacings are required. For Naon the Bragg spacing increases with the water volume fraction (fw), typically from 3 nm (fw ¼ 0.1) to 5 nm (fw ¼ 0.3).9 Atomistic simulations for large system size O(104 nm3) containing several million atoms have been performed,12,24 but these require pre410-1118, 1107-2 sano, Belle Crea 502, Susono-shi, Shizuoka-ken, Japan. E-mail: [email protected]
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assumed starting morphologies,24 or several cycles of system size increase by including periodic images that are again equilibrated.12 Examples of modeling approaches used to generate equilibrium morphologies of Naon of size O(104 nm3) are coarse grained MD (CGMD),25,26 the bond uctuation model,27 selfconsistent approaches28,29 and dissipative particle dynamics (DPD).30–36 Yamamoto and Hyodo30 calculated for the system size 2 104 nm3, the pore structures for hydrated Naon1200 (1200 is the equivalent weight (EW) of 1200 g polymer per sulfonic site). In their DPD study molecular fragments are coarse grained into beads. From the pair correlation between water (W) beads the corresponding X-ray scattering proles were estimated. The wave factor qmax, at which a maximum in the scattering occurs, was found to decrease with water uptake. This agrees with experimental observations9 and indicates that water clusters increase in size and get separated further apart with hydration level.
Fig. 1
Repeat unit of Nafion.
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Naon for which the side chains are assumed to be uniformly distributed along the polymer backbones was studied later as a function of EW.31 By mapping the hydrophilic pore phase onto a cubic grid, and by restricting tracer particles to move through the hydrophilic phase, water diffusion was mimicked by Monte Carlo (MC) simulation. Water diffusion constants derived from the particles' mean square displacement (MSD) match experimental values well when the water mobility within the pores is assumed the same as in pure water. This is an oversimplication, but quasi elastic neutron scattering (QENS) studies37,38 on Naon revealed that within the pores the water diffusion constant is close to that of pure water. At a hydration level of l ¼ 5 water molecules per SO3H site a local water mobility of 1.6 105 cm2 s1 was measured,38 which is close to the pure water value of 2.3 105 cm2 s1. Various studies34,39–44 in which DPD is combined with MC tracer diffusion calculations reveal for model polymeric membranes of the same ion exchange capacity (IEC) that pore morphology and water diffusion depend strongly on architecture. The main trends deduced from these studies are overviewed below. When for block polymers successive hydrophilic fragments are separated by x and y (y > x) connected hydrophobic backbone beads (i.e. a bimodal distribution of C beads within the backbone) pore size and water diffusion increase with bimodal character (or asymmetry parameter y/x).40 For graed polymers with side chains arranged at regular intervals along the backbone (uni-modal distribution) and with a single pendant hydrophilic fragment, the pore size and water diffusion coefficients increase with side chain length.41 When the side chains are kept at the same length but the inter-branching distance along the backbone is alternatingly long and short (y > x, bimodal distribution), while keeping the IEC xed, pore size and diffusion increases with increase of bimodal character, or y/x.34,42,43 Diffusion can be increased further when instead of one side chain branching off from a backbone bead, two side chains, each of the same length, are allowed to branch off.44 Such a kind of architecture can be considered as a bimodal side chain distribution with an innite asymmetry parameter value (y/x, x ¼ 0). For the graed model membranes diffusion and morphology depend thus strongly on y/x (xed side chain length) and on side chain length (for xed y/x ¼ 1). The purpose here is to know how pore size and diffusion are affected for model architectures with a uniform side chain distribution (y/x ¼ 1) but for which the length of the side chains along the backbones are alternatingly short (l1-1 A beads) and long (l2-1 A beads). When the lengths of both side chains are varied while simultaneously their total lengths l1 + l2 are kept constant, does then a long side chain in combination with a short side chain result in the largest cluster size and diffusion constant, or does a unimodal distribution (l1 ¼ l2) result in the largest cluster size/water diffusion? In Computational details the polymeric architectures are dened and the DPD simulation method and its current parameterization are outlined. Pore morphologies obtained for seven architectures and diffusion through the pores are
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Soft Matter
presented in Results and analysis. In Discussion the results are explained and by comparing with other DPD studies a consistent picture evolves.
Computational details A
Polymer chain architecture
In DPD single beads represent molecular fragments. The architecture type is given in Fig. 2. Each architecture consists of a hydrophobic backbone that contains A beads from which at regular intervals (corresponding to x A–A bond distances) side chains branch off. The lengths of the side chains are alternatingly short and long and contain respectively p and q A beads, with q $ p. Each side chain is end-linked with a hydrophilic C bead. The DPD formulae of the architectures are listed in Table 1 and can be split into two groups for which the C bead fractions are the same. Within each group the distance between branching points and the total length of both side chains (p + q) is kept the same. The number of repeat units is 3. All bead volumes correspond to that of the volume of 4 water molecules and are 0.12 nm3. Water is represented by W beads. For architectures I–III the acidic site density |C| ¼ 1/6.5 and the ion exchange capacity (IEC) expressed in mmol g1 for a polymer with a mass density of 1 g cm3 would correspond to 2.13 mmol g1 or an EW of 470 g mol1. For architectures IV–VII the acidic site density |C| ¼ 1/8 and the IEC corresponds to 1.73 mmol g1 or an EW of 578 g mol1. B Dissipative particle dynamics DPD has been applied to model phase separation,46 membrane rupture,47 vesicles48,49 and drug delivery.50 DPD was introduced by Hoogerbrugge and Koelman.45 The developed DPD version by Groot and Warren51 is followed here. Beads i and j interact with R each other via conservative FCij , dissipative FD ij , and random Fij forces. The force acting on bead i is: X FijC þFijR þ FijD (1) fi ¼ jsi
The sum is over all particles j that are located within a distance rc that denes the length scale. The conservative force is so repulsive and decreases with distance: 8 > < aij 1 rij ^rij rij \rc C r c (2a) Fij ¼ > : 0 rij $ rc
Fig. 2 Repeat units of the polymer architectures listed in Table 1. In the present work p + q ¼ 5 for x ¼ 3 architectures and p + q ¼ 6 for x ¼ 4 architectures, and q $ p.
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Table 1 Architectures Ax[ApC]Ax[AqC] with p + q ¼ 5 for x ¼ 3 (architectures I, II, III) and p + q ¼ 6 for x ¼ 4 (architectures IV, V, VI, VII). hNbondi is the average number of bonds between A beads towards a nearest C bead. DCl–Cl, DMC(W) and DMC(W + C) are given in the 7th, 8th and 9th columns, respectively. Values in parenthesis were obtained from additional DPD runs with DMC(W) and DCl–Cl sampled over time window 5 104Dt to 6 104Dt and DMC(W + C) sampled over 4 104Dt to 6 104Dt
Architecture
|C|
p+q
(p, q)
qp
hNbondi
I: A3[C]A3[A5C]-3 II: A3[AC]A3[A4C]-3 III: A3[A2C]A3[A3C]-3 IV: A4[C]A4[A6C]-3 V: A4[AC]A4[A5C]-3 VI: A4[A2C]A4[A4C]-3 VII: A4[A3C]A4[A3C]-3
1/6.5 1/6.5 1/6.5 1/8 1/8 1/8 1/8
5 5 5 6 6 6 6
(0, 5) (1, 4) (2, 3) (0, 6) (1, 5) (2, 4) (3, 4)
5 3 1 6 4 2 0
30/11 ¼ 32/11 ¼ 34/11 ¼ 45/14 ¼ 48/14 ¼ 51/14 ¼ 52/14 ¼
rij ¼ ri rj, rij ¼ |rij|, ^rij ¼ rij/|rij|
(2b)
ri and vi, are position and velocity of bead i, and ^rij is the unit vector in the direction of bead j towards bead i: the random force (eqn (3)) introduces thermal noise. The dissipative force is proportional to the beads' relative velocity (eqn (4)). R 0.5 FR (kBT)1^rij ij ¼ su (rij)zij(Dt)
(3)
D rij$vij)^rij FD ij ¼ gu (rij)(^
(4)
vij ¼ vi vj, uD and uR are weight functions, randomness is introduced by a randomly uctuating variable with Gaussian statistics with zero mean and unit variance: hzij(t)i ¼ 0, and hzij(t)zkl(t0 )i ¼ (dikdjl + dild4k)d(t t0 ). uD and uR are given by eqn (5). 8 2 rij > < rij \rc 1 2 rc uD rij ¼ uR rij ¼ (5) > : 0 rij $ rc Noise (s) and friction (g) parameters are related as s ¼ 2gkBT,52 with s ¼ 3, g ¼ 4.5, kB the Boltzmann constant and T the temperature. The three forces conserve linear and angular momentum. Adjacent polymer beads are joined by a spring force with spring constant C ¼ 50 and equilibrium bond distance R0 ¼ 0.85 rc:
2.727 2.909 3.0909 3.214 3.429 3.643 3.714
DCl–Cl (nm)
DMC(W)
DMC(W + C)
4.31 (4.45) 4.51 (4.52) 4.66 (4.65) 4.77 (4.81) 5.03 (4.98) 4.98 (5.03) 5.09 (5.1)
0.029 (0.026) 0.052 (0.047) 0.077 (0.084) 0.004 (0.008) 0.032 (0.031) 0.035 (0.035) 0.056 (0.049)
0.090 (0.093) 0.139 (0.130) 0.184 (0.196) 0.015 (0.028) 0.061 (0.064) 0.086 (0.074) 0.102 (0.102)
parameters between beads of given chemical formulae were not performed. For the current architectures (Table 1) the adapted c parameter set is expected to result in well phase separated morphologies. These are identical as in ref. 34, 39–44 and 53 and comparable with those derived from calculated binding energies of the molecular fragments in the DPD parameterization of hydrated Naon.30 DPD repulsions and the c-parameters are related by eqn (7).40 Daij ¼ (4.16 0.15) cij
The water volume fraction is given by eqn (8), where NA, NW and NC are the total numbers of A, W and C beads, respectively. fw ¼
(6)
Bead motions evolve as: dri/dt ¼ vi, midvi/dt ¼ fii. The bead masses mi, unit of energy (kBT) and of time s ¼ rc(m/kBT)0.5 are all scaled to 1. The temperature is kept at kBT ¼ 1.0 by using a modied Verlet integration scheme51 with an empirical factor 0.65 and a time step Dt ¼ 0.05s. The bead density r is 3. The repulsions aij are the same as in ref. 40, 44 and 53 and are listed in Table 2. Repulsions between similar beads are 104 and reproduce the compressibility of water.30,34 The other repulsions correspond with the adapted Flory–Huggins c values (Table 2). For this set of c parameters already a database has been created and predicted trends in this work can be compared with previous work. Atomistic calculations to obtain c
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NW
NW þ NA þ NC
(8)
The simulations are performed for fw ¼ 0.161 The water content expressed in terms of l is given by eqn (9). l¼
2
FSij ¼ C(rij R0)^rij
(7)
NW 4 NC
(9)
Since fw ¼ 0.161, the l values are thus 5.0 for architectures I–III and 6.14 for architectures IV–VII. The system is cubic with an edge length L ¼ 24. One DPD length unit corresponds to rc ¼ (3 0.12 nm3)1/3 ¼ 0.71 nm. Edge lengths are L rc 17 nm, which correspond to a volume of 5 103 nm. The simulation box contains 892 (architectures I–III) or 725 (architectures IV–VII) polymer chains. The total amount of beads is rL3 ¼ 41 472. Periodic boundary conditions are applied. The physical time to which s corresponds for aww ¼ 104 is 130 ps. Morphologies are generated up to 4 104Dt which correspond to a time of O(0.2 ms). DPD bead
Table 2
cij aij
DPD repulsions and c-parameters A–A
W–W
C–C
A–W
A–C
C–W
0 104
0 104
0 104
4.9 124.4
4.9 124.4
2.6 93.2
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Fig. 3 Pore morphologies sampled at 4 104Dt obtained for architectures I–VII. C beads are colored yellow (light grey) and W beads are colored blue (dark).
coordinates are stored on le every 2000Dt. A total of 140 morphologies are stored. The VMD-visual molecular dynamics package54 is used to display the morphologies.
