Letter Cite This: ACS Macro Lett. 2018, 7, 59−64
pubs.acs.org/macroletters
Fluctuation Effects in Semiflexible Diblock Copolymers Shifan Mao,† Quinn MacPherson,‡ and Andrew J. Spakowitz*,†,§,∥,⊥ †
Department of Chemical Engineering, ‡Department of Physics, §Department of Materials Science and Engineering, ∥Department of Applied Physics, and ⊥Biophysics Program, Stanford University, Stanford, California 94305, United States S Supporting Information *
ABSTRACT: We present a simulation study of the equilibrium thermodynamic behavior of semiflexible diblock copolymer melts. Using discretized wormlike chains and field-theoretic Monte Carlo, we find that concentration fluctuations play a critical role in controlling phase transitions of semiflexible diblock copolymers. Polymer flexibility and aspect ratio control the order− disorder transition Flory−Huggins parameter χODTN. For polymers with low aspect ratios, fluctuations strongly elevate the phase transition χODTN at finite molecular weights. For high aspect-ratio polymers, chain semiflexibility decreases the phase transition χODTN. We find that the simulated phase behavior agrees well with our recently developed fluctuation theory based on wormlike chain configurations and a one-loop treatment of concentration fluctuations.
C
qualitatively predict the phase diagram of diblock copolymer melts. However, experiments observe an elevated order− disorder transition χODT over the mean-field predictions. Such quantitative discrepancy is partially corrected by incorporating finite molecular weight effects with concentration fluctuations.18−20 Molecular simulations and experiments show agreement with the fluctuation-corrected theories at relatively high molecular weights.19,21,22,22−25 However, nonuniversal phase behavior at low molecular weights is also noted by recent experiments.26 One source of nonuniversality is the finite polymer flexibility at low molecular weights. Recently, we developed a theory that accounts for polymer semiflexibility and fluctuation effects using the wormlike chain model.27 We show that phase transitions of semiflexible diblock copolymers with finite aspect ratios strongly deviate from the Gaussian chain theory at molecular weights lower than N ≈ 100. Such nontrivial strong deviations show that the incorporation of density fluctuations is critical in studying phase transitions of semiflexible copolymers. In this work, we aim to leverage field-theoretic Monte Carlo simulations to study the phase behavior of semiflexible copolymers in the presence of density fluctuations. This work
opolymers composed of incompatible chemical blocks self-assemble into microstructures with mesoscale domain sizes ranging from a few angstroms to many nanometers.1−4 Such microstructures endow such polymeric materials with ideal mechanical, interfacial, and transport properties.5−8 Furthermore, copolymer self-assembly can be used as an informative model for the structural organization of biomolecules.9 For several decades, numerous experimental and theoretical works have established our foundational understanding of polymeric self-assembly.1,10−14 In the theoretical study of the phase behavior of copolymers, polymer field theory has been the primary theoretical framework,3 and the majority of the work implementing polymer field theory uses the Gaussian chain model and mean-field theory. The Gaussian chain model assumes polymer configurations exhibit random walk statistics.15 Mean-field theory assumes that the chemical monomers interact through spatially averaged density fields, neglecting the impact of correlated concentration fluctuations.15−17 A classical example of a copolymer is the diblock copolymer, which consists of two covalently bonded segments with chemically different monomers (i.e., ...-A-A-A-B-B-B-...). Experiments map out the phase diagram of diblock copolymers at different A-segment fraction fA and interaction strength between A and B monomers, controlled by the Flory−Huggins parameter χ.1,15 The coarse-grained model assuming Gaussian chain configurations and mean-field interactions is able to © XXXX American Chemical Society
Received: August 21, 2017 Accepted: December 13, 2017
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DOI: 10.1021/acsmacrolett.7b00638 ACS Macro Lett. 2018, 7, 59−64
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ACS Macro Letters
where σαij = 1 if the chemical identity of jth bead on ith chain is of type α, and σαij = 0 otherwise. The vector R⃗ k is the location of the kth bin grid vertex, and w is a weighting factor that gives a linear interpolation of the density in three dimensions.32 The interaction energy can then be written as34
