Article Cite This: Macromolecules XXXX, XXX, XXX-XXX
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Complex Morphology of the Intermediate Phase in Block Copolymers and Semicrystalline Polymers As Revealed by 1H NMR Spin Diffusion Experiments Horst Schneider,† Kay Saalwac̈ hter,† and Matthias Roos*,†,‡ †
Institut für Physik - NMR, Martin-Luther-Universität Halle-Wittenberg, Betty-Heimann-Str. 7, 06120 Halle (Saale), Germany Department of Chemistry, Massachusetts Institute of Technology, 170 Albany St, Cambridge, Massachusetts 02139-4208, United States
‡
S Supporting Information *
ABSTRACT: Nanostructured multiphase polymers exhibiting a mobile and a rigid phase also contain a phase of intermediate mobility that is usually assumed to be a continuous, uninterrupted interphase layer. This assumption is contrary to recent molecular-resolution micrographs and contradicts results from NMR spin diffusion experiments, all of which suggest a nontrivial interface structure. In this contribution, we reconsider our previous 1H NMR spin diffusion data sets (Roos et al. Colloid. Polym. Sci. 2014, 292, 1825) and perform optimized 2D and 3D numerical spin diffusion calculations to characterize the basic intermediate-phase morphological pattern, thus overcoming previous inconsistencies in data fitting. For the diblock copolymer poly(butadiene)-poly(styrene), PS-b-PB, we demonstrate that the interphase region comprises nanometersize intermixed immobile, intermediate and mobile subregions. In contrast, for the semicrystalline polymer poly(ε-caprolactone), PCL, the spin diffusion data are best reproduced by an intermediate phase that is fully embedded within the rigid phase, which is attributed to an imperfect crystalline structure. For both samples, the new findings reveal a complex discontinuous, dynamically inhomogeneous structure of the intermediate phase.
1. INTRODUCTION Heterogeneous polymers comprising phases with distinct molecular mobility are often employed as high-performance materials to combine sample rigidity with elastic properties. These polymeric materials can either be semicrystalline, or consist of phase-separated block copolymers that phase separate into a rigid and a mobile component. In these materials, the rigid phase consists of polymer crystallites or has a glass transition temperature well above room temperature, while the mobile phase results from an amorphous polymer fraction with a low glass transition temperature, Tg. Differential scanning calorimetry (DSC),1−3 nuclear magnetic resonance (NMR),4−8 and dielectric spectroscopy9,10 measurements have revealed a polymer fraction with intermediate mobility, or increased Tg, compared to the mobile phase. This polymer fraction is commonly attributed to the transition region between the rigid and the mobile majority phases and is referred to as an interphase, or the rigid−amorphous fraction in semicrystalline polymers. For microphase-separated copolymers in the strong segregation limit, such as poly(styrene)-blockpoly(butadiene), PS-b-PB, the transition region is affected by composition fluctuations11 and is asymmetrically enriched12,13 with rigid-phase monomers.14,15 Heterogeneous polymers are typically assumed to consist of well-organized structures exhibiting a continuous interphase © XXXX American Chemical Society
layer. This interphase layer involves a smooth mobility gradient29,16−18 in the range of 1 to 5 nm.27,30−32,19−21 Noting that the interphase affects the thermal and mechanical behavior of the material, knowledge of the structural and dynamical organization of the interphase region is of high interest for engineering applications and in the field of polymer crystal growth. In addition, the interphase impacts the dispersion of filler particles, such as carbon black22 or carbon nanotubes,23 which are widely used nowadays in high-performance polymer applications to increase the stiffness, longevity, or charge conductivity of the polymeric material.24−26 Techniques used to elucidate the structure and dynamics of the interphase include, but are not limited to, small-angle X-ray scattering,27,28 neutron scattering,29 electron microscopy,30 (deuterium) NMR,14,15 and dielectric spectroscopy,9,10 as well as Monte Carlo31,32 and molecular-dynamics33−35 simulations. Most experimental techniques, however, do not provide insights on a molecular level, but provide data that are spatially averaged. In transmission electron microscopy (TEM), for instance, the electron beam is transmitted through a sample layer as thin as tens of nanometers; nevertheless, such situations Received: April 3, 2017 Revised: October 22, 2017
A
DOI: 10.1021/acs.macromol.7b00703 Macromolecules XXXX, XXX, XXX−XXX