Results and analysis A
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The distance between water clusters, DCl–Cl, is estimated from the pair correlation function g(r) between W beads. The g(r) calculated for these seven morphologies are shown in Fig. 4(a and b). DCl–Cl is derived from the position of the maximum which is located between 4 nm and 6 nm. When comparing g(r) obtained for architectures for which the IEC are the same, the distances where the maxima in g(r) occur are the smallest for architectures I and IV. For these two architectures the difference in side chain lengths is the highest and p ¼ 0. This means that the combination of a short side chain and a long side chain alternatingly branching off from the backbone beads results in signicant lower DCl–Cl as compared with two side chains of the same length branching off from the backbone beads. When comparing Fig. 4(a) with Fig. 4(b) the distances at which the maxima occur are larger for the lower IEC architectures displayed in Fig. 4(b). Fig. 5(a) displays the dependence of DCl–Cl on DPD time. Aer an initial increase DCl–Cl seems to stabilize beyond around 2 104Dt. DCl–Cl obtained by averaging over the time interval 3 104Dt to 4 104Dt are listed in Table 1. The apparent convergence of DCl–Cl observed in Fig. 5(a) was conrmed by additional runs up to 6 104Dt, for which DCl–Cl averaged over 5 104Dt to 6 104Dt are on average within one percent (or 0.04 nm) of the values listed in Table 1. This is in line with DPD modelling on Naon at fw ¼ 0.2, for which the equilibrium of DCl–Cl was obtained at 2 104Dt (or s ¼ 1000).30,31 DCl–Cl is plotted against difference in side chain lengths, or q p, in Fig. 5(b). From Fig. 5(b) a clear trend appears that DCl–Cl decreases with difference in side chain lengths, q p. An increase in side chain length difference thus results in smaller DCl–Cl values.
Pore structures
Snapshots of morphologies obtained for the architectures listed in Table 1 are displayed in Fig. 3. For all structures the C beads are located at the borders of the water clusters and together with the W beads they form a hydrophilic pore network. When a comparison is made among the architectures that are of the same IEC it is the one for which one of the side chains are of equal (architecture III) or near equal length (architecture VII) for which the pores seem to be larger and more elongated than the architectures I and IV for which p ¼ 0.
B Diffusion through frozen W and W + C pore networks Previous DPD simulations on graed architectures showed that pore morphologies change with DPD time.44 This is caused by the motion of the polymer beads. Actually during the DPD simulations polymer motions are higher than in reality for several reasons. All beads have the same mass (reduced mass m ¼ 1). Since a W bead contains 4 water molecules its mass would correspond to 72 amu, while the mass of a polymer bead
W bead pair correlation functions for the architectures Ax[ApC]Ax[AqC]. (a) x ¼ 3, and p + q ¼ 5 and (b) x ¼ 4 and p + q ¼ 6. DPD time t ¼ 4 104Dt.
Fig. 4
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Fig. 5
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(a) DCl–Cl plotted vs. DPD time. (b) DCl–Cl plotted against side chain length difference.
(e.g. –CF2CF2CF2CF2–) of the same volume is typically a factor 2 or more higher. The introduction of higher masses for the polymer beads (e.g. mA ¼ mC ¼ 3; mw ¼ 1) will decrease the polymer bead diffusivity. Also bond crossings are allowed to occur in (conventional) DPD, which lis up entanglements and speeds up polymer dynamics. A third reason for the fast polymer motion is that in the current work the backbone lengths are relatively short (up to 28 beads for architecture VII). Real polymers can be much longer resulting in slow reptation motions. Since polymer motions are overestimated, it is instructive to derive water diffusion coefficients within frozen pore networks. Water clusters (pores) may become temporally less connected, or even disconnected. The time evolution of the pore connectivity cannot be derived from the W bead motions. But it can be studied by MC tracer diffusion calculations through frozen morphologies that are generated at dened DPD time intervals.39–44,53 Since C beads are distributed along the pore edges (Fig. 3), their pending motion might increase the volume fraction accessible for water molecules during the diffusion process, and enhance water diffusion. For this reason, in this section, two MC tracer diffusion constants, DMC(W) and DMC(W + C) are derived for diffusion through frozen hydrophilic “W” and “W + C” pore networks, respectively. These should be considered representative for the hypothetical case that all polymeric beads (DMC(W)) or only A beads (DMC(W + C)) are immobile. Since these diffusion constants uctuate with DPD time, the obtained averages sampled over consecutive snapshots will be compared with W bead diffusion coefficients in Section C. Another reason to calculate diffusion through frozen lattices is that the MC derived diffusion constants can be compared with those derived earlier for 60 other architectures (see Discussion). In the MC tracer diffusion calculations the particle movement is restricted to the hydrophilic pore phase.31 Diffusion coefficients are determined by following the random movement of several thousand particles on a grid of size that contains (10L)3 ¼ 2403 nodes. DPD bead positions sampled at time intervals of 2000Dt are used to construct pore networks. From the bead positions two pore networks are constructed on the grid: a “W” and a “W + C” pore network. For the W(W + C) pore network each node for which the nearest bead is a W bead (W or C bead) belongs to the pore phase. All other nodes belong to the
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polymer phase. For the 140 DPD morphologies saved on le thus 280 pore networks are constructed. The fraction of nodes that joins the W(W + C) pore network is close to the fraction of W(W + C) beads. For architectures I–III these are 0.161(¼(NW)(NW + NC + NA)1) and 0.29(¼(NC + NW)(NW + NC + NA)1). For architectures IV–VII the node fractions are 0.161(W) and 0.266(W + C). As in previous studies34,40–44,53 Ntracer ¼ 2000 particles are put at MC time t0 ¼ 0 at randomly selected nodes that belong to the pore phase. A jump trial towards a nearest node in one of the six orthogonal directions is randomly selected during every Monte Carlo Step (MCS). A jump trial will only be accepted when the aimed site also belongs to the pore phase. Usually molecular diffusion constants are derived from the ensemble average of the mean square displacement (MSD) of N molecules, given by eqn (10a). By plotting the MSD against time the diffusion constant D can be derived from the slopes d(MSD)/dt0 , eqn (10b). MSD ¼
D¼
2 N 1 d X ~ 0 ~i ð0Þ lim ðt Þ R R i N t0 /N dt0 i¼1
2 N 1 d X ~ 0 ~i ð0Þ ¼ 1 dðMSDÞ=dt0 lim 0 Ri ðt Þ R 0 6N t /N dt i¼1 6
(10a)
(10b)
In eqn (10a) and 10(b) ~ Ri(t0 ) is the position of molecule i at 0 MC time t . The term within the sum is thus the squared displacement of molecule i at t0 . When sampling is performed over a sufficient number of particles then, for long enough time, the slopes converge to the value 6D (eqn (10b)). MSD curves obtained for diffusion through the W pore networks sampled at DPD time t ¼ 40 000Dt are shown in Fig. 6(a) (architecture I–III) and Fig. 6(b) (architectures IV–VII). Also the pure water case for which every jump trial is accepted is included in Fig. 6(b). Since the MSD are expressed in units of (inter node distance)2 and time is given in MCS, the slopes d(MSD)/dt0 for the pure water case is (per denition) equal to 1. Therefore the diffusion constant, DMC, within the membrane relative to that of pure water is obtained from the slopes. The slopes of the curves (D(MSD)/D(MCS)) are calculated by performing a linear t over the time window t0 ¼ 1 106–2 106 MCS. DMC is then given by eqn (11).
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MSD curves obtained for diffusion within the morphologies generated at a DPD time t ¼ 4 104Dt. (a) Diffusion through W networks for architectures I–III. (b) Diffusion through W networks for architectures IV–VII. The pure water case (fw ¼ 1.0), for which the slope is 1, is indicated by crosses. (c) Diffusion through W + C networks for architectures I–III. (d) Diffusion through W + C networks for architectures IV–VII. Fig. 6
DMC ¼
MSDðt0 ¼ 2 106 Þ MSDðt0 ¼ 106 Þ 106
(11)
For both sets, diffusion is the lowest for architectures for which difference in side chain length is the highest. These are respectively architectures I (A3[C]A3[A5C]-3) (Fig. 6(a)) and IV (A4[C]A4[A6C]-3) (Fig. 6(b)). MSD curves for diffusion through the C + W networks are shown in Fig. 6(c) and (d). The same trends are obtained as for diffusion through the W pore networks, but diffusion is signicantly higher. In Fig. 7(a) and (b) DMC(W) and DMC(W + C) are plotted against DPD time, respectively. DMC values uctuate
signicantly with time because connections between pores can become better (resulting in enhanced DMC) or worse as DPD time proceeds. However, beyond 2 104Dt, DMC seems to uctuate around average values. DMC(W)(DMC(W + C)) averages over the DPD time window 3 104Dt to 4 104Dt(2 104Dt to 4 104Dt) are given in Table 1. From additional calculations it is conrmed that these averages are close to those obtained from averages (listed in parenthesis in Table 1) taken over the DPD time window 5 104Dt to 6 104Dt(4 104Dt to 6 104Dt). C
DPD W bead diffusion through dynamic pore networks
The water diffusion constants can also be derived from the W bead displacements during the DPD simulations. The diffusion
Fig. 7 (a) DMC(W) against DPD time. (b) DMC(W + C) against DPD time. Datapoints connected with solid lines were obtained for architectures with equal (p ¼ q ¼ 3) and near equal side chain (p ¼ 2; q ¼ 3) lengths. Dashed lines connect datapoints obtained for architectures with one of the side chains containing no hydrophobic A beads at all (p ¼ 0).
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MSD of the W beads for architectures for which x ¼ 3 (filled symbols) and architectures for which x ¼ 4 (open symbols). MSD of W beads for a system solely composed of W beads is also included. Fig. 8
coefficient of water molecules through the membrane, DMembrane , can be obtained by comparing DMembrane with the W W-bead H2 O water bead diffusion coefficient DPure for a system composed of W-bead Pure water W beads. DMembrane is then given by eqn (12a), with D , H2 O H2 O the experimental pure water diffusion coefficient (2.3 105 cm2 s1). ¼ DMembrane H2 O
DMembrane water W-bead DPure H2 O water DPure W-Bead
DDPD ¼
DMembrane W-bead water DPure W-Bead
(12a)
(12b)
During a DPD simulation pores can rearrange. Therefore W bead diffusivities are higher than DMC(W).40,44 In Fig. 8 the MSD of the W beads with respect to their positions at t ¼ 20 000Dt are plotted vs. DPD time. The MSD calculated for a system composed of only water beads is also included. From the slope water it follows that DPure ¼ 0.128. W-Bead For each of the 7 architectures, the DDPD values (calculated from the ratio of the slopes, eqn (12b)) are compared with DMC(W) and DMC(W + C) in Fig. 9. The trends obtained for DDPD and the MC derived diffusion constants are consistent with each other. The absolute diffusivities obtained by each of the three methods are not the same: DMC(W + C) is always larger then DMC(W) because there is much more volume fraction available for diffusion through the C + W network. Also the water bead diffusivities are larger than DMC(W). This is because
Fig. 9 (a) Comparison between W bead diffusion constants (DDPD) and MC derived diffusion constants (DMC(W) and DMC(W + C)).
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Fig. 10 Tracer diffusion coefficients DMC(W) and DMC(W + C) and W bead diffusion coefficients (DDPD) plotted vs. inter cluster distance DCl–Cl. Solid (dashed) lines serve as eye guides for architectures I–III (IV–VII).