provides a concrete assessment of the impact of concentration fluctuations on copolymer phase behavior as the molecular weight is reduced. The technological trend for copolymer materials is to assemble into morphologies with smaller domain sizes (i.e., nanometer length scales), making our predictions crucial for practical applications of copolymer assembly. We consider a diblock copolymer melt consisting of np polymers. Each polymer has A- and B-type segments, with Asegment fraction fA. The total energy of the system is the sum of polymer conformation energy and monomer interaction energy βE = βEpoly + βE int
n
βE int =
(7)
where Δ is the discretization length of each bin, and the bead volume v = l0A. The compressibility factor κ is chosen to be large to approximately enforce melt incompressibility. The simulation used to generate Figure 1 has a cube size Lbox = 20Δ with periodic boundary conditions, filled by np = L3box/ (νnb) = 4000 polymers each consisting of nb = 20 beads. We implement the same type of Monte Carlo moves and parallel tempering method as in our previous work.35 In each simulation, we use a total of 31 simulations spanning a range of χ values that result in an average replica-exchange probability of 12−22% between replicas. We focus on symmetric diblock copolymers with fA = 0.5, with varying number of Kuhn steps per polymer N = L/(2lp) = nbl0/(2lp), and the aspect ratio between a Kuhn segment and cubic root of a Kuhn volume α = 2lp/ 3 2lpA = 2lp/ 3 2lpv /l0 . We obtain the desired N and α by choosing appropriate persistence length lp and discretization length l0 simultaneously (see Supporting Information). A Gaussian chain is infinitely flexible, corresponding to chain with N → ∞, and a rigid rod is infinitely stiff with N → 0. The chain aspect ratio α gives the thickness of the Kuhn segments within the chain. The top panel of Figure 1 shows the morphologies of symmetric, semiflexible diblock copolymers with N = 10 and
(1)
where β = 1/(kBT) and kB is the Boltzmann constant. To capture polymer semiflexibility, we model each chain as a wormlike chain whose conformational energy is proportional to the square of the chain curvature28,29 βEpoly =
lp 2
np
L
∑∫
0
i=1
⎛ ∂u ⃗ ⎞2 ds ⎜ i ⎟ ⎝ ∂s ⎠
(2)
where lp is the persistence length and L is the chain contour length. For the ith chain at arclength position s, ri⃗ (s) defines the ∂ r (⃗ s)
monomer position, and ui⃗ (s) = ∂i s is the chain tangent vector. Each Kuhn segment monomer has volume 2lpA, where A is the chain cross-sectional area. The chemical incompatibility between the monomers is calculated based on a coarse-grained Flory−Huggins parameter χ βE int =
χ 2lpA
∫ d r ϕ⃗ A( r ⃗)ϕB( r ⃗)
(3)
where local monomer densities of each species are given by np
ϕA ( r ⃗) = A ∑ i=1
fA L
∫0
np
ϕB( r ⃗) = A ∑ i=1
dsδ[ r ⃗ − ri (⃗ s)]
(4)
L
∫f L dsδ[r ⃗ − ri (⃗ s)] A
(5)
In addition, the copolymer melt is effectively incompressible, requiring space filling condition of monomers everywhere, that is, ϕA(r)⃗ + ϕB(r)⃗ = 1 for all spatial positions r.⃗ In our Monte Carlo simulations, we discretize the polymer chains into beads and spatial coordinates r ⃗ into bins. We use a discrete shearable-stretchable wormlike chain model30,31 with each chain represented by nb beads spaced an arclength l0 apart. The discretized wormlike chain consists of quadratic energy penalties associated with stretch, shear, bend, and coupling between the bend and shear degrees of freedom. This model is shown to reproduce the statistical behavior of the continuous inextensible wormlike chain model over the entire range of structural rigidity.30,31 To capture the monomer interactions in eq 3 and effective melt incompressibility, we discretize the spacial coordinates r ⃗ into nbin lattice bins. We obtain the monomer density field of chemical species α in the kth bin ϕ(k) α from first-order particleto-mesh interpolations32,33 ϕα(k) =
v Δ3
np
n
bin χl0 3 bin (k) (k) Δ ∑ ϕA ϕB + κ Δ3 ∑ (ϕA(k) + ϕB(k) − 1)2 2lpv k = 1 k=1
Figure 1. Phase transitions of semiflexible diblock copolymer melts with N = 10 and α = 2. The top panel shows the morphologies of diblock copolymer melts with increasing χ, and the bottom panel shows the degree of monomer mixing (black) and heat capacity (orange). The value of χ at the order−disorder transition χODT are identified, based on mean-field theory (blue-dashed vertical line), fluctuation theory (red dashed line), and from our simulations (red solid line).
