Article pubs.acs.org/Macromolecules
Phase Coexistence Calculations of Reversibly Bonded Block Copolymers: A Unit Cell Gibbs Ensemble Approach Zoltan Mester,† Nathaniel A. Lynd,‡ Kris T. Delaney,⊥ and Glenn H. Fredrickson*,⊥,§ †
Department of Chemical and Biological Engineering, Princeton University, Princeton, New Jersey 08540, United States Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, United States ⊥ Materials Research Laboratory, University of California, Santa Barbara, California, 93106, United States § Department of Chemical Engineering and Materials, University of California, Santa Barbara, California, 93106, United States ‡
ABSTRACT: A self-consistent field theory (SCFT) technique for calculating the compositions of coexisting phases in melt blends of reversibly bonding block polymers is presented. This new method is obtained by deriving a canonical model for reaction equilibrium and incorporating it into the previously reported Gibbs ensemble unit-cell SCFT for investigating macrophase separation. We use our new technique to attack the phase behavior of an incompressible melt blend of AB diblock copolymers that reversibly react at their B termini with monofunctional B homopolymers to produce longer ABB diblock copolymers. We find that reaction equilibrium favoring the product polymer stabilizes a rich variety of ordered phases with little macrophase separation. Reaction equilibrium favoring the reactants, however, leads to macrophase separation over a broad range of compositions.
1. INTRODUCTION Interest in block copolymers stems from their ability to selfassemble into 5−100 nm scale mesophases in the melt and solution states.1,2 These nanostructures can be used as thermoplastic elastomers,3 lithographic templates,4,5 adhesives,6 and medical devices.7 The immiscibility of polymer blocks induces phase separation while the connectivity between blocks limits the length scale of segregation. Previous investigations in the self-assembly of block copolymers have focused primarily on covalently connected polymer blocks. More recently, dynamic and reversible bonding (e.g., hydrogen bonds, metallo-ligand interactions, etc.) between blocks have been used to impart stimuli-responsive properties to block copolymer melts. The reversibility of bonds in supramolecular assemblies allows for environmental (e.g., temperature, pH, etc.) control over reaction equilibrium. This can be utilized to tune viscosity, chain length, morphology, and composition.8 Specific bonding between polymer blend components has also been introduced to achieve long-range order where there is otherwise an overwhelming tendency to macrophase separate.9−11 Experimental studies on reversibly bonded block copolymers include supramolecular chains and networks derived from di- and trifunctional polymers,8 ureido-pyrimidone polymer networks,12 supramolecular crystalline networks,13 self-healing rubbers,14,15 and hydrogen-bonding AB + BC diblock copolymers.11 Numerical self-consistent field theory (SCFT)2,16−18 is a remarkably successful theory for predicting the melt phase © 2014 American Chemical Society
behavior of block polymers. Previous studies of block polymers have focused on a variety of fixed architectures such as diblock copolymers,19−23 triblock copolymers,24−26 mixtures of block copolymers and homopolymers,27−29 and blends of block copolymers that are distinct in length or composition.30−32 SCFT has been more recently applied to supramolecular assembly of block copolymers including reversibly bonding homopolymers that form diblock33 and triblock copolymers,34 single component reversibly bonded polymer networks,35 binary reversibly bonded polymer networks,36 and binary telechelic polymers.37 The addition of dynamic and reversible bonding to multicomponent polymer mixtures affects phase behavior by allowing the relative concentrations of various bonded and nonbonded polymeric species to change subject to reaction equilibrium. An ability to predict the phase behavior of supramolecular polymers is valuable because environmental and stoichiometric control over the equilibrium composition of the melt can produce a large variety of morphologies including ordered mesophases, disordered macrophases, emulsions, and micellar phases. When the reactions produce a mixture of sufficiently incompatible polymer species, coexistence between morphologically and compositionally distinct ordered phases can occur, which in an SCFT theoretical study, necessitates Received: December 20, 2013 Revised: February 5, 2014 Published: February 25, 2014 1865
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copolymer and homopolymer melt blend,27,28,41 the funcionalized AB diblock copolymer and B homopolymer system is ideal for investigating the effect on phase behavior of introducing a polymer with a new length and symmetry through equilibrium reaction.