C beads are always located near the W pore network (Fig. 3). The continuous rearrangement of the polymer A and C beads during a simulation effectively increases the space accessible to the W beads. Now imagine that all A beads are xed in space, the W beads can then access at most the whole hydrophilic phase which for architectures I–III and IV–VII constitute a volume fraction of fW+C ¼ 0.29 and 0.266, respectively. When supposing that C beads do not obstruct W bead diffusion, then DDPD is expected to approach DMC(W + C). Actually, the W bead diffusivities are higher than the DMC(W + C) values. This is because A beads also rearrange.44
Discussion Among the Ax[ApC]Ax[AqC]-3 architectures two parameters were varied. These are the inter-branching distance (equal to x A–A bond lengths) and the lengths (p + 1 and q + 1) of two consecutive side chains within a single repeat unit. DCl–Cl and the diffusion constants, DDPD, DMC and DMC(W + C) depend on these two parameters. Diffusion correlates strongly with the inter-cluster distance as illustrated in Fig. 10 where DMC(W), DMC(W + C) and DDPD are plotted vs. DCl–Cl. For membranes of the same IEC or C bead fraction diffusion increases with DCl–Cl. Larger pores become
Fig. 11 DMC(W), DMC(W + C), and DDPD vs. difference in side chain
lengths.
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Fig. 12 (a) DCl–Cl as a function of hNbondi and (b) DMC(W) vs. hNbondi. The drawn lines are linear fits for data obtained for similar IEC membranes. Inset: DMC(W) plotted against hNbondi |C|1 |A|.
thus better connected with regards to diffusion. This is in line with DPD-MC modeling studies on graed polymers that contain Y-shaped53 and linear31,34,41–44 side chains and block polymers.39,40 These studies also revealed that at the same IEC (or C bead fraction, |C|) an increase in DCl–Cl, or pore size, correlates with increased diffusion. As shown in Fig. 11 for a similar inter branching distance (given by x) diffusion decreases with difference in side chain lengths, q p. An increase in side chain length difference thus results in reduced water diffusion and generally also in lower DCl–Cl values (Fig. 5(b)). The reason for these trends lies in the fact that an increase in DCl–Cl also implies that the hydrophobic A region becomes larger in size. For the current set of interaction parameters (Table 2) the micro-phase separation process maximizes the number of hydrophilic–hydrophilic (W–C and W–W) and hydrophobic–hydrophobic (A–A) contacts and minimizes hydrophilic–hydrophobic (W–A) and (C–A) contacts. When bonds between polymer beads would be absent then this process nally results in a morphology in which a hydrophilic pool composed of C and W beads is fully phase separated from a majority A bead phase (as illustrated in Fig. 12 of ref. 40). For the present architectures, the system also tries to make as large as possible pores with minimal interface area. This is achieved by expelling the C beads from the hydrophobic A regions. The C beads accordingly arrange themselves along the pore walls, as indeed observed in Fig. 3. Due to the bonding constraints imposed onto the polymer beads, the hydrophobic A regions can only be large for architectures for which a prominent fraction of A beads is topologically far away (in terms of number of A–A and A–C bonds) from the nearest C bead in that architecture. The hydrophobic A region and DCl–Cl can only be large for that architecture for which the average number of bonds, Nbond, towards the nearest C bead is high. The average number of bonds, hNbondi towards the nearest C bead for each architecture is listed in Table 1. In Fig. 12(a) a clear correlation between DCl–Cl and hNbondi is observed. For architectures of the same IEC, diffusion generally increases with DCl–Cl (Fig. 10) and therefore increases also with hNbondi as shown in Fig. 12(b). The inset in Fig. 12(b) demonstrates that for the current architectures DMC(W) increases linearly with hNbondi |C| |A|1, where |C|(|A|) is the C(A) bead fraction within each architecture.
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The correlations that appear in Fig. 12 can also be deduced from DPD studies on graed membranes for which the c parameters are the same as here. The database is compiled in Table 3. DCl–Cl values from ref. 34 and 41 were obtained for two times smaller bead volumes and have been rescaled in Table 3 by multiplying them with a factor (0.12/0.06)0.333 1.26. In Fig. 13 DCl–Cl is plotted against hNbondi. Similar to Fig. 12(a) DCl–Cl increases linearly with hNbondi. In Fig. 14 the DMC(W) values in Table 1 are compared with those architectures listed in Table 3 for which C beads are only bonded with A beads (discarding architectures A3[A8[C][C]]-6, A3[A8C2]-6, A11[C][C]). When again a linear tting is performed on architectures for which the IEC is the same, DMC(W) increases with hNbondi. Also for xed hNbondi diffusion increases when the C bead fraction, |C|, is increased. Fig. 14(b) reveals that DMC(W) increases approximately linearly hNbondi |C| |A|1. This is in line with the inset in Fig. 12(b). A similar analysis can be performed for the percolation thresholds for diffusion listed in Table 3. The percolation threshold fp is dened as the water volume fraction at which the onset of diffusion occurs.41,42,55 fp has been determined for graed polymers with linear41,42 and Y-shaped55 side chains. The dependence of fp on hNbondi and IEC is shown in Fig. 15. The lines in Fig. 15 connect data obtained for architectures of the same C bead fractions (|C|1). Since fp decreases systematically with hNbondi, control and increase of hNbondi is an interesting parameter to achieve low percolation thresholds. From Fig. 15 it can also be observed that for architectures with similar hNbondi values the one with the highest IEC (largest |C|) is expected to reveal the lowest percolation threshold. When fp is plotted against hNbondi + 4 |1.612C|1 the data condense almost onto a single curve that is given by eqn (13). fp ¼ 0.389/[hNbondi + 4 (1.612 |C|)1]
(13)
For the architectures |C|1 is equal to 6.5 (I–III) and 8 (IV– VII) with hNbondi in the range 2.72–3.09 (I–III) and 3.21–3.71 (IV– VII). From Fig. 15 and eqn (13) the percolation thresholds for all these architectures are therefore expected to be within the range of 0.13 < fp < 0.18.
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DCl–Cl (5th column) and DMC(W) (6th column) at fw ¼ 0.16 for various grafted architectures. Percolation thresholds, fp, in 7th columns labeled with an asterisk were calculated for architectures that contain 12 repeat units and the corresponding hNbondi values are given within parenthesis in the 4th column. C and A bead fractions are given in the 2nd and 3rd column
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Table 3
Architecture
|C|
|A|
hNbondi
DCl–Cl [ref]
DMC(W) [ref]
fp [ref]
A[AC]A5[AC]-6 A2[AC]A4[AC]-6 A3[AC]A3[AC]-6 A[AC]A7[AC]-6 A2[AC]A6[AC]-6 A3[AC]A5[AC]-6 A4[AC]A4[AC]-6 A[AC]A9[AC]-6 A2[AC]A8[AC]-6 A3[AC]A7[AC]-6 A4[AC]A6[AC]-6 A5[AC]A5[AC]-6 A[AC]A11[AC]-6 A2[AC]A10[AC]-6 A3[AC]A9[AC]-6 A4[AC]A8[AC]-6 A5[AC]A7[AC]-6 A6[AC]A6[AC]-6 A[AC]A15[AC]-6 A2[AC]A14[AC]-6 A8[AC]A8[AC]-6
1/5 1/5 1/5 1/6 1/6 1/6 1/6 1/7 1/7 1/7 1/7 1/7 1/8 1/8 1/8 1/8 1/8 1/8 1/10 1/10 1/10
4/5 4/5 4/5 5/6 5/6 5/6 5/6 6/7 6/7 6/7 6/7 6/7 7/8 7/8 7/8 7/8 7/8 7/8 9/10 9/10 9/10
2.5 2.375 2.271 3 2.8 2.617 2.633 3.5 (3.5) 3.25 (3.25) 3.014 (3.006) 2.94 (2.931) 2.89 (2.861) 4 3.71 3.44 3.31 3.19 3.21 5 4.67 3.78
4.13 4.06 4.04 4.55 4.41 4.33 4.35 4.96 4.78 4.66 4.61 4.62 5.33 5.11 4.98 4.91 4.86 4.83
(ref. 34) (ref. 34) (ref. 34) (ref. 34) (ref. 34) (ref. 34) (ref. 34) (ref. 34) (ref. 34) (ref. 34) (ref. 34) (ref. 34) (ref. 34) (ref. 34) (ref. 34) (ref. 34) (ref. 34) (ref. 34)
0.06 (ref. 34) 0.061 (ref. 34) 0.052 (ref. 34) 0.07 (ref. 34) 0.044 (ref. 34) 0.037 (ref. 34) 0.034 (ref. 34) 0.073 (ref. 34) 0.038 (ref. 34) 0.024 (ref. 34) 0.022 (ref. 34) 0.017 (ref. 34) 0.066 (ref. 34) 0.021 (ref. 34) (0.029 (ref. 42)) 0.009 (ref. 34) 0.007 (ref. 34) 0.001 (ref. 34) 0.008 (ref. 34)
— — — 0.122 (ref. 42) 0.128 (ref. 42)
A4[AC]-5 A5[AC]-5 A6[AC]-5 A7[AC]-5 A8[AC]-5 A[A4C]-5 A[A5C]-5 A[A6C]-5 A[A7C]-5 A[A8C]-5 A4[A3C]-5 A5[A4C]-5
1/6 1/7 1/8 1/9 1/10 1/6 1/7 1/8 1/9 1/10 1/8 1/10
5/6 6/7 7/8 8/9 9/10 5/6 6/7 7/8 8/9 9/10 7/8 9/10
2.68 2.97 3.31 3.6 3.93 3 3.5 4 4.5 5 3.77 4.64
4.41 4.67 4.94 5.19 5.38 4.64 4.98 5.33 5.61 5.96 — —
(ref. 41) (ref. 41) (ref. 41) (ref. 41) (ref. 41) (ref. 41) (ref. 41) (ref. 41) (ref. 41) (ref. 41)
0.047 (ref. 41) 0.02 (ref. 41) 0.005 (ref. 41) 0.005 (ref. 41) 0 (ref. 41) 0.076 (ref. 41) 0.078 (ref. 41) 0.065 (ref. 41) 0.05 (ref. 41) 0.04 (ref. 41) 0.04 (ref. 41) 0.02 (ref. 41)
0.135 (ref. 41) — 0.165 (ref. 41) 0.2 (ref. 41) 0.23 (ref. 41) 0.11 (ref. 41) — 0.12 (ref. 41) 0.125 (ref. 41) 0.14 (ref. 41) 0.135 (ref. 41) 0.165 (ref. 41)
A3[A8[C][C]]-6 A3[A6[AC][AC]]-6 A3[A4[A2C][A2C]]-6 A3[A2[A3C][A3C]]-6 A3[A4C][A4C]-6 A3[A8C2]-6
1/6.5 1/6.5 1/6.5 1/6.5 1/6.5 1/6.5
5.5/6.5 5.5/6.5 5.5/6.5 5.5/6.5 5.5/6.5 5.5/6.5
5.924 5.015 4.287 3.742 3.38 5.924
6.53 (ref. 53) 6.1 (ref. 53) 5.4 (ref. 53) 4.93 (ref. 53) 4.73 (ref. 53) 6.71 (ref. 53)
0.188 0.188 0.143 0.103 0.084 0.134
(ref. 53) (ref. 53) (ref. 53) (ref. 53) (ref. 53) (ref. 53)
0.07 (ref. 55) 0.077 (ref. 55) 0.095 (ref. 55) 0.109 (ref. 55) — —
A5[C]A5[C]-3 A4[AC][A4[AC]-3 A3[A2C]A3[A2C]-3 A2[A3C]A3[A3C]-3 A9[C]A[C]-3 A7[AC]A[AC]-3 A5[A2C]A[A2C]-3 A3[A3C]A[A3C]-3
1/6 1/6 1/6 1/6 1/6 1/6 1/6 1/6
5/6 5/6 5/6 5/6 5/6 5/6 5/6 5/6
2.33 2.67 2.83 3 3.53 3.3 3.13 3.03
3.92 (ref. 43) 4.31 (ref. 43) 4.39 (ref. 43) 4.54 (ref. 43) 4.86 (ref. 43) 4.6 (ref. 43) 4.5 (ref. 43) 4.56 (ref. 43)
0.013 0.042 0.074 0.075 0.121 0.081 0.076 0.099
(ref. 43) (ref. 43) (ref. 43) (ref. 43) (ref. 43) (ref. 43) (ref. 43) (ref. 43)
A3[A4C][A4C]-6 A3[A3C][A5C]-6 A3[A2C][A6C]-6 A3[AC][A7C]-6 A3[C][A8C]-6 A5[A3C][A3C]-6 A5[A2C][A4C]-6 A5[AC][A5C]-6 A5[C][A6C]-6 A7[A2C][A2C]-6
1/6.5 1/6.5 1/6.5 1/6.5 1/6.5 1/6.5 1/6.5 1/6.5 1/6.5 1/6.5
5.5/6.5 5.5/6.5 5.5/6.5 5.5/6.5 5.5/6.5 5.5/6.5 5.5/6.5 5.5/6.5 5.5/6.5 5.5/6.5
3.38 3.2 3.01 2.83 2.65 3.51 3.15 2.79 2.36 3.68
4.766 (ref. 56) 4.766 (ref. 56) 4.694 (ref. 56) 4.464 (ref. 56) 4.262 (ref. 56) 4.896 (ref. 56) 4.709 (ref. 56) 4.507 (ref. 56) 4.190 (ref. 56) 4.82 (ref. 56)