α = 2. Each snapshot of morphology is obtained at the χN values shown in the bottom panel of Figure 1 with corresponding letters A−D. Qualitatively, the diblock copolymer melt transitions from a homogeneous phase to a disordered phase, and then to a lamellar phase. We quantify the phase transitions of diblock copolymers by looking at the degree of monomer mixing ⟨ϕAϕB⟩. Its response
nb
∑ ∑ σijαw(R⃗k − rij⃗ ) i=1 j=1
(6) 60
DOI: 10.1021/acsmacrolett.7b00638 ACS Macro Lett. 2018, 7, 59−64
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ACS Macro Letters to increasing χN is the χ component of the heat capacity cχ = −∂⟨ϕAϕB⟩/∂(χN). The bottom panel in Figure 1 shows the degree of mixing ⟨ϕAϕB⟩ and heat capacity cχ with increasing χN. At χN = 0, the degree of mixing ⟨ϕAϕB⟩ ≈ 0.23, smaller than the perfectly homogeneous value of 0.25 because of the connectivity of same-type beads along polymers and density fluctuations. With increasing χN, chemical incompatibility strengthens the degree of segregation, resulting in a decreasing ⟨ϕAϕB⟩. The dashed blue vertical line indicates the mean-field limit of stability of the well-mixed phase at χMFN = 8.93, roughly corresponding to the χN value above which phase segregation starts to take place. The snapshots A and B clearly show a qualitative segregation into local domains. However, these domains remain disordered,26,36−38 and no clear lamellar phase emerges at the mean-field theory prediction χMFN. Crossing from χN = 15.0 (C) to χN = 16.0 (D), an abrupt decrease of the degree of mixing is seen with phase transition from a disordered phase (C) to a lamellar phase (D). This phase transition also corresponds to the maximum of the heat capacity cχ. The value of χN at which the phase transition to a lamellar phase occurs is indicated by the vertical solid red line, whereas fluctuation theory predicts the phase transition at χODTN = 14.5 (dashed red line). Comparing the simulation with theory, we argue that the fluctuation correction to meanfield theory is important. To quantify the density−density correlations at different length scales, we calculate the correlation function of local monomer density variation ψ(r)⃗ = ϕA(r)⃗ − fA, which in Fourier space is written as S(q) = ⟨ψ̃ (q ⃗)ψ̃ ( −q ⃗)⟩ np
=
n
b 1 ⟨| ∑ ∑ (σilA − fA )exp(iq ⃗ · rij⃗ )|2 ⟩ n pnb i = 1 j = 1
(8)
where ψ̃ (q)⃗ is the Fourier transform of density variation ψ(r)⃗ , and the average is taken over all possible q⃗ orientations under periodic boundary conditions. We refer to the density−density correlation S(q) as the structure factor because it is proportional to the scattering intensity in neutron and X-ray scattering experiments. Next, we compare our simulation results of the structure factor and phase transition χODTN with our recent theoretical predictions.27 Figure 2A shows the simulated (symbols) and theoretical (lines) structure factors of semiflexible diblock copolymer melts with chain length N = 10 and aspect ratio α = 2 as in Figure 1. When χN = 0 (blue), the simulation and meanfield theory (dashed blue line) agree at all wavevector q. This indicates the approximate validity of the mean-field theory in the absence of monomer interactions. Both simulation and theory exhibit a maximum in the structure factor derived from the correlated monomer densities on connected polymers. At high q, the structure factor has a characteristic q−1 falloff because of the local chain rigidity. At χN = 0, our fluctuation theory (solid blue line) does not capture the scattering pattern, because the Brazovskii approximation39 is only intended to predict the scattering intensity at q* and assumes a highly peaked scattering pattern.27 With increasing χN, mean-field theory (dashed) overpredicts the scattering intensity at all q. In addition, simulations show a left shift of the peak location, indicating an expansion of correlation size during phase transition. This shift is absent in
Figure 2. (A) Structure factors of semiflexible diblock copolymers with N = 10 and chain aspect ratio α = 2. The color legend indicates the values of the increasing Flory−Huggins parameter χN. Open symbols are simulation results, dashed lines are mean-field predictions, and solid lines are fluctuation-theory predictions. (B) Inverse peak intensity of the structure factors of the same system as in (A). Dashed and solid lines show mean-field and fluctuation theory predictions. The filled symbols are simulation results, with the colored symbols corresponding those in (A). The x-axis intercept is indicated by the blue-dashed vertical line, identifying the mean-field prediction. The red-dashed vertical line shows the fluctuation theory prediction.27 The vertical red line corresponds to simulated phase transition χODTN to the lamellar phase, obtained from the peak location in heat capacity (see Figure 1).