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mediate displacements on a subnanometer scale, which restricts spatial resolution to about 0.5 nm. At the same time, the mobility gradient between the rigid and the mobile phase may comprise only a couple of monomers.44 For such regions, the signal of the intermediate phase would be indistinguishable from that of the mobile one. Direct contact between the rigid and the mobile phase should thus be regarded as being a consequence of a coarse-grained representation of a steep mobility gradient. As a transition area between structurally and dynamically distinct regions, and due to confinement effects by the immobilized phase, the intermediate phase might be fairly inhomogeneous. Recently, for block copolymers comprising rigid and mobile regions, and for semicrystalline polymers as well, it has been suggested that the size of dynamic polymeric interphases might be commonly related to the length scale of cooperative rearrangements, i.e. dynamical inhomogeneity on the nanometer scale.45 Nanoscale dynamical inhomogeneity is reminiscent of materials close to Tg. Further, thermal fluctuations cause the rigid−mobile interface in lamella-forming block copolymers to exhibit roughness on the nanometer-scale, as characterized recently.46 Accounting for these facts, along with indications of direct contract between the rigid and mobile phase, we tentatively evaluated the consequences of a discontinuous, “island-like” structure of the interphase on the spin diffusion behavior in our previous publication. Our model could reproduce the experimental findings on a qualitative level, yet with systematic deviations.41 In this contribution, we reconsider the data presented in ref 41 that was obtained from 1H FID detection and decomposition,47 where observable spin diffusion was initiated by creating a selective magnetization of either the rigid or the mobile phase. Using optimized 2D and 3D spin-diffusion simulations that account for dynamical inhomogeneity within the rigid−mobile transition zone, the experimental data is successfully reproduced for both mobile- and rigid-phasefiltered experiments. In particular, we rely on re-evaluated 1H spin diffusion coefficients and perform a grid search that accounts for more-than-one-dimensional structures of the interphase. The data analysis reveals a qualitatively different nature of the intermediate phase in semicrystalline polymers compared to phase-separated diblock copolymers, the latter exhibiting features of materials close to Tg. Performing the calculations on a three-dimensional simulation grid, the spin diffusion simulations account for the three-dimensional nature of nanoscale dynamical inhomogeneity. Longitudinal relaxation during the spin diffusion delay is considered as well. Unlike advanced morphological models, alternative attempts such as local T1 sinks distributed within the interphase, phase boundary orientation effects, or more sophisticated pathways of spin diffusion, turn out to be not sufficient for reproducing the experimental data, even on a qualitative level. Details of the spin diffusion simulations, including optimized simulation parameters for quick calculations in the three-dimensional case, are discussed and provide valuable insights on the information accessible by (low-field 1H) spin diffusion NMR for structure investigation of heterogeneous polymers.
still provide an average over regions larger than, or at least comparable to, structural patterns characterizing the interphase. Although molecular scale resolution can be achieved with atomic force microscopy (AFM), this technique is still challenged by tip convolution effects such as adhesive interactions between the cantilever and the polymer, indentation artifacts, and the size of the cantilever tip as compared to the characteristic length scale of the investigated structure.36,37 Beyond these technical aspects, spatially clustered disordered regions often show little structural heterogeneity, while differences in dynamics can be much larger. It is noted that direct visualization of nanometer-scale regions of intermediate mobility, including stiffness contrast by tappingmode AFM or related techniques, is nearly impossible36 due to the molecular flexibility and lack of contrast between the mobile and the intermediate phasesa problem that is often overlooked in such studies. Entanglement effects and restricted motions oppose the formation of equilibrated structures. Polymer crystal growth in the solid phase differs from crystallization in solution by the increased importance of structural and dynamical constraints, which likely leads to increased complexity of structural patterns at the rigid-amorphous interface. Importantly, as the current state of the art, molecular-resolution micrographs of semicrystalline polymers38−40 are in stark contrast to a uniform interphase structure, and the crystal−amorphous interphase is found to be “unexpectedly disorganized when compared to textbook schematics”.40 Dynamic contrast between intermediate, rigid and rather mobile regions can provide complementary insights to structure-based findings. NMR