0.092 (ref. 56) 0.096 (ref. 56) 0.076 (ref. 56) 0.056 (ref. 56) 0.013 (ref. 56) 0.092 (ref. 56) 0.083 (ref. 56) 0.044 (ref. 56) 0.004 (ref. 56) 0.1 (ref. 56)
2802 | Soft Matter, 2015, 11, 2794–2805
0.133 (ref. 42) 0.121* (ref. 42) 0.133* (ref. 42) 0.142* (ref. 42) 0.145* (ref. 42) 0.145* (ref. 42) 0.124 (ref. 42) 0.143 (ref. 42) — 0.162 (ref. 42) — 0.164 (ref. 42) 0.133 (ref. 42) 0.16 (ref. 42) 0.27 (ref. 42)
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Table 3
Soft Matter (Contd. )
Architecture
|C|
|A|
hNbondi
DCl–Cl [ref]
DMC(W) [ref]
A7[AC][A3C]-6 A7[C][A4C]-6 A9[AC][AC]-6 A9[C][A2C]-6 A11[C][C]-6
1/6.5 1/6.5 1/6.5 1/6.5 1/6.5
5.5/6.5 5.5/6.5 5.5/6.5 5.5/6.5 5.5/6.5
3.13 2.59 3.88 3.15 4.11
4.766 (ref. 56) 4.306 (ref. 56) 5.083 (ref. 56) 4.81 (ref. 56) 5.155 (ref. 56)
0.07 (ref. 56) 0.014 (ref. 56) 0.126 (ref. 56) 0.076 (ref. 56) 0.097 (ref. 56)
A4[AC]-6 A7[AC]A[AC]-3 A8[AC][AC]-3 A8[AC][AC]-6
1/6 1/6 1/6 1/6
5/6 5/6 5/6 5/6
2.67 3.3 3.8 3.6
4.25 4.62 5.02 4.75
0.048 0.085 0.131 0.108
Percolation thresholds (Table 3 and Fig. 15) were obtained for diffusion of water through the frozen “W” pore networks. When water particles are restricted to move through the frozen “W + C” network, diffusion increases signicantly (DMC(W + C) > DMC(W) (Table 1)). It was found that lower fp for diffusion through the “W + C” pore network41 is obtained as compared to diffusion through the W pore network.41 DDPD values are in general larger than DMC(W + C) (Fig. 11 in this work, Fig. 9(b)) in ref. 44, Fig. 9(a) in ref. 53, Fig. 11(a–d) in ref. 43). Therefore fp as derived from the W bead motions within the dynamically evolving pore networks maybe lower than those derived from DMC(W + C). The high W bead diffusivities as compared to DMC(W + C) are caused by the structural evolution of the polymer beads. The polymers are assisting the W beads in their diffusion. Interestingly, Vishnyakov and Neimark15 noted from their MD simulations of hydrated Naon that water diffusion is enhanced by “temporal bridges” that open and close again (due to polymeric motion) for water transport. When a mapping onto a grid would be performed similar to this work, then those temporal bridges would lead to a DMC that varies with time, since water clusters are not continuously connected. This is in line with the time variation of the diffusivities DMC(W + C) and DMC(W) (Fig. 7) which is also caused by structural changes.
DCl–Cl vs. hNbondi for grafted architectures in Tables 3 and 1. Data indicated with crosses were obtained from ref. 34, 41–44, 53 and 56. DCl–Cl for architectures I, II, and III (IV, V, VI, and VII) are indicated by open squares (triangles). The water contents fw ¼ 0.16. Fig. 13
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(ref. 44) (ref. 44) (ref. 44) (ref. 44)
fp [ref]
(ref. 44) (ref. 44) (ref. 44) (ref. 44)
The effective diffusivity, DDPD, of W beads is expected to be affected by structural changes caused by the polymer bead motion. This was also found in a DPD study for several architectures in ref. 44 by increasing the polymer mass. It would be worthwhile to study dynamical percolation57,58 to see how DDPD (at xed fw) and percolation thresholds for the W beads are affected by reducing the polymer mobility. This can be realized by increase of polymer bead masses, increasing polymer lengths, and/or forbidding bond crossings. Percolation through a dynamical varying medium might already occur at fw approaching zero.57 In practice it may be difficult to determine W bead percolation thresholds from the effective DPD diffusion coefficients. This is because it requires too much computing time for a W bead to probe the whole system size. Instead of determining dynamic percolation thresholds it is more practical to compare in future for various architectures the water contents at which a nite DDPD value (e.g. DDPD ¼ 0.05) is obtained. It should be noted that no bending constraints were imposed on the polymer beads, whilst bending rigidity depends on the molecular fragment that each bead represents. As mentioned in Computational details the aim was not to model polymers with known chemical formula, but to predict trends how morphology and diffusion are affected by their architecture, while keeping the interactions between building blocks (represented by DPD beads) the same. With bending constraints imposed on the polymer backbone and side chain beads, the morphologies may well depend on polymer length. Real polymers can be longer than in this modeling study. Modeling phase separation by DPD for Naon composed of 5 side chains per polymer was in earlier studies sufficient to reproduce the experimental Bragg spacing as a function of water contents30,31 and EW,31 despite that Naon polymers are composed of several hundreds of side chains.59 In the present work the side chain lengths and their distribution along the polymer backbones could be easily controlled in a computer experiment. A bimodal side chain length distribution results in lower hNbondi, smaller DCl–Cl, and decreased water diffusion as compared with the case for which the side chain lengths are the same. A well-dened (mono disperse) side chain length during synthesis of novel membranes is therefore expected to result in a more optimal pore network that favors water and proton transport. However, the control of side chain
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Fig. 14 (a). DMC vs. hNbondi for grafted architectures listed in Tables 3 and 1. Fractional numbers are the corresponding C bead fractions within the architectures for which the fitting was performed. (b) Same data as in (a) plotted against hNbondi|C||A|1. The water content is fw ¼ 0.16.
Fig. 15 fp plotted against hNbondi. Open symbols were obtained for architectures with [AC] side chains distributed along the backbone. Filled symbols were obtained for equidistant distribution of [AxC] side chains for various x values. Crosses were obtained for Y-shaped architectures. Lines and curves serve as eye guides that connect architectures for which the C bead fractions are the same. Inset: fp plotted against 4 (1/1.612|C|) + hNbondi).
lengths and their distribution along the backbones is more difficult to achieve during synthesis. It is therefore worthwhile to verify whether the obtained trends derived from this and other34,41–44,53 coarse grained studies can also be obtained from large scale MD simulations. If so, then this may motivate experimentalist to synthesize new membranes to verify predicted trends such as deduced from Fig. 15. For fuel cell applications it is important to have a good ion-channel connectivity with low percolation thresholds. From Fig. 15 architectures with low percolation thresholds are obtained by the combination of large hNbondi values and high IEC, such as e.g. Y-shaped membranes.53,55
Conclusions For 7 polymeric architectures micro-phase separation was modeled by DPD. The architectures contain backbones constructed from hydrophobic A fragments from which at regular intervals side chains branch off. The side chains are composed of A and/or C beads, where C represents a hydrophilic fragment.
2804 | Soft Matter, 2015, 11, 2794–2805
Along the backbone, the lengths of the side chains alternatingly vary as [ApC] and [AqC] (q $ p). The water volume fraction is xed at 0.16. Diffusion through the hydrophilic pores was calculated by Monte Carlo tracer diffusion calculations and derived from the W bead motions. When comparing membranes of the same IEC, the inter cluster distance DCl–Cl and water diffusion constants decrease with increasing difference in side chain lengths, or q p. The results can be explained by the parameter hNbondi which is the average number of bonds between A fragments towards the nearest C fragment within each architecture. Consistency is obtained with numerical data from more than 60 graed polymers. The characteristic distance, DCl–Cl, increases approximately linearly with hNbondi, whilst water diffusion increases approximately linearly with hNbondi|C||A|1, where |C|(|A|) is the C(A) bead fraction within the architecture. Percolation thresholds from previous studies decrease with increase of hNbondi for architectures for which the IEC are the same, and decrease with IEC for architectures for which hNbondi are the same. These insights may be of value to design pore networks with low percolation thresholds for solvent and/or proton diffusion.
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33 G. Dorenbos and K. Morohoshi, J. Chem. Phys., 2011, 134, 044133. 34 G. Dorenbos and K. Morohoshi, J. Chem. Phys., 2013, 138, 064902. 35 D. Wu, S. J. Paddison and J. A. Elliott, Macromolecules, 2009, 42, 3358–3367. 36 D. Wu, S. J. Paddison, J. A. Elliott and S. J. Hamrock, Langmuir, 2010, 26, 14308–14315. 37 J.-C. Perrin, S. Lyonnard and F. Volino, J. Phys. Chem. C, 2007, 111, 3393–3404. 38 A. M. Pivovar and B. S. Pivovar, J. Phys. Chem. B, 2005, 109, 785–793. 39 G. Dorenbos and K. Morohoshi, Energy Environ. Sci., 2010, 3, 1326–1338. 40 G. Dorenbos, Polymer, 2013, 54, 5024–5034. 41 G. Dorenbos and K. Morohoshi, J. Mater. Chem., 2011, 21, 13503–13515. 42 G. Dorenbos, RSC Adv., 2013, 3, 18630–18642. 43 G. Dorenbos, J. Power Sources, 2015, 276, 328–339. 44 G. Dorenbos, J. Power Sources, 2014, 270, 536–546. 45 P. J. Hoogerbrugge and J. M. V. A. Koelman, Europhys. Lett., 1992, 19, 155–160. 46 R. D. Groot and T. J. Madden, J. Chem. Phys., 1998, 108, 8713– 8724. 47 R. D. Groot and K. L. Rabone, Biophys. J., 2001, 81, 725–736. 48 S. Yamamoto, Y. Murayama and S. Hyodo, J. Chem. Phys., 2002, 116, 5842. 49 H. Wang, Y.-T. Liu, H.-J. Qian and Z.-Y. Lu, Polymer, 2011, 52, 2094–2101. 50 X. D. Guo, L. J. Zhang, Z. M. Wu and Y. Qian, Macromolecules, 2010, 43, 7839–7844. 51 R. D. Groot and P. B. Warren, J. Chem. Phys., 1997, 107, 4423– 4435. 52 P. Espanol and P. B. Warren, Europhys. Lett., 1995, 230, 191– 196. 53 G. Dorenbos, RSC Adv., 2014, 4, 51038–51046. 54 W. Humphrey, A. Dalke and K. J. Schulten, J. Mol. Graphics, 1996, 14, 33–38. 55 G. Dorenbos, J. Power Sources, 2015, submitted. 56 G. Dorenbos, Polymer, 2014, under review. 57 M. Sahimi, J. Phys. C: Solid State Phys., 1986, 19, 1311–1317. 58 O. Durr, W. Dieterich and A. Nitzan, Solid State Ionics, 2002, 149, 125–130. 59 H. G. Haubolt, T. Vad, H. Jungbluth and P. Hiller, Electrochim. Acta, 2001, 46, 1559–1563.