the mean-field theory predictions. Both the suppression of scattering intensity and the peak shift can be attributed to correlated density fluctuation effects. These observations are consistent with previous studies for flexible diblock copolymers.19,37,40,41 The mean-field structure factors diverge and are not plotted above χMFN = 8.93. Approaching the phase transition to the lamellar phase (χN = 15.0), fluctuation theory captures the approximate peak intensity. Our theory does not predict the left shift of the peak, because we assume a stationary peak location. In the scope of fluctuation theory, previous studies use self-consistent equations involving peak location40 and fluctuation regularization techniques19 to address the peak shift. 61
DOI: 10.1021/acsmacrolett.7b00638 ACS Macro Lett. 2018, 7, 59−64
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ACS Macro Letters
interpolate between the mean-field and fluctuation theories with increasing χN. Notably, the deviation from the mean-field theory at high χN is smaller for polymers with larger lengths (e.g., N = 100). For longer polymer (in units of Kuhn steps), each polymer interacts with more neighboring polymers. In this case, concentration fluctuations around each chain is suppressed, and the mean-field theory adequately captures the behavior. Next, we look at Figure 3C,D displaying inverse peak intensities for chains with number of Kuhn steps per chain N = 10 and 100 and aspect ratio α = 4. At each N value, the meanfield theory becomes more accurate in comparison with Figure 3A,B. For high aspect ratio polymers (thin polymers), each polymer interacts with more neighbors in a space filling melt, decreasing the impact of correlated fluctuations that render the mean-field theory inaccurate. Finally, across the range of explored N and α, fluctuation theory gives better predictions of the simulated phase transition χODTN than mean-field theory. Especially in the case of low N and low α, fluctuations strongly elevate the phase transition χMFN above the mean-field predictions. Figure 4 shows the order−disorder transition χODTN of semiflexible diblock copolymers at different number of Kuhn
We summarize the comparison between the simulation and theory by focusing on the structure factor peak intensities (filled symbols in Figure 2A). Figure 2B shows the inverse peak intensities versus χN. Mean-field theory (dashed line) predicts a linear decrease of inverse peak intensity versus χN, with an xaxis intercept (blue dashed line), indicating the mean-field phase transition χMFN with diverging peak intensity. Fluctuation theory predicts a phase transition to a lamellar phase (solid red line) when the free energy of the fluctuating homogeneous phase matches that of a lamellar phase. The simulated inverse peak intensities (filled symbols) interpolate between the meanfield and fluctuation theories with increasing χN. This suggests the failure of mean-field theory at strong segregation and increasing validity of the fluctuation theory. The vertical red line at χN = 15.0 indicates the simulated phase transition χODTN obtained from the peak in the heat capacity and validated with qualitative morphological transition to a lamellar phase (see Figure 1). The location of the simulated phase transition χODTN is much more aligned with the fluctuationtheory prediction, showing the importance of fluctuations present in semiflexible diblock copolymers. We argue that in studying the phase transitions of semiflexible copolymer melts, the collective phenomena approaching the phase transition becomes more important, rendering fluctuation corrections indispensable. We now look at diblock copolymers across a range of chain flexibilities and aspect ratios. Figure 3A,B shows inverse peak intensities for number of Kuhn steps per chain N = 10 and 100 with common aspect ratio α = 2 (Figure 3A is identical to Figure 2B). In both cases, the inverse peak intensities
Figure 4. Phase transition χODTN vs chain length N = L/(2lp). The solid lines are theoretical predictions at different chain aspect ratios, and the squares are simulation predictions at corresponding chain aspect ratios.