spin diffusion (SD) experiments are well suited for this purpose as they can combine dynamical contrast with structural information on the nanometer scale. A selective excitation of either the rigid or the mobile phase can be achieved by proper spin polarization filters and will be followed by naturally occurring transfer of spin polarization toward neighboring regions (“spin diffusion”). The process of spin diffusion, in turn, facilitates the determination of connectivity patterns and nanoscale sizes of the rigid, intermediate and mobile regions. The spin polarization transfer between one region and another behaves diffusively; a continuous, uninterrupted interphase layer between the rigid and the mobile phase thus requires the interphase to be crossed first by the polarization gradient. Only thereafter, the rigid phase is expected to be polarized. Experimentally, however, it appears that a selective excitation of the mobile phase is followed by an almost simultaneous rise of both the intermediate and rigid-phase signals.41 This experimental observation cannot be explained on the basis of a simple interfacial layer with a mobility gradient, and suggests that the mobile phase has, apparently, direct contact with both the rigid and the intermediate phase. Notably, similar conclusions for one specific case were already drawn as early as 1985 in previous SD studies performed by Havens and VanderHart.42 From an experimental point of view, direct contact between the rigid and the mobile phase might be explained as follows. NMR signal decomposition is most conveniently performed by making use of differences in the molecular mobility;43,47,49 periodic structures are not required. In practice, this approach utilizes the transverse relaxation time, T2, which is in the case of solid polymers of the order of several tens to hundreds microseconds. During this period, local segmental motions will
2. EXPERIMENTS 2.1. Samples. Two polymers with distinct origins of the rigid, mobile and intermediate phase were investigated: (i) a phase-separated diblock copolymer, poly(styrene)-block-poly(butadiene) (PS-b-PB), and (ii) a semicrystalline polymer, poly(ε-caprolactone) (PCL). PS-bB
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Macromolecules PB has a molecular weight, Mn, of 20 kg/mol and a polydispersity of 1.1 and was purchased from Polymer Standards Service, Mainz, Germany. The 1H spin diffusion measurements41 of this sample discussed here were performed at T = 315 K, where microphase separation provides a glassy PS and a mobile PB phase that are accompanied by a PS-enriched transition zone with intermediate mobility.14,15 PCL has a molecular weight of 52.5 kg/mol with a polydispersity of 1.5 and was purchased from Sigma-Aldrich, Germany. Measurements41 of PCL were performed at T = 300 K. Both samples were characterized using small-angle X-ray scattering (SAXS) experiments,21,41 revealing a lamellar morphology with a long period of L = 18.5 nm and L = 15.0 nm for PS-b-PB and PCL, respectively. The NMR-based determination of the volume fraction of each phase relies on the 1H NMR intensities and requires knowledge of the proton densities. These are 0.081 and 0.100 g/cm3 for PS and PB, respectively, and 0.094 and 0.100 g/cm3 for the amorphous and crystalline phase of PCL, respectively.41,48 The proton density of the intermediate-mobility phase has been assumed to be an average of the rigid and mobile phases (systematic errors related to this choice are small if not negligible). For both samples, the region of intermediate mobility amounts to about 10% of all protons inside the sample. Along the direction of the rigid−mobile mobility gradient, the domain size of the interphase is on the order of 1 nm,21,41 reflecting a steep mobility gradient between the rigid (PS, crystalline PCL) and the mobile (PB, amorphous PCL) majority phases. 2.2. Spin Diffusion Experiments and Indicators for the Local Morphology. NMR spin diffusion experiments make use of energyconserving flip-flop processes that will cause a spin polarization gradient to re-equilibration over the space of the sample. Applications include the determination of domain sizes in heterogeneous polymers,49−53 and yield high sensitivity to the interphase morphology.19,21,41,54,42 1H spin diffusion is most efficient, but spectral resolution might be low. 13C-detection can overcome this issue; however, there is no 1H → 13C polarization transfer mechanism that could be considered quantitative. Even though intensity correction factors might be used to compensate differences of the 1H → 13C transfer efficiency in the rigid, mobile and intermediate phase, a precise quantification of the interphase signal is essentially impossible. Transfer mechanisms utilizing dipolar couplings are further biased by orientation selectivity. This situation is further complicated by a strong intensity loss due to intermediate mobility in the interphase. 