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Water diffusion within hydrated model grafted polymeric membranes with bimodal side chain length distributions G. Dorenbos* The effect of bimodal side chain length distributions on pore morphology and solvent diffusion within hydrated amphiphilic polymeric membranes is predicted. Seven polymeric architectures are constructed from hydrophobic backbones from which at regular intervals side chains branch off that are alternatingly short (composed of p hydrophobic A fragments or beads) and long (q A fragments, q > p). The side chains are end-linked with a hydrophilic C fragment. Pore morphologies at a water volume fraction of 0.16 are calculated by dissipative particle dynamics (DPD). Water diffusion through the water containing pores is calculated by tracer diffusion calculations through 140 selected snapshots and from the water bead motions. Diffusion constants decrease with difference in side chain lengths, q p. Overall, the distance between pores also decreases with q p. The results are explained by counting for every
Received 5th January 2015 Accepted 11th February 2015
architecture the average number of bonds hNbondi between an A and the nearest C fragment. These results are in line with a database that contains more than 60 architectures. Diffusion constants tend to
DOI: 10.1039/c5sm00016e
increase linearly with hNbondi|C|1|A|, where |C| and |A| are the C and A bead fractions within the architecture. hNbondi is therefore expected to be an interesting design parameter for obtaining low
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percolation thresholds for solvent and/or proton diffusion.
Introduction Proton exchange membranes (PEMs) are used in fuel cells where their purpose is to separate the anode from the cathode and to allow protons to ow from the anode to the cathode. Naon® is mostly used as a PEM. Naon polymers are constructed from hydrophobic (–CF2–CF2–) backbones to which side chains are attached with pendant hydrophilic SO3H groups, see Fig. 1. When exposed to humid environments water is absorbed1–9 and microphase separation occurs with the acidic sites and water forming a hydrophilic pore network. The pores are a few nm in diameter and provide the diffusive pathway for protons.6–8 One route to increase proton transport in a PEM is therefore by optimizing the pore morphology. Molecular dynamics (MD)10–24 has been frequently applied to model microphase separation within Naon. Due to the computational demand MD simulations are mostly performed for small system size. In order to gain insight in pore connectivity simulations for length scales longer than that of corresponding Bragg spacings are required. For Naon the Bragg spacing increases with the water volume fraction (fw), typically from 3 nm (fw ¼ 0.1) to 5 nm (fw ¼ 0.3).9 Atomistic simulations for large system size O(104 nm3) containing several million atoms have been performed,12,24 but these require pre410-1118, 1107-2 sano, Belle Crea 502, Susono-shi, Shizuoka-ken, Japan. E-mail: [email protected]
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assumed starting morphologies,24 or several cycles of system size increase by including periodic images that are again equilibrated.12 Examples of modeling approaches used to generate equilibrium morphologies of Naon of size O(104 nm3) are coarse grained MD (CGMD),25,26 the bond uctuation model,27 selfconsistent approaches28,29 and dissipative particle dynamics (DPD).30–36 Yamamoto and Hyodo30 calculated for the system size 2 104 nm3, the pore structures for hydrated Naon1200 (1200 is the equivalent weight (EW) of 1200 g polymer per sulfonic site). In their DPD study molecular fragments are coarse grained into beads. From the pair correlation between water (W) beads the corresponding X-ray scattering proles were estimated. The wave factor qmax, at which a maximum in the scattering occurs, was found to decrease with water uptake. This agrees with experimental observations9 and indicates that water clusters increase in size and get separated further apart with hydration level.
Fig. 1
Repeat unit of Nafion.
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Naon for which the side chains are assumed to be uniformly distributed along the polymer backbones was studied later as a function of EW.31 By mapping the hydrophilic pore phase onto a cubic grid, and by restricting tracer particles to move through the hydrophilic phase, water diffusion was mimicked by Monte Carlo (MC) simulation. Water diffusion constants derived from the particles' mean square displacement (MSD) match experimental values well when the water mobility within the pores is assumed the same as in pure water. This is an oversimplication, but quasi elastic neutron scattering (QENS) studies37,38 on Naon revealed that within the pores the water diffusion constant is close to that of pure water. At a hydration level of l ¼ 5 water molecules per SO3H site a local water mobility of 1.6 105 cm2 s1 was measured,38 which is close to the pure water value of 2.3 105 cm2 s1. Various studies34,39–44 in which DPD is combined with MC tracer diffusion calculations reveal for model polymeric membranes of the same ion exchange capacity (IEC) that pore morphology and water diffusion depend strongly on architecture. The main trends deduced from these studies are overviewed below. When for block polymers successive hydrophilic fragments are separated by x and y (y > x) connected hydrophobic backbone beads (i.e. a bimodal distribution of C beads within the backbone) pore size and water diffusion increase with bimodal character (or asymmetry parameter y/x).40 For graed polymers with side chains arranged at regular intervals along the backbone (uni-modal distribution) and with a single pendant hydrophilic fragment, the pore size and water diffusion coefficients increase with side chain length.41 When the side chains are kept at the same length but the inter-branching distance along the backbone is alternatingly long and short (y > x, bimodal distribution), while keeping the IEC xed, pore size and diffusion increases with increase of bimodal character, or y/x.34,42,43 Diffusion can be increased further when instead of one side chain branching off from a backbone bead, two side chains, each of the same length, are allowed to branch off.44 Such a kind of architecture can be considered as a bimodal side chain distribution with an innite asymmetry parameter value (y/x, x ¼ 0). For the graed model membranes diffusion and morphology depend thus strongly on y/x (xed side chain length) and on side chain length (for xed y/x ¼ 1). The purpose here is to know how pore size and diffusion are affected for model architectures with a uniform side chain distribution (y/x ¼ 1) but for which the length of the side chains along the backbones are alternatingly short (l1-1 A beads) and long (l2-1 A beads). When the lengths of both side chains are varied while simultaneously their total lengths l1 + l2 are kept constant, does then a long side chain in combination with a short side chain result in the largest cluster size and diffusion constant, or does a unimodal distribution (l1 ¼ l2) result in the largest cluster size/water diffusion? In Computational details the polymeric architectures are dened and the DPD simulation method and its current parameterization are outlined. Pore morphologies obtained for seven architectures and diffusion through the pores are
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Soft Matter
presented in Results and analysis. In Discussion the results are explained and by comparing with other DPD studies a consistent picture evolves.
Computational details A
Polymer chain architecture
In DPD single beads represent molecular fragments. The architecture type is given in Fig. 2. Each architecture consists of a hydrophobic backbone that contains A beads from which at regular intervals (corresponding to x A–A bond distances) side chains branch off. The lengths of the side chains are alternatingly short and long and contain respectively p and q A beads, with q $ p. Each side chain is end-linked with a hydrophilic C bead. The DPD formulae of the architectures are listed in Table 1 and can be split into two groups for which the C bead fractions are the same. Within each group the distance between branching points and the total length of both side chains (p + q) is kept the same. The number of repeat units is 3. All bead volumes correspond to that of the volume of 4 water molecules and are 0.12 nm3. Water is represented by W beads. For architectures I–III the acidic site density |C| ¼ 1/6.5 and the ion exchange capacity (IEC) expressed in mmol g1 for a polymer with a mass density of 1 g cm3 would correspond to 2.13 mmol g1 or an EW of 470 g mol1. For architectures IV–VII the acidic site density |C| ¼ 1/8 and the IEC corresponds to 1.73 mmol g1 or an EW of 578 g mol1. B Dissipative particle dynamics DPD has been applied to model phase separation,46 membrane rupture,47 vesicles48,49 and drug delivery.50 DPD was introduced by Hoogerbrugge and Koelman.45 The developed DPD version by Groot and Warren51 is followed here. Beads i and j interact with R each other via conservative FCij , dissipative FD ij , and random Fij forces. The force acting on bead i is: X FijC þFijR þ FijD (1) fi ¼ jsi
The sum is over all particles j that are located within a distance rc that denes the length scale. The conservative force is so repulsive and decreases with distance: 8 > < aij 1 rij ^rij rij \rc C r c (2a) Fij ¼ > : 0 rij $ rc
Fig. 2 Repeat units of the polymer architectures listed in Table 1. In the present work p + q ¼ 5 for x ¼ 3 architectures and p + q ¼ 6 for x ¼ 4 architectures, and q $ p.
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Table 1 Architectures Ax[ApC]Ax[AqC] with p + q ¼ 5 for x ¼ 3 (architectures I, II, III) and p + q ¼ 6 for x ¼ 4 (architectures IV, V, VI, VII). hNbondi is the average number of bonds between A beads towards a nearest C bead. DCl–Cl, DMC(W) and DMC(W + C) are given in the 7th, 8th and 9th columns, respectively. Values in parenthesis were obtained from additional DPD runs with DMC(W) and DCl–Cl sampled over time window 5 104Dt to 6 104Dt and DMC(W + C) sampled over 4 104Dt to 6 104Dt
Architecture
|C|
p+q
(p, q)
qp
hNbondi
I: A3[C]A3[A5C]-3 II: A3[AC]A3[A4C]-3 III: A3[A2C]A3[A3C]-3 IV: A4[C]A4[A6C]-3 V: A4[AC]A4[A5C]-3 VI: A4[A2C]A4[A4C]-3 VII: A4[A3C]A4[A3C]-3
1/6.5 1/6.5 1/6.5 1/8 1/8 1/8 1/8
5 5 5 6 6 6 6
(0, 5) (1, 4) (2, 3) (0, 6) (1, 5) (2, 4) (3, 4)
5 3 1 6 4 2 0
30/11 ¼ 32/11 ¼ 34/11 ¼ 45/14 ¼ 48/14 ¼ 51/14 ¼ 52/14 ¼
rij ¼ ri rj, rij ¼ |rij|, ^rij ¼ rij/|rij|
(2b)
ri and vi, are position and velocity of bead i, and ^rij is the unit vector in the direction of bead j towards bead i: the random force (eqn (3)) introduces thermal noise. The dissipative force is proportional to the beads' relative velocity (eqn (4)). R 0.5 FR (kBT)1^rij ij ¼ su (rij)zij(Dt)
(3)
D rij$vij)^rij FD ij ¼ gu (rij)(^
(4)
vij ¼ vi vj, uD and uR are weight functions, randomness is introduced by a randomly uctuating variable with Gaussian statistics with zero mean and unit variance: hzij(t)i ¼ 0, and hzij(t)zkl(t0 )i ¼ (dikdjl + dild4k)d(t t0 ). uD and uR are given by eqn (5). 8 2 rij > < rij \rc 1 2 rc uD rij ¼ uR rij ¼ (5) > : 0 rij $ rc Noise (s) and friction (g) parameters are related as s ¼ 2gkBT,52 with s ¼ 3, g ¼ 4.5, kB the Boltzmann constant and T the temperature. The three forces conserve linear and angular momentum. Adjacent polymer beads are joined by a spring force with spring constant C ¼ 50 and equilibrium bond distance R0 ¼ 0.85 rc:
2.727 2.909 3.0909 3.214 3.429 3.643 3.714
DCl–Cl (nm)
DMC(W)
DMC(W + C)
4.31 (4.45) 4.51 (4.52) 4.66 (4.65) 4.77 (4.81) 5.03 (4.98) 4.98 (5.03) 5.09 (5.1)
0.029 (0.026) 0.052 (0.047) 0.077 (0.084) 0.004 (0.008) 0.032 (0.031) 0.035 (0.035) 0.056 (0.049)
0.090 (0.093) 0.139 (0.130) 0.184 (0.196) 0.015 (0.028) 0.061 (0.064) 0.086 (0.074) 0.102 (0.102)
parameters between beads of given chemical formulae were not performed. For the current architectures (Table 1) the adapted c parameter set is expected to result in well phase separated morphologies. These are identical as in ref. 34, 39–44 and 53 and comparable with those derived from calculated binding energies of the molecular fragments in the DPD parameterization of hydrated Naon.30 DPD repulsions and the c-parameters are related by eqn (7).40 Daij ¼ (4.16 0.15) cij
The water volume fraction is given by eqn (8), where NA, NW and NC are the total numbers of A, W and C beads, respectively. fw ¼
(6)
Bead motions evolve as: dri/dt ¼ vi, midvi/dt ¼ fii. The bead masses mi, unit of energy (kBT) and of time s ¼ rc(m/kBT)0.5 are all scaled to 1. The temperature is kept at kBT ¼ 1.0 by using a modied Verlet integration scheme51 with an empirical factor 0.65 and a time step Dt ¼ 0.05s. The bead density r is 3. The repulsions aij are the same as in ref. 40, 44 and 53 and are listed in Table 2. Repulsions between similar beads are 104 and reproduce the compressibility of water.30,34 The other repulsions correspond with the adapted Flory–Huggins c values (Table 2). For this set of c parameters already a database has been created and predicted trends in this work can be compared with previous work. Atomistic calculations to obtain c
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NW
NW þ NA þ NC
(8)
The simulations are performed for fw ¼ 0.161 The water content expressed in terms of l is given by eqn (9). l¼
2
FSij ¼ C(rij R0)^rij
(7)
NW 4 NC
(9)
Since fw ¼ 0.161, the l values are thus 5.0 for architectures I–III and 6.14 for architectures IV–VII. The system is cubic with an edge length L ¼ 24. One DPD length unit corresponds to rc ¼ (3 0.12 nm3)1/3 ¼ 0.71 nm. Edge lengths are L rc 17 nm, which correspond to a volume of 5 103 nm. The simulation box contains 892 (architectures I–III) or 725 (architectures IV–VII) polymer chains. The total amount of beads is rL3 ¼ 41 472. Periodic boundary conditions are applied. The physical time to which s corresponds for aww ¼ 104 is 130 ps. Morphologies are generated up to 4 104Dt which correspond to a time of O(0.2 ms). DPD bead
Table 2
cij aij
DPD repulsions and c-parameters A–A
W–W
C–C
A–W
A–C
C–W
0 104
0 104
0 104
4.9 124.4
4.9 124.4
2.6 93.2
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Fig. 3 Pore morphologies sampled at 4 104Dt obtained for architectures I–VII. C beads are colored yellow (light grey) and W beads are colored blue (dark).