implementation of arbitrary phase coexistence that includes dynamic and reversible bonding between polymeric species. SCFT for supramolecular assembly was originally formulated in the grand canonical ensemble27,28,33−37 to easily implement the constraints imposed by reaction equilibrium on the product and reactant chemical potentials (or the equivalent activities). These constraints are used to obtain the chemical potentials of product species from the chemical potentials of reactant species that specify the system. The average compositions of coexisting phases are determined by equating the chemical potentials and pressures of each separately calculated phase. The grand canonical formulation overcomes the need to account for the effect of interfaces between phases on bulk properties, which is a problematic issue in direct canonical simulations of coexisting phases. The grand canonical technique still has some limitations. The method can be tedious if at least one of the coexisting phases is ordered because one must compare the chemical potentials and pressures of numerous ordered phases to locate coexistence. Typically, root-finding algorithms for determining the chemical potentials (set equal between the candidate phases) that guarantees equal pressures exhibit fast convergence when good initial values for the chemical potentials are used, but otherwise suffer from stability issues. In addition, the theory is cast in terms of chemical potentials, which are nonobservable variables, and can only be converted to experimentally measured values of composition via a full SCFT calculation. The Gibbs ensemble description of phase coexistence originally developed by Panagiotopoulos38,39 for Monte Carlo simulations enables direct canonical calculations of phase coexistence without interfaces. The problem of interfaces is solved by placing each coexisting phase in a separate simulation cell. Chemical and mechanical equilibrium between coexisting phases is achieved by allowing the exchange of mass and volume, respectively. Since the overall Gibbs ensemble system forms a canonical ensemble, the distribution of mass and volume is constrained by the total mass and volume with which the system is specified. Riggleman and Fredrickson40 adapted the Gibbs ensemble to an SCFT and field-theoretic description of polymer phase behavior. Mester et al.41 further developed Gibbs ensemble SCFT using an intensive formulation in order to simulate multicomponent polymer mixtures of arbitrary ordered state symmetries. Mesostructured phases were expressed as minimally representative volumes, i.e., unit cells, and the resultant formalism represents a powerful platform for the calculation of equilibrium coexistence of microphase separated morphologies. The formalism does not suffer from the rootfinding stability issues of the grand canonical method because coexistence is obtained by finding the minimum of the free energy of the total system with respect to composition in each phase via conjugate gradient minimization. To combine the advantages of a unit-cell description of microphase separation with a canonical description of macrophase separation and supramolecular assembly, we adapt the Gibbs ensemble unit cell SCFT formalism of Mester et al.41 to reversibly bonding block polymer systems. Since the overall Gibbs ensemble system forms a canonical ensemble, we develop a method for calculating supramolecular assembly in the canonical ensemble. We demonstrate our method on a melt blend of AB diblock copolymer that reversibly reacts with B homopolymer functionalized at one of its ends to form “ABB” diblock copolymer. Because of similarities with the previously studied diblock copolymer melt 19−23 and the diblock