1 H-detected SD experiments thus represent the only feasible approach. Experiments41 were carried out at low magnetic field (0.5 T) using complementary rigid- and mobile-phase filtered magnetization gradients, achieved by a double-quantum (DQ)55,56 and a magicand-polarization echo (MAPE)48,52 polarization filter, respectively. Both filters sustain about 85% of the Boltzmann polarization in the selected phase. DQ excitation and reconversion, in turn, intrinsically retains only 50% this filtered polarization. Technical aspects of lowfield 1H NMR spin diffusion experiments have recently been reviewed in ref 47 and will not be discussed here. Notably, MAPE and DQ-filtered 1H spin diffusion experiments performed on the same sample and under the same conditions reveal a distinct asymmetry in the phase-resolved spin diffusion build-up curves.41 Studying the magnetization flow from the rigid to the mobile and intermediate phases shows that the interphase gains polarization faster than the mobile phase does; see Figure 1, DQ curves. This situation suggests that regions of intermediate mobility are situated between the rigid and mobile phases. In contrast, if the magnetization transfer takes place in the opposite direction (from the mobile to the rigid and intermediate phases), the rigid and intermediate phase have been observed to gain polarization almost simultaneously (cf. Figure 1, MAPE curves). This initial buildup of the sink phase signals is faster than what is caused by longitudinal relaxation, so the above behavior needs to be explained in terms of spin diffusion. Thus, the MAPE-filtered SD data suggest that there are regions in which mobile and rigid regions are in close proximity to each other rather than being well separated by a phase of intermediate mobility.41
Figure 1. Experimental spin diffusion curves of the rigid (R), intermediate (I) and mobile (M) phase of PS-b-PB (a) and PCL (b) after applying a DQ (top) or MAPE polarization filter (bottom). For details on data representation, see section 2.3. Assuming that part of the rigid phase is in direct contact with the mobile one, the different behaviors in DQ and MAPE-filtered 1H spin diffusion experiments are explained as follows. Spin diffusion is mediated by dipolar couplings, the interaction strength of which is decreased with an increasing distance between the spins, and by molecular motions in the kHz regime or faster. Therefore, spin diffusion is more efficient in the rigid phase than in the mobile regions, with diffusion coefficients of about 0.3 and 0.1 nm2/ms for the glassy PS and the molecularly mobile PB, respectively.21,41 Note that spin diffusion under static conditions can be slower57 than the commonly referenced value of 0.8 nm2/ms which was determined using slow MAS.51 Starting from the rigid phase (DQ filter), the excess of spin polarization redistributes over the mobile and intermediate regions, where the interphase will be polarized more efficiently due to its small domain size combined with the higher spin diffusivity. This is why the interphase signal rises first. In contrast, if there is selective polarization of the mobile phase (MAPE filter), the rate with which the intermediate and rigid regions will be polarized is limited by the reduced diffusivity of magnetization in the mobile phase, while any magnetization delivered to the rigid and the intermediate phases redistributes quickly. As a consequence, the rigid and intermediate regions can be polarized (nearly) simultaneously. 2.3. Data Representation. It is noted that if there is a welldefined interface between domains, along with an initial step function of the magnetization gradient, and assuming uniform diffusion coefficients and no influence of T1 relaxation, the initial, timeintegrated flux of magnetization will be proportional to the square-root of the diffusion time. On the basis of this relationship it has been established to plot spin diffusion curves as a function of the square root of the mixing time43 and to evaluate the initial, linear regime of the SD build-up and decay curves. However, in the case of a heterogeneous, discontinuous interphase layer and given a nonideal shape of the initial magnetization gradient as caused by MAPE filtering,48 this relationship will be violated. Therefore, a nonlinear initial intensity rise is often observed, rendering the initial-rate approximation51,49,58 problematic. Moreover, the SD data will be represented in terms of the average polarization level of the sink phases relative to the initial polarization of the source phase, thus C
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Macromolecules directly reflecting the process of polarization transfer. For this purpose, the NMR intensities as observed after FID decomposition were renormalized by the proton fraction f k of each phase. Thus, we account for the volume fraction and (relative) proton density of each phase.