coordinates are stored on le every 2000Dt. A total of 140 morphologies are stored. The VMD-visual molecular dynamics package54 is used to display the morphologies.
Results and analysis A
Soft Matter
The distance between water clusters, DCl–Cl, is estimated from the pair correlation function g(r) between W beads. The g(r) calculated for these seven morphologies are shown in Fig. 4(a and b). DCl–Cl is derived from the position of the maximum which is located between 4 nm and 6 nm. When comparing g(r) obtained for architectures for which the IEC are the same, the distances where the maxima in g(r) occur are the smallest for architectures I and IV. For these two architectures the difference in side chain lengths is the highest and p ¼ 0. This means that the combination of a short side chain and a long side chain alternatingly branching off from the backbone beads results in signicant lower DCl–Cl as compared with two side chains of the same length branching off from the backbone beads. When comparing Fig. 4(a) with Fig. 4(b) the distances at which the maxima occur are larger for the lower IEC architectures displayed in Fig. 4(b). Fig. 5(a) displays the dependence of DCl–Cl on DPD time. Aer an initial increase DCl–Cl seems to stabilize beyond around 2 104Dt. DCl–Cl obtained by averaging over the time interval 3 104Dt to 4 104Dt are listed in Table 1. The apparent convergence of DCl–Cl observed in Fig. 5(a) was conrmed by additional runs up to 6 104Dt, for which DCl–Cl averaged over 5 104Dt to 6 104Dt are on average within one percent (or 0.04 nm) of the values listed in Table 1. This is in line with DPD modelling on Naon at fw ¼ 0.2, for which the equilibrium of DCl–Cl was obtained at 2 104Dt (or s ¼ 1000).30,31 DCl–Cl is plotted against difference in side chain lengths, or q p, in Fig. 5(b). From Fig. 5(b) a clear trend appears that DCl–Cl decreases with difference in side chain lengths, q p. An increase in side chain length difference thus results in smaller DCl–Cl values.
Pore structures
Snapshots of morphologies obtained for the architectures listed in Table 1 are displayed in Fig. 3. For all structures the C beads are located at the borders of the water clusters and together with the W beads they form a hydrophilic pore network. When a comparison is made among the architectures that are of the same IEC it is the one for which one of the side chains are of equal (architecture III) or near equal length (architecture VII) for which the pores seem to be larger and more elongated than the architectures I and IV for which p ¼ 0.
B Diffusion through frozen W and W + C pore networks Previous DPD simulations on graed architectures showed that pore morphologies change with DPD time.44 This is caused by the motion of the polymer beads. Actually during the DPD simulations polymer motions are higher than in reality for several reasons. All beads have the same mass (reduced mass m ¼ 1). Since a W bead contains 4 water molecules its mass would correspond to 72 amu, while the mass of a polymer bead
W bead pair correlation functions for the architectures Ax[ApC]Ax[AqC]. (a) x ¼ 3, and p + q ¼ 5 and (b) x ¼ 4 and p + q ¼ 6. DPD time t ¼ 4 104Dt.
Fig. 4
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Fig. 5
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(a) DCl–Cl plotted vs. DPD time. (b) DCl–Cl plotted against side chain length difference.
(e.g. –CF2CF2CF2CF2–) of the same volume is typically a factor 2 or more higher. The introduction of higher masses for the polymer beads (e.g. mA ¼ mC ¼ 3; mw ¼ 1) will decrease the polymer bead diffusivity. Also bond crossings are allowed to occur in (conventional) DPD, which lis up entanglements and speeds up polymer dynamics. A third reason for the fast polymer motion is that in the current work the backbone lengths are relatively short (up to 28 beads for architecture VII). Real polymers can be much longer resulting in slow reptation motions. Since polymer motions are overestimated, it is instructive to derive water diffusion coefficients within frozen pore networks. Water clusters (pores) may become temporally less connected, or even disconnected. The time evolution of the pore connectivity cannot be derived from the W bead motions. But it can be studied by MC tracer diffusion calculations through frozen morphologies that are generated at dened DPD time intervals.39–44,53 Since C beads are distributed along the pore edges (Fig. 3), their pending motion might increase the volume fraction accessible for water molecules during the diffusion process, and enhance water diffusion. For this reason, in this section, two MC tracer diffusion constants, DMC(W) and DMC(W + C) are derived for diffusion through frozen hydrophilic “W” and “W + C” pore networks, respectively. These should be considered representative for the hypothetical case that all polymeric beads (DMC(W)) or only A beads (DMC(W + C)) are immobile. Since these diffusion constants uctuate with DPD time, the obtained averages sampled over consecutive snapshots will be compared with W bead diffusion coefficients in Section C. Another reason to calculate diffusion through frozen lattices is that the MC derived diffusion constants can be compared with those derived earlier for 60 other architectures (see Discussion). In the MC tracer diffusion calculations the particle movement is restricted to the hydrophilic pore phase.31 Diffusion coefficients are determined by following the random movement of several thousand particles on a grid of size that contains (10L)3 ¼ 2403 nodes. DPD bead positions sampled at time intervals of 2000Dt are used to construct pore networks. From the bead positions two pore networks are constructed on the grid: a “W” and a “W + C” pore network. For the W(W + C) pore network each node for which the nearest bead is a W bead (W or C bead) belongs to the pore phase. All other nodes belong to the
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polymer phase. For the 140 DPD morphologies saved on le thus 280 pore networks are constructed. The fraction of nodes that joins the W(W + C) pore network is close to the fraction of W(W + C) beads. For architectures I–III these are 0.161(¼(NW)(NW + NC + NA)1) and 0.29(¼(NC + NW)(NW + NC + NA)1). For architectures IV–VII the node fractions are 0.161(W) and 0.266(W + C). As in previous studies34,40–44,53 Ntracer ¼ 2000 particles are put at MC time t0 ¼ 0 at randomly selected nodes that belong to the pore phase. A jump trial towards a nearest node in one of the six orthogonal directions is randomly selected during every Monte Carlo Step (MCS). A jump trial will only be accepted when the aimed site also belongs to the pore phase. Usually molecular diffusion constants are derived from the ensemble average of the mean square displacement (MSD) of N molecules, given by eqn (10a). By plotting the MSD against time the diffusion constant D can be derived from the slopes d(MSD)/dt0 , eqn (10b). MSD ¼
D¼
2 N 1 d X ~ 0 ~i ð0Þ lim ðt Þ R R i N t0 /N dt0 i¼1
2 N 1 d X ~ 0 ~i ð0Þ ¼ 1 dðMSDÞ=dt0 lim 0 Ri ðt Þ R 0 6N t /N dt i¼1 6
(10a)
(10b)
In eqn (10a) and 10(b) ~ Ri(t0 ) is the position of molecule i at 0 MC time t . The term within the sum is thus the squared displacement of molecule i at t0 . When sampling is performed over a sufficient number of particles then, for long enough time, the slopes converge to the value 6D (eqn (10b)). MSD curves obtained for diffusion through the W pore networks sampled at DPD time t ¼ 40 000Dt are shown in Fig. 6(a) (architecture I–III) and Fig. 6(b) (architectures IV–VII). Also the pure water case for which every jump trial is accepted is included in Fig. 6(b). Since the MSD are expressed in units of (inter node distance)2 and time is given in MCS, the slopes d(MSD)/dt0 for the pure water case is (per denition) equal to 1. Therefore the diffusion constant, DMC, within the membrane relative to that of pure water is obtained from the slopes. The slopes of the curves (D(MSD)/D(MCS)) are calculated by performing a linear t over the time window t0 ¼ 1 106–2 106 MCS. DMC is then given by eqn (11).
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Soft Matter
MSD curves obtained for diffusion within the morphologies generated at a DPD time t ¼ 4 104Dt. (a) Diffusion through W networks for architectures I–III. (b) Diffusion through W networks for architectures IV–VII. The pure water case (fw ¼ 1.0), for which the slope is 1, is indicated by crosses. (c) Diffusion through W + C networks for architectures I–III. (d) Diffusion through W + C networks for architectures IV–VII. Fig. 6
DMC ¼
MSDðt0 ¼ 2 106 Þ MSDðt0 ¼ 106 Þ 106
(11)
For both sets, diffusion is the lowest for architectures for which difference in side chain length is the highest. These are respectively architectures I (A3[C]A3[A5C]-3) (Fig. 6(a)) and IV (A4[C]A4[A6C]-3) (Fig. 6(b)). MSD curves for diffusion through the C + W networks are shown in Fig. 6(c) and (d). The same trends are obtained as for diffusion through the W pore networks, but diffusion is signicantly higher. In Fig. 7(a) and (b) DMC(W) and DMC(W + C) are plotted against DPD time, respectively. DMC values uctuate
signicantly with time because connections between pores can become better (resulting in enhanced DMC) or worse as DPD time proceeds. However, beyond 2 104Dt, DMC seems to uctuate around average values. DMC(W)(DMC(W + C)) averages over the DPD time window 3 104Dt to 4 104Dt(2 104Dt to 4 104Dt) are given in Table 1. From additional calculations it is conrmed that these averages are close to those obtained from averages (listed in parenthesis in Table 1) taken over the DPD time window 5 104Dt to 6 104Dt(4 104Dt to 6 104Dt). C
DPD W bead diffusion through dynamic pore networks
The water diffusion constants can also be derived from the W bead displacements during the DPD simulations. The diffusion
Fig. 7 (a) DMC(W) against DPD time. (b) DMC(W + C) against DPD time. Datapoints connected with solid lines were obtained for architectures with equal (p ¼ q ¼ 3) and near equal side chain (p ¼ 2; q ¼ 3) lengths. Dashed lines connect datapoints obtained for architectures with one of the side chains containing no hydrophobic A beads at all (p ¼ 0).