steps per chain N and aspect ratio α. The solid lines show predictions from the fluctuation theory of semiflexible diblock copolymers,27 and the solid circles are simulation results. The classical mean-field prediction of χODTN = 10.5 (valid for N → ∞, α → ∞) based on the Gaussian-chain theory is shown as the black dashed line. In the infinitely thin chain limit, chain semiflexibility decreases the phase transition χODTN with decreasing N due to the diminishing impact of elastic chain entropy for shorter chains. The mean-field phase transition of semiflexible diblock copolymers χODTN(α → ∞) is recovered42,43 and is shown in blue. The trends of the simulated χODTN are in agreement with theoretical predictions. For lowaspect-ratio polymers (α = 2), the elevation of phase transition χODTN at low N due to fluctuation effects is very pronounced. When polymers have chain aspect ratios of α = 4, fluctuation effects become less important, and chain semiflexibility
Figure 3. Inverse peak intensities of semiflexible diblock copolymers. Top panels show inverse peak intensities for chains with aspect ratios α = 2 and chain lengths N = 10 (A) and N = 100 (B). Bottom panels show inverse peak intensities for chains with aspect ratios α = 4 and chain lengths N = 10 (C) and N = 100 (D). The identifiers of the phase transition predications are the same as in Figure 2B. See Supporting Information for noninverted peak intensities. 62
DOI: 10.1021/acsmacrolett.7b00638 ACS Macro Lett. 2018, 7, 59−64
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ACS Macro Letters decreases the phase transition χODTN at low N. For both lowand high-aspect-ratio polymers, the fluctuation theory predicts a slightly lower χODTN than simulation results. This may be due to the underestimation of the fluctuation correction using the Brazovskii approximation in calculating the divergent one-loop integral.44 The pronounced elevation of the phase transition χODTN at low N and aspect ratio α can be understood in terms of polymer concentration. In the perspective of each chain, there are a total of C = R3/(LA) polymers in its pervaded volume R3, where R is end-to-end distance (scales with the radius of gyration). In the fluctuation theory, we predict that the elevation of the mean-field phase transition scales as Δ(χODTN) = χODTN − χMFN ∼ C−2/3. At high C, each polymer interacts with many neighbors, and mean-field approximation is valid. In the high-N limit, chain concentration can be related to the invariant degree of polymerization N̅ = C2, that is, Δ(χODTN) ∼ C−2/3 ∼ N−1/3α−2. In the low-N limit, Δ(χODTN) ∼ C ∼ N−4/3α−2. Generally speaking, the fluctuation correction Δ(χODTN) becomes important for a combination of low N and low α. Depending on α, the phase transition χODTN can either increase (low α) or decrease (high α) with decreasing N. In conclusion, we use a particle-field simulation with wormlike chains to specifically address the role of chain semiflexibility and density fluctuations in the phase behavior of diblock copolymers. We find that symmetric diblock copolymers first segregate into a disordered phase, and then transition to a lamellar phase. This is in agreement with previous theoretical prediction of fluctuation-induced first-order phase transition.37,41 Quantitative comparisons of structure factors show that a mean-field description of density correlation is sufficient at zero χN, and fluctuation corrections become essential as the system approaches a phase transition. The qualitative dependences of phase transition χODTN on N and α agree between the simulation and theory. In the future, a wavelength-cutoff independent theory will be developed. Additionally, such theory will be developed in conjunction with simulations of semiflexible copolymers that address the effect of small-scale spatial cutoff in a field-theoretic simulation. Here, we do not intend to fully address the result of discretization on renormalizing χ in simulation, due to a lack of a regularized theory for semiflexible diblock copolymers. A resolution to the unregularized fluctuation theory and simulations of semiflexible copolymers will be valuable to our understanding of the cumulative effects of microscopic details in governing the macroscopic properties of polymers.
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Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS We are grateful to Jian Qin, Zhen-Gang Wang, David Morse, and Jay Schieber for valuable discussions. This work was supported by the NSF Interfacial Processes and Thermodynamics Program, Award 1511373. Quinn MacPherson acknowledges support from the National Science Foundation Graduate Research Fellowship under Grant No. DGE-1656518.
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ASSOCIATED CONTENT
S Supporting Information *
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acsmacrolett.7b00638. A table of the simulation parameters for the results presented in this manuscript and a plot of the structure factors for the simulations presented in Figure 3 (PDF).
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REFERENCES
AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. ORCID
Andrew J. Spakowitz: 0000-0002-0585-1942 63
DOI: 10.1021/acsmacrolett.7b00638 ACS Macro Lett. 2018, 7, 59−64
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