intermediate and mobile phase, respectively. Apart from the initial stage of the spin diffusion simulation in which phase boundaries (variables with asterisk, x*) are accompanied by abrupt steps in the polarization profile, the polarization profile will be a continuous function, that is
3. THEORY, MODELS, AND ASSUMPTIONS 3.1. Spin-Diffusion Simulations. For the sake of completeness, we briefly recall the primary physical concepts underlying the spin diffusion simulations. Spin diffusion in the presence of longitudinal relaxation can be calculated on the basis of48
* , t ) = mB(xAB * , t) ∀ t > 0 mA (xAB
(3)
The flux of magnetization across the phase boundary is determined by ρA DA
∂m(r, t ) 1 [m0 − m(r, t )] = ∇·[D(r)∇m(r, t )] + T1(r) ∂t
* , t) ∂mA (xAB ∂m (x* , t ) = ρB DB B AB ∂x ∂x
(4)
where ρA,B is the (average) spin density of the phases A and B. At the phase boundary, it follows that
(1) T
where m(r,t) is the time, t, and position, r = (x,y,z) , dependent magnetization, D is the spin diffusion coefficient, and T1 is the longitudinal relaxation time that is characteristic for the recovery of the equilibrium magnetization m0 = lim m(r, t ).
* , t j) = m(xAB
ρA DA m(xAB − 1 , t j − 1) + ρB DBm(xAB + 1 , t j − 1) ρA DA + ρB DB (5)
where xAB is located at half the distance between the two neighboring lattice points xAB‑1 and xAB+1; cf. Figure 2a. The distance between xAB∓1 and xAB is only Δx/2 instead of Δx; thus, eq 2 has to be replaced by
t →∞
In eq 1, we have used the nomenclature of Demco et al.52 but recognize that our use of the term m refers to magnetization as an intensive quantity, like spin polarization or temperature. Thus, this term does not depend on spin density. In the absence of T1 effects, the spin diffusion process conserves the product miρi summed over all i components, but it is the local gradient in m that is the driving force for observable spin diffusion. We note that the spin density will come into play when calculating the flux of magnetization across phase boundaries. If T1 ≫ τ ∼d2/D is valid, where τ is the characteristic spin diffusion period dependent on the average domain size d, the influence of longitudinal spin relaxation on the spin diffusion experiment will be negligible. For low-field studies, in contrast, T1 is on the order of only 100 ms, and spin diffusion competes with longitudinal relaxation. In solid polymers, T1 is usually shorter in mobile regions than in the rigid phase, so that longitudinal relaxation affects the shape and the evolution of the magnetization gradient. Therefore, spin diffusion and longitudinal relaxation are intermixed in low-field phase-resolved SD experiments,21,48 and the later regime of the spin diffusion curves, in which T1 dominates the signal, is still affected by spin diffusion. In our experiments, T1 relaxation is not as short as to challenge sensitivity to spin diffusion; nevertheless, it occurs on a time scale similar to the period required for re-equilibration of the spin polarization gradient. The diffusive re-equilibration of spin polarization across the sample can be treated analytically for lamellar morphologies or morphologies with cylindrical or spherical symmetry.51,56,59 Accounting for a rather complex morphological pattern and longitudinal relaxation, we here rely on a numerical solution of eq 1,48 which is implemented and evaluated in 3D for the first time. For the sake of demonstration, expressing eq 1 in terms of finite differences in one dimension (x-coordinates), we have
m(xAB ∓ 1 , t j + 1) = m(xAB ∓ 1 , t j) +
4 D k Δt 3 (Δx)2
* , t j) − 3m(xAB ∓ 1 , t j)] [m(xAB ∓ 2 , t j) + 2m(xAB +
Δt [m0 − m(xAB ∓ 1 , t j)] T1,k
(6)
For 2D or 3D Cartesian coordinate systems, the same equations apply to the y and z coordinates. For each of the nx × ny × nz points of the simulation grid the solution of the above equations is calculated successively for each dimension. Thus, the total number of iteration steps in an N-dimensional simulation grid is nCPU = N nx ny nz tmax/Δt, where tmax is the longest spin diffusion mixing time. [The factor of N results from the repetitive calculation of the diffusion process for each dimension and each lattice point.] Note that the relaxation term is applied only once per time step Δt, i.e., longitudinal relaxation will be accounted for only after computing the (three-dimensional) diffusion process. In order to obtain numerically stable results, the condition ⎡ 1 1 1 ⎤ 1 ⎥Δt < , Dmax ⎢ + + 2 2 2 2 (Δy) (Δz) ⎦ ⎣ (Δx) Dmax = max{Dm , Di , Dr }
(7)
must be fulfilled, thus restricting the choice of Δt. At phase boundaries the distance between XAB and XAB∓1 is only Δx/2 instead of Δx, followed by Δt/4 in eq 7. However, shortening Δt by a factor of 1.1 already yields correct simulations results, resulting in only 10% longer CPU times. While spin diffusion curves are commonly plotted as a function of the square-root of time,43 the simulation algorithm provides linear time increments. Simulated SD curves that comprise nt equidistant time points on a t1/2-scale thus require at least nt2 linear iteration steps. Each of the simulation curves presented herein consists of at least 100 equidistant time points, i.e., Δt ≤ 10−4 tmax must be fulfilled as well. In the MAPE-filtered experiments, the onset of spin diffusion at t = 0 might not be accompanied by an instantaneous increase 59