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MSD of the W beads for architectures for which x ¼ 3 (filled symbols) and architectures for which x ¼ 4 (open symbols). MSD of W beads for a system solely composed of W beads is also included. Fig. 8
coefficient of water molecules through the membrane, DMembrane , can be obtained by comparing DMembrane with the W W-bead H2 O water bead diffusion coefficient DPure for a system composed of W-bead Pure water W beads. DMembrane is then given by eqn (12a), with D , H2 O H2 O the experimental pure water diffusion coefficient (2.3 105 cm2 s1). ¼ DMembrane H2 O
DMembrane water W-bead DPure H2 O water DPure W-Bead
DDPD ¼
DMembrane W-bead water DPure W-Bead
(12a)
(12b)
During a DPD simulation pores can rearrange. Therefore W bead diffusivities are higher than DMC(W).40,44 In Fig. 8 the MSD of the W beads with respect to their positions at t ¼ 20 000Dt are plotted vs. DPD time. The MSD calculated for a system composed of only water beads is also included. From the slope water it follows that DPure ¼ 0.128. W-Bead For each of the 7 architectures, the DDPD values (calculated from the ratio of the slopes, eqn (12b)) are compared with DMC(W) and DMC(W + C) in Fig. 9. The trends obtained for DDPD and the MC derived diffusion constants are consistent with each other. The absolute diffusivities obtained by each of the three methods are not the same: DMC(W + C) is always larger then DMC(W) because there is much more volume fraction available for diffusion through the C + W network. Also the water bead diffusivities are larger than DMC(W). This is because
Fig. 9 (a) Comparison between W bead diffusion constants (DDPD) and MC derived diffusion constants (DMC(W) and DMC(W + C)).
2800 | Soft Matter, 2015, 11, 2794–2805
Fig. 10 Tracer diffusion coefficients DMC(W) and DMC(W + C) and W bead diffusion coefficients (DDPD) plotted vs. inter cluster distance DCl–Cl. Solid (dashed) lines serve as eye guides for architectures I–III (IV–VII).
C beads are always located near the W pore network (Fig. 3). The continuous rearrangement of the polymer A and C beads during a simulation effectively increases the space accessible to the W beads. Now imagine that all A beads are xed in space, the W beads can then access at most the whole hydrophilic phase which for architectures I–III and IV–VII constitute a volume fraction of fW+C ¼ 0.29 and 0.266, respectively. When supposing that C beads do not obstruct W bead diffusion, then DDPD is expected to approach DMC(W + C). Actually, the W bead diffusivities are higher than the DMC(W + C) values. This is because A beads also rearrange.44
Discussion Among the Ax[ApC]Ax[AqC]-3 architectures two parameters were varied. These are the inter-branching distance (equal to x A–A bond lengths) and the lengths (p + 1 and q + 1) of two consecutive side chains within a single repeat unit. DCl–Cl and the diffusion constants, DDPD, DMC and DMC(W + C) depend on these two parameters. Diffusion correlates strongly with the inter-cluster distance as illustrated in Fig. 10 where DMC(W), DMC(W + C) and DDPD are plotted vs. DCl–Cl. For membranes of the same IEC or C bead fraction diffusion increases with DCl–Cl. Larger pores become
Fig. 11 DMC(W), DMC(W + C), and DDPD vs. difference in side chain
lengths.
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Fig. 12 (a) DCl–Cl as a function of hNbondi and (b) DMC(W) vs. hNbondi. The drawn lines are linear fits for data obtained for similar IEC membranes. Inset: DMC(W) plotted against hNbondi |C|1 |A|.
thus better connected with regards to diffusion. This is in line with DPD-MC modeling studies on graed polymers that contain Y-shaped53 and linear31,34,41–44 side chains and block polymers.39,40 These studies also revealed that at the same IEC (or C bead fraction, |C|) an increase in DCl–Cl, or pore size, correlates with increased diffusion. As shown in Fig. 11 for a similar inter branching distance (given by x) diffusion decreases with difference in side chain lengths, q p. An increase in side chain length difference thus results in reduced water diffusion and generally also in lower DCl–Cl values (Fig. 5(b)). The reason for these trends lies in the fact that an increase in DCl–Cl also implies that the hydrophobic A region becomes larger in size. For the current set of interaction parameters (Table 2) the micro-phase separation process maximizes the number of hydrophilic–hydrophilic (W–C and W–W) and hydrophobic–hydrophobic (A–A) contacts and minimizes hydrophilic–hydrophobic (W–A) and (C–A) contacts. When bonds between polymer beads would be absent then this process nally results in a morphology in which a hydrophilic pool composed of C and W beads is fully phase separated from a majority A bead phase (as illustrated in Fig. 12 of ref. 40). For the present architectures, the system also tries to make as large as possible pores with minimal interface area. This is achieved by expelling the C beads from the hydrophobic A regions. The C beads accordingly arrange themselves along the pore walls, as indeed observed in Fig. 3. Due to the bonding constraints imposed onto the polymer beads, the hydrophobic A regions can only be large for architectures for which a prominent fraction of A beads is topologically far away (in terms of number of A–A and A–C bonds) from the nearest C bead in that architecture. The hydrophobic A region and DCl–Cl can only be large for that architecture for which the average number of bonds, Nbond, towards the nearest C bead is high. The average number of bonds, hNbondi towards the nearest C bead for each architecture is listed in Table 1. In Fig. 12(a) a clear correlation between DCl–Cl and hNbondi is observed. For architectures of the same IEC, diffusion generally increases with DCl–Cl (Fig. 10) and therefore increases also with hNbondi as shown in Fig. 12(b). The inset in Fig. 12(b) demonstrates that for the current architectures DMC(W) increases linearly with hNbondi |C| |A|1, where |C|(|A|) is the C(A) bead fraction within each architecture.
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The correlations that appear in Fig. 12 can also be deduced from DPD studies on graed membranes for which the c parameters are the same as here. The database is compiled in Table 3. DCl–Cl values from ref. 34 and 41 were obtained for two times smaller bead volumes and have been rescaled in Table 3 by multiplying them with a factor (0.12/0.06)0.333 1.26. In Fig. 13 DCl–Cl is plotted against hNbondi. Similar to Fig. 12(a) DCl–Cl increases linearly with hNbondi. In Fig. 14 the DMC(W) values in Table 1 are compared with those architectures listed in Table 3 for which C beads are only bonded with A beads (discarding architectures A3[A8[C][C]]-6, A3[A8C2]-6, A11[C][C]). When again a linear tting is performed on architectures for which the IEC is the same, DMC(W) increases with hNbondi. Also for xed hNbondi diffusion increases when the C bead fraction, |C|, is increased. Fig. 14(b) reveals that DMC(W) increases approximately linearly hNbondi |C| |A|1. This is in line with the inset in Fig. 12(b). A similar analysis can be performed for the percolation thresholds for diffusion listed in Table 3. The percolation threshold fp is dened as the water volume fraction at which the onset of diffusion occurs.41,42,55 fp has been determined for graed polymers with linear41,42 and Y-shaped55 side chains. The dependence of fp on hNbondi and IEC is shown in Fig. 15. The lines in Fig. 15 connect data obtained for architectures of the same C bead fractions (|C|1). Since fp decreases systematically with hNbondi, control and increase of hNbondi is an interesting parameter to achieve low percolation thresholds. From Fig. 15 it can also be observed that for architectures with similar hNbondi values the one with the highest IEC (largest |C|) is expected to reveal the lowest percolation threshold. When fp is plotted against hNbondi + 4 |1.612C|1 the data condense almost onto a single curve that is given by eqn (13). fp ¼ 0.389/[hNbondi + 4 (1.612 |C|)1]
(13)
For the architectures |C|1 is equal to 6.5 (I–III) and 8 (IV– VII) with hNbondi in the range 2.72–3.09 (I–III) and 3.21–3.71 (IV– VII). From Fig. 15 and eqn (13) the percolation thresholds for all these architectures are therefore expected to be within the range of 0.13 < fp < 0.18.
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DCl–Cl (5th column) and DMC(W) (6th column) at fw ¼ 0.16 for various grafted architectures. Percolation thresholds, fp, in 7th columns labeled with an asterisk were calculated for architectures that contain 12 repeat units and the corresponding hNbondi values are given within parenthesis in the 4th column. C and A bead fractions are given in the 2nd and 3rd column
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Table 3
Architecture
|C|
|A|
hNbondi
DCl–Cl [ref]
DMC(W) [ref]
fp [ref]
A[AC]A5[AC]-6 A2[AC]A4[AC]-6 A3[AC]A3[AC]-6 A[AC]A7[AC]-6 A2[AC]A6[AC]-6 A3[AC]A5[AC]-6 A4[AC]A4[AC]-6 A[AC]A9[AC]-6 A2[AC]A8[AC]-6 A3[AC]A7[AC]-6 A4[AC]A6[AC]-6 A5[AC]A5[AC]-6 A[AC]A11[AC]-6 A2[AC]A10[AC]-6 A3[AC]A9[AC]-6 A4[AC]A8[AC]-6 A5[AC]A7[AC]-6 A6[AC]A6[AC]-6 A[AC]A15[AC]-6 A2[AC]A14[AC]-6 A8[AC]A8[AC]-6
1/5 1/5 1/5 1/6 1/6 1/6 1/6 1/7 1/7 1/7 1/7 1/7 1/8 1/8 1/8 1/8 1/8 1/8 1/10 1/10 1/10
4/5 4/5 4/5 5/6 5/6 5/6 5/6 6/7 6/7 6/7 6/7 6/7 7/8 7/8 7/8 7/8 7/8 7/8 9/10 9/10 9/10
2.5 2.375 2.271 3 2.8 2.617 2.633 3.5 (3.5) 3.25 (3.25) 3.014 (3.006) 2.94 (2.931) 2.89 (2.861) 4 3.71 3.44 3.31 3.19 3.21 5 4.67 3.78
4.13 4.06 4.04 4.55 4.41 4.33 4.35 4.96 4.78 4.66 4.61 4.62 5.33 5.11 4.98 4.91 4.86 4.83
(ref. 34) (ref. 34) (ref. 34) (ref. 34) (ref. 34) (ref. 34) (ref. 34) (ref. 34) (ref. 34) (ref. 34) (ref. 34) (ref. 34) (ref. 34) (ref. 34) (ref. 34) (ref. 34) (ref. 34) (ref. 34)
0.06 (ref. 34) 0.061 (ref. 34) 0.052 (ref. 34) 0.07 (ref. 34) 0.044 (ref. 34) 0.037 (ref. 34) 0.034 (ref. 34) 0.073 (ref. 34) 0.038 (ref. 34) 0.024 (ref. 34) 0.022 (ref. 34) 0.017 (ref. 34) 0.066 (ref. 34) 0.021 (ref. 34) (0.029 (ref. 42)) 0.009 (ref. 34) 0.007 (ref. 34) 0.001 (ref. 34) 0.008 (ref. 34)
— — — 0.122 (ref. 42) 0.128 (ref. 42)
A4[AC]-5 A5[AC]-5 A6[AC]-5 A7[AC]-5 A8[AC]-5 A[A4C]-5 A[A5C]-5 A[A6C]-5 A[A7C]-5 A[A8C]-5 A4[A3C]-5 A5[A4C]-5
1/6 1/7 1/8 1/9 1/10 1/6 1/7 1/8 1/9 1/10 1/8 1/10
5/6 6/7 7/8 8/9 9/10 5/6 6/7 7/8 8/9 9/10 7/8 9/10
2.68 2.97 3.31 3.6 3.93 3 3.5 4 4.5 5 3.77 4.64
4.41 4.67 4.94 5.19 5.38 4.64 4.98 5.33 5.61 5.96 — —
(ref. 41) (ref. 41) (ref. 41) (ref. 41) (ref. 41) (ref. 41) (ref. 41) (ref. 41) (ref. 41) (ref. 41)
0.047 (ref. 41) 0.02 (ref. 41) 0.005 (ref. 41) 0.005 (ref. 41) 0 (ref. 41) 0.076 (ref. 41) 0.078 (ref. 41) 0.065 (ref. 41) 0.05 (ref. 41) 0.04 (ref. 41) 0.04 (ref. 41) 0.02 (ref. 41)
0.135 (ref. 41) — 0.165 (ref. 41) 0.2 (ref. 41) 0.23 (ref. 41) 0.11 (ref. 41) — 0.12 (ref. 41) 0.125 (ref. 41) 0.14 (ref. 41) 0.135 (ref. 41) 0.165 (ref. 41)
A3[A8[C][C]]-6 A3[A6[AC][AC]]-6 A3[A4[A2C][A2C]]-6 A3[A2[A3C][A3C]]-6 A3[A4C][A4C]-6 A3[A8C2]-6
1/6.5 1/6.5 1/6.5 1/6.5 1/6.5 1/6.5
5.5/6.5 5.5/6.5 5.5/6.5 5.5/6.5 5.5/6.5 5.5/6.5
5.924 5.015 4.287 3.742 3.38 5.924
6.53 (ref. 53) 6.1 (ref. 53) 5.4 (ref. 53) 4.93 (ref. 53) 4.73 (ref. 53) 6.71 (ref. 53)
0.188 0.188 0.143 0.103 0.084 0.134
(ref. 53) (ref. 53) (ref. 53) (ref. 53) (ref. 53) (ref. 53)
0.07 (ref. 55) 0.077 (ref. 55) 0.095 (ref. 55) 0.109 (ref. 55) — —
A5[C]A5[C]-3 A4[AC][A4[AC]-3 A3[A2C]A3[A2C]-3 A2[A3C]A3[A3C]-3 A9[C]A[C]-3 A7[AC]A[AC]-3 A5[A2C]A[A2C]-3 A3[A3C]A[A3C]-3
1/6 1/6 1/6 1/6 1/6 1/6 1/6 1/6
5/6 5/6 5/6 5/6 5/6 5/6 5/6 5/6
2.33 2.67 2.83 3 3.53 3.3 3.13 3.03
3.92 (ref. 43) 4.31 (ref. 43) 4.39 (ref. 43) 4.54 (ref. 43) 4.86 (ref. 43) 4.6 (ref. 43) 4.5 (ref. 43) 4.56 (ref. 43)
0.013 0.042 0.074 0.075 0.121 0.081 0.076 0.099
(ref. 43) (ref. 43) (ref. 43) (ref. 43) (ref. 43) (ref. 43) (ref. 43) (ref. 43)
A3[A4C][A4C]-6 A3[A3C][A5C]-6 A3[A2C][A6C]-6 A3[AC][A7C]-6 A3[C][A8C]-6 A5[A3C][A3C]-6 A5[A2C][A4C]-6 A5[AC][A5C]-6 A5[C][A6C]-6 A7[A2C][A2C]-6
1/6.5 1/6.5 1/6.5 1/6.5 1/6.5 1/6.5 1/6.5 1/6.5 1/6.5 1/6.5
5.5/6.5 5.5/6.5 5.5/6.5 5.5/6.5 5.5/6.5 5.5/6.5 5.5/6.5 5.5/6.5 5.5/6.5 5.5/6.5
3.38 3.2 3.01 2.83 2.65 3.51 3.15 2.79 2.36 3.68
4.766 (ref. 56) 4.766 (ref. 56) 4.694 (ref. 56) 4.464 (ref. 56) 4.262 (ref. 56) 4.896 (ref. 56) 4.709 (ref. 56) 4.507 (ref. 56) 4.190 (ref. 56) 4.82 (ref. 56)
0.092 (ref. 56) 0.096 (ref. 56) 0.076 (ref. 56) 0.056 (ref. 56) 0.013 (ref. 56) 0.092 (ref. 56) 0.083 (ref. 56) 0.044 (ref. 56) 0.004 (ref. 56) 0.1 (ref. 56)
2802 | Soft Matter, 2015, 11, 2794–2805
0.133 (ref. 42) 0.121* (ref. 42) 0.133* (ref. 42) 0.142* (ref. 42) 0.145* (ref. 42) 0.145* (ref. 42) 0.124 (ref. 42) 0.143 (ref. 42) — 0.162 (ref. 42) — 0.164 (ref. 42) 0.133 (ref. 42) 0.16 (ref. 42) 0.27 (ref. 42)
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Table 3
Soft Matter (Contd. )
Architecture
|C|
|A|
hNbondi
DCl–Cl [ref]
DMC(W) [ref]
A7[AC][A3C]-6 A7[C][A4C]-6 A9[AC][AC]-6 A9[C][A2C]-6 A11[C][C]-6
1/6.5 1/6.5 1/6.5 1/6.5 1/6.5
5.5/6.5 5.5/6.5 5.5/6.5 5.5/6.5 5.5/6.5
3.13 2.59 3.88 3.15 4.11
4.766 (ref. 56) 4.306 (ref. 56) 5.083 (ref. 56) 4.81 (ref. 56) 5.155 (ref. 56)
0.07 (ref. 56) 0.014 (ref. 56) 0.126 (ref. 56) 0.076 (ref. 56) 0.097 (ref. 56)
A4[AC]-6 A7[AC]A[AC]-3 A8[AC][AC]-3 A8[AC][AC]-6
1/6 1/6 1/6 1/6
5/6 5/6 5/6 5/6
2.67 3.3 3.8 3.6
4.25 4.62 5.02 4.75
0.048 0.085 0.131 0.108
Percolation thresholds (Table 3 and Fig. 15) were obtained for diffusion of water through the frozen “W” pore networks. When water particles are restricted to move through the frozen “W + C” network, diffusion increases signicantly (DMC(W + C) > DMC(W) (Table 1)). It was found that lower fp for diffusion through the “W + C” pore network41 is obtained as compared to diffusion through the W pore network.41 DDPD values are in general larger than DMC(W + C) (Fig. 11 in this work, Fig. 9(b)) in ref. 44, Fig. 9(a) in ref. 53, Fig. 11(a–d) in ref. 43). Therefore fp as derived from the W bead motions within the dynamically evolving pore networks maybe lower than those derived from DMC(W + C). The high W bead diffusivities as compared to DMC(W + C) are caused by the structural evolution of the polymer beads. The polymers are assisting the W beads in their diffusion. Interestingly, Vishnyakov and Neimark15 noted from their MD simulations of hydrated Naon that water diffusion is enhanced by “temporal bridges” that open and close again (due to polymeric motion) for water transport. When a mapping onto a grid would be performed similar to this work, then those temporal bridges would lead to a DMC that varies with time, since water clusters are not continuously connected. This is in line with the time variation of the diffusivities DMC(W + C) and DMC(W) (Fig. 7) which is also caused by structural changes.
DCl–Cl vs. hNbondi for grafted architectures in Tables 3 and 1. Data indicated with crosses were obtained from ref. 34, 41–44, 53 and 56. DCl–Cl for architectures I, II, and III (IV, V, VI, and VII) are indicated by open squares (triangles). The water contents fw ¼ 0.16. Fig. 13
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(ref. 44) (ref. 44) (ref. 44) (ref. 44)
fp [ref]
(ref. 44) (ref. 44) (ref. 44) (ref. 44)
The effective diffusivity, DDPD, of W beads is expected to be affected by structural changes caused by the polymer bead motion. This was also found in a DPD study for several architectures in ref. 44 by increasing the polymer mass. It would be worthwhile to study dynamical percolation57,58 to see how DDPD (at xed fw) and percolation thresholds for the W beads are affected by reducing the polymer mobility. This can be realized by increase of polymer bead masses, increasing polymer lengths, and/or forbidding bond crossings. Percolation through a dynamical varying medium might already occur at fw approaching zero.57 In practice it may be difficult to determine W bead percolation thresholds from the effective DPD diffusion coefficients. This is because it requires too much computing time for a W bead to probe the whole system size. Instead of determining dynamic percolation thresholds it is more practical to compare in future for various architectures the water contents at which a nite DDPD value (e.g. DDPD ¼ 0.05) is obtained. It should be noted that no bending constraints were imposed on the polymer beads, whilst bending rigidity depends on the molecular fragment that each bead represents. As mentioned in Computational details the aim was not to model polymers with known chemical formula, but to predict trends how morphology and diffusion are affected by their architecture, while keeping the interactions between building blocks (represented by DPD beads) the same. With bending constraints imposed on the polymer backbone and side chain beads, the morphologies may well depend on polymer length. Real polymers can be longer than in this modeling study. Modeling phase separation by DPD for Naon composed of 5 side chains per polymer was in earlier studies sufficient to reproduce the experimental Bragg spacing as a function of water contents30,31 and EW,31 despite that Naon polymers are composed of several hundreds of side chains.59 In the present work the side chain lengths and their distribution along the polymer backbones could be easily controlled in a computer experiment. A bimodal side chain length distribution results in lower hNbondi, smaller DCl–Cl, and decreased water diffusion as compared with the case for which the side chain lengths are the same. A well-dened (mono disperse) side chain length during synthesis of novel membranes is therefore expected to result in a more optimal pore network that favors water and proton transport. However, the control of side chain
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Fig. 14 (a). DMC vs. hNbondi for grafted architectures listed in Tables 3 and 1. Fractional numbers are the corresponding C bead fractions within the architectures for which the fitting was performed. (b) Same data as in (a) plotted against hNbondi|C||A|1. The water content is fw ¼ 0.16.
Fig. 15 fp plotted against hNbondi. Open symbols were obtained for architectures with [AC] side chains distributed along the backbone. Filled symbols were obtained for equidistant distribution of [AxC] side chains for various x values. Crosses were obtained for Y-shaped architectures. Lines and curves serve as eye guides that connect architectures for which the C bead fractions are the same. Inset: fp plotted against 4 (1/1.612|C|) + hNbondi).
lengths and their distribution along the backbones is more difficult to achieve during synthesis. It is therefore worthwhile to verify whether the obtained trends derived from this and other34,41–44,53 coarse grained studies can also be obtained from large scale MD simulations. If so, then this may motivate experimentalist to synthesize new membranes to verify predicted trends such as deduced from Fig. 15. For fuel cell applications it is important to have a good ion-channel connectivity with low percolation thresholds. From Fig. 15 architectures with low percolation thresholds are obtained by the combination of large hNbondi values and high IEC, such as e.g. Y-shaped membranes.53,55
Conclusions For 7 polymeric architectures micro-phase separation was modeled by DPD. The architectures contain backbones constructed from hydrophobic A fragments from which at regular intervals side chains branch off. The side chains are composed of A and/or C beads, where C represents a hydrophilic fragment.
2804 | Soft Matter, 2015, 11, 2794–2805
Along the backbone, the lengths of the side chains alternatingly vary as [ApC] and [AqC] (q $ p). The water volume fraction is xed at 0.16. Diffusion through the hydrophilic pores was calculated by Monte Carlo tracer diffusion calculations and derived from the W bead motions. When comparing membranes of the same IEC, the inter cluster distance DCl–Cl and water diffusion constants decrease with increasing difference in side chain lengths, or q p. The results can be explained by the parameter hNbondi which is the average number of bonds between A fragments towards the nearest C fragment within each architecture. Consistency is obtained with numerical data from more than 60 graed polymers. The characteristic distance, DCl–Cl, increases approximately linearly with hNbondi, whilst water diffusion increases approximately linearly with hNbondi|C||A|1, where |C|(|A|) is the C(A) bead fraction within the architecture. Percolation thresholds from previous studies decrease with increase of hNbondi for architectures for which the IEC are the same, and decrease with IEC for architectures for which hNbondi are the same. These insights may be of value to design pore networks with low percolation thresholds for solvent and/or proton diffusion.
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