D k Δt [m(xi − 1, t j) + m(xi + 1, t j) (Δx)2 Δt − 2m(xi , t j)] + [m0 − m(xi , t j)] T1,k (2)
m(xi , t j + 1) = m(xi , t j) +
where Δt is the period between the discrete time points tj+1 and tj, and Δx gives the distance between the equidistant positions xi of the simulation grid. The subscript k = {r,i,m} relates the diffusion coefficient and relaxation time to the rigid, D
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converts a step in the polarization into a smooth gradient within a few simulation steps, with the detailed shape of the polarization gradient being not critical for the simulation result.48 Thus, the actual shape of the initial magnetization gradient is not particularly important, and consequences of the basic sample morphology (or alternative scenarios) should be accounted for in order to mimic the asymmetry between DQ and MAPE-filtered 1H spin diffusion experiments. 3.2. Verification of Assumptions Underlying the Spin Diffusion Approach. Throughout our discussion on the physical nature of the intermediate phase, we consider this region to be inhomogeneous and structurally complex. This situation is opposed to representing the intermediate phase by average values, i.e., average spin density, average 1H spin diffusion coefficient, and average T1 relaxation time. The assumption of a single spin density is least critical for the simulations, given that this parameter mainly governs the volume fractions in the morphological model. Using an average spin density instead of the actual density distribution does not change the size of the rigid, mobile and intermediate phases. The same holds for local spin topology effects: weak dipolar couplings along with the dipolar truncation effect60 and the interaction with third spins57,61−63 can cause local bottlenecks for spin diffusion, and can render spin polarization transfer anisotropic.64 In the approach presented herein, we rely on 1H spin diffusion on the nanometer scale, where local effects, including the impact of preferential SD pathways, are largely averaged out. Recent isotope dilution experiments performed by Rössler and co-workers reveal that intermolecular dipolar couplings contribute considerably to dipolar mechanisms such as dipolar-mediated spin relaxation,65,66 which will reduce anisotropy even on a local scale in the mobile phase. Our simulation results are not challenged by a spread of longitudinal relaxation times: if the period required by spin diffusion to cover the length scale characteristic of T1 inhomogeneities is comparable to, or even shorter than, the longitudinal relaxation times T1, longitudinal relaxation will be (almost) singly exponential.67,68 The average relaxation time, ⟨T1⟩, serves a good measure for the loss or recovery of spin polarization per unit time. The issue of using average values is discussed in greater detail in the Supporting Information (section SI2), including additional simulations that address the effect of localized T1 sinks in the intermediate phase. These considerations verify the assumptions underlying our applied treatment of spin diffusion, meaning that we ought to get reasonable agreement with the experimental data regardless of the direction of the magnetization gradient. The standard morphology cannot account for any asymmetry between the MAPE and DQ-filtered curves; therefore, advanced morphological models need to be considered. 3.3. Models on the Sample Morphology. Figure 2a shows the basic 1D simulation grid comprising three phases in a symmetric lamellar structure. Higher-dimensional morphological patterns such as cylindrical or spherical arrangement can be readily realized in 1D by the use of cylindrical and spherical coordinates, respectively, see e.g. ref. 48 In these models the phase of intermediate mobility (I) always consists of a smooth and continuous layer that reflects the mobility gradient region between the rigid (R) and the mobile (M) phase. The volume fractions φk on an N-dimensional simulation grid are determined by the normalized equilibrium NMR intensities, f k, in the absence of magnetization filters,
Figure 2. Morphological models (xy−planes) comprising rigid (R), intermediate (I), and mobile (M) regions: standard, one-dimensional model (a), island model (b), and mixed-interphase model (c) in which the rigid−mobile transition area consists of intermixed rigid (RI), mobile (MI), and intermediate (IR, IM) regions. For the model in part c, a cross-section of the transition region parallel to the yz-plane is shown, where the IR and IM layers are superimposed and surrounded by RI and MI regions (not shown). Symmetry planes are represented by red dash-dotted lines. Only the reduced, gray areas surrounded by symmetry planes have been used in the simulations. For the sake of clarity, the asterisks indicating phase boundaries were omitted from the variables.
of the overall sink (i.e., the rigid plus the intermediate) phase signal. This situation can be explained by a small (≤1 nm) depolarized region (“lag phase”) close to the phase boundary that is located inside the mobile phase,48 indicating a residual mobility gradient within the mobile phase. In the spin diffusion simulations, for practical reasons only, this time shift was incorporated by starting the diffusion procedure only after a short delay. Despite not being physically accurate, this treatment results in virtually the same simulation results as those relying on a depleted magnetization layer close to the phase boundary of the mobile phase (see the Supporting Information, section SI1). In fact, the process of diffusion E
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Macromolecules φk =
fk ρk−N /3
, where ∑ fj = −N /3
∑j f j ρj
j
and mobile (MI) regions. The total signal of intermediatemobility regions is comprised of both IR and IM interfacial regions, while the RI and MI regions contribute to the signal of the rigid and the mobile phase, respectively. The size of these regions is scaled by the following parameters (cf. again Figure 2c): Q = (x2* − x1*)/(x1* − x0*) is the size ratio between the mixed IM−MI phase and the purely mobile phase; P = (x3* − x2*)/(xt* − x3*) is the analogous ratio for the mixed IR−RI region and the purely rigid phase; S = φIR/φI is the relation of the intermediate fraction φIR in the mixed layer between x2* and x3* relative to the total amount of intermediate phases φI = φIR + φIM. Note that S corresponds by definition to the parameter r ≤ 1 of the island model. The mixed-interphase model may also be extended to three dimensions (cf. bottom graph in Figure 2c) in order to account for limited sizes along the z-direction. As before, scaling relations to calculate the positions of the phase boundaries are given in the Supporting Information, eqs S.5−S.11. Finally, modeling appreciable thickness fluctuations of the intermediate phase, or considering dynamical heterogeneity in the rigid−mobile transition region, will lead to models that are qualitatively similar to those discussed above, depending on the level of coarsening and the way the rigid, mobile and intermediate signals are defined. We also note that combining MAPE and DQ-filtered spin diffusion curves considerably restricts the parameter space: while the DQ-filtered curves yield a quick build-up for the intermediate phase, thus indicating considerable contact between the rigid and the intermediate phase, the observed MAPE-filtered curves require some direct contact between the rigid and the mobile phase. Hence, simultaneous fits to the DQ and MAPE-filtered data must account for opposing tendencies, rendering the fits to the experimental data more stable.
∑ φj = 1 j
(8)
Lamellar sizes were calibrated by means of the long period from SAXS and the φk assuming a simple stack model.41 To account for the asymmetry in the magnetization flow between DQ and MAPE-filtered experiments, the above model has been generalized assuming a disconnected, island-like distribution of the intermediate phase;41 see Figure 2b. We again emphasize that limited spatial resolution and thus a coarse-grained representation of the actual sample morphology will result in apparent direct contact between the rigid and the mobile phase at places where the mobility gradient is steep. Regions of “direct contact” next to regions with intermediate mobility thus indicate pronounced dynamical inhomogeneity in the rigid−mobile transition zone. It is also worth mentioning that the coarse representation of the interphase in this model provides the same simulation results as a model in which randomly shaped phase boundaries are accounted for, representing more realistic sample morphologies.41 The “island” model can be optimized by adjusting the relative amount of the mobile-rigid contact area (parameter 0 < qy < 1) and by varying the immersion depth of the intermediate phase within the rigid region (parameter 0 ≤ r ≤ 1), while the lateral size yt mimics the characteristic length scale of the dynamical inhomogeneity along the rigid-mobile transition area. If the lateral dimension yt were extended to infinity, the 2D island model would effectively reflect a superposition of 1D slices (Figure 2a) with and without presence of the interphase, i.e. a superposition of a twophase and three-phase model, the weighting factor of which is determined by the choice of qy. For a discussion on the lateral size, yt, vide inf ra. The island model can be readily generalized to three dimensions in order to account for a limited size of the intermediate phase along the z-direction (parameter 0 < qz < 1). This in effect reflects the three-dimensional nature of dynamical or structural inhomogeneity. The equations used to calculate the phase boundaries based upon the parameters qy, qz, and r and the volume fractions φk are presented in the Supporting Information, eqs S.2−S.4. The island model provides simulation results41 that are in qualitative agreement with the experimental data; nevertheless, for a combined data set of MAPE and DQ-filtered 1H spin diffusion experiments, systematic deviations remain.41 These deviations may either result from imperfect spin diffusion coefficients or from model imperfections, both of which will be addressed in the following. These imperfections, and the fact that the intrinsically assumed direct contact between the rigid and the mobile phase is a matter of debate, motivated us to develop a model to overcome this inadequacy, resulting in the so-called “mixed-interphase” model introduced below. Nonetheless, the island model is useful to explore and mimic basic features of the sample morphology, particularly in regards to a non-uniform distribution of the intermediate phase. As indicated by its name, the mixed-interphase model assumes that the overall interphase volume consists of intermixed rigid, mobile and intermediate regions (Figure 2c, top panel). To mimic an apparent mobility gradient within the interphase, yet keeping the model simple, we have subdivided the rigid− mobile transition region into two sublayers: the layer close to the rigid phase contains intermediate (IR) and rigid (RI) fractions, while the second layer consists of intermediate (IM)
4. RESULTS AND DISCUSSION 4.1. Optimized Simulation Setup. The minimum number of lattice points within each unit cell is determined by the precision with which the experimentally fixed volume fractions can be reproduced. The deviation of the simulated phase fractions from the desired values can be computed quickly for many values of nx, ny and nz (usually, nx > ny = nz), and is optimized for each set of morphology parameters (qy, qz, r or P, Q, S). Only after this optimization procedure can the timeconsuming spin diffusion calculations be performed efficiently. Note that the mirror planes (positions rS) permit the further reduction of the simulation grid by a factor of 2N by making use of ∇m(rS, t ) ·eS = 0
(9)
where es is a direction perpendicular to the symmetry plane. As an example of 3D simulations, the grid sizes shown in Table 1 were found by applying the above optimization procedure for the case of S = 0.8, Q = 1.4 and P = 3.3 (mixedinterphase model), where the triple {nx, ny, nz} = {30, 23, 23} provides the same simulation results as for the denser simulation grid using {nx, ny, nz} = {63, 45, 45}, but yielding 35-fold faster simulations. 4.2. Main Simulation Results for PS-b-PB. Figure 3 shows the combined data from the MAPE and DQ-filtered PSb-PB 1H spin diffusion experiments together with their best-fit simulation results using the parameters (Dk, T1,k) summarized in Table 2. In order to identify the optimal parameter set, the F
DOI: 10.1021/acs.macromol.7b00703 Macromolecules XXXX, XXX, XXX−XXX
Article
Macromolecules Table 1. CPU Time tCPU as a Function of the Number of Lattice Points nx and ny = nz Using a Reduced Unit Cell by Incorporating Symmetry Planesa nx 26 30 63 191
ny = nz 7 23 45 49
Δt/μs 60 43.7 10.7 2.07
δf k/%