2. METHODS 2.1. Theoretical Methods. In this paper, we examine an AB diblock copolymers with polymerincompressible melt of n(0),T AB B ization index N and volume fraction of A segments f and n(0),T Bh homopolymers of polymerization index αN that react to form ABB diblock copolymers with polymerization index (1 + α)N. The term α denotes the ratio of the polymerization index of the B homopolymer to the AB diblock copolymer. When the melt forms two coexisting phases K = I, II of volumes VK, the number of polymers in each phase of the melt is denoted by nKBh, nKAB, and nKABB for the B homopolymer, AB diblock copolymer, and ABB diblock copolymer, respectively. The reaction in each phase K enforces molar balances given by K (0), K rxn, K nBh = nBh − nBh
(1)
K (0), K rxn, K nAB = nAB − nBh
(2)
rxn, K K nABB = nBh
(3)
where the superscript (0) indicates the unreacted parent compositions and nrxn,K Bh is the number of B homopolymer molecules reacted in phase K. The molar balance between phases I and II involving reactant and product polymers can be expressed in terms of B homopolymer building blocks whose numbers are given by the unreacted parent (0),T compositions n(0),K Bh in each phase K and nBh in the overall melt. The molar balance between the phases is (0),II (0),T (0),I nBh = nBh − nBh
(4)
The volumes are related by VII = VT − VI
(5)
where VT is the total volume of both cells. Incompressibility specifies the AB diblock copolymer unreacted parent compositions as (0), K = nAB
VK ρ0 N
(0), K − αnBh
(6)
where ρ0 is the average monomer density. To obtain the partition function for this system, we employ the standard model of Gaussian thread polymers interacting via contact interactions and an incompressibility constraint. The canonical partition function describing chains divided between two simulation cells connected by volume and molar balances and incompressibility (eq 1−6) is given by (0),T (0),T , nAB , VT , T ) Z(nBh
=
∫0
VT
dVI VT
(0),T nBh
∑ ∏ (0),I = 0 K = I,II nBh
(0), K (0), K min[nAB , nBh ]
×
∑ rxn, K nBh =0
1 (0), K (0), K 3(nBh N) αN + nAB
Λ
K nBh
∏∫
K nAB
j,K +r Bh
j=1
× δ[ρÂK + ρB̂ K − ρ0 ]
∏∫ l=1
K nABB
,K +r lAB
,K ∏ ∫ +r kABB k=1
1 K K K ! nAB ! nABB ! nBh
AB , K , r nABB , K ]] × exp[− U0K [r nBhBh , K , r nAB ABB AB , K , r nABB , K ] + F n rxn, K ] × exp[− U1K [r nBhBh , K , r nAB ABB b Bh
(7)
where Λ is the thermal wavelength, incompressibility is enforced by the delta functional δ [ρ̂KA + ρ̂KB − ρ0], the location of a segment along the backbone of each chain is indicated by a continuous contour 1866
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l,K k,K variable s, and rj,K Bh(s), rAB(s), and rABB(s) denote the positions of segment s of polymer j, l, and k in cell K for the B homopolymer, AB diblock copolymer, and ABB diblock copolymer, respectively. The harmonic stretching energy is given by K nBh
U0K [r nBhBh , K ,
AB , K , r nAB
ABB , K ] r nABB
=
∑ j=1
K nAB
+
∑ l=1 K nABB
+
∑ k=1
3 2b2
∫0
3 2b2
,K dr lAB (s) ds
N
∫0
3 2b2
j,K dr Bh (s) ds
AB , K , r nAB
ABB , K ] r nABB
⎛ Q 0, K ⎞ ⎛ Q 0, K ⎞ ⎛ Q 0, K ⎞ K K K AB ⎟ ABB ⎟ ⎟ ⎜ ⎜⎜ ln⎜⎜ 3Bh ln ln − nBh − n − n AB ABB ⎟ ⎜ ⎟ 3N αN 3(1 + α)N ⎟ ⎝Λ ⎠ ⎝Λ ⎠ ⎝Λ ⎠
ds
K K K rxn, K } + ln(nBh ! nAB ! nABB ! ) − FbnBh
ds (8)
∫V
drK ρÂK (rK )ρB̂ K (rK )
K Q AB N
(9)
=
where χ is the A−B Flory−Huggins parameter and ν0 = 1/ρ0 is the statistical segment volume. The A segment density in cell K as a function of position rK is given by
ρÂK (rK ) =
∑∫ l=1
fN
0
K nABB
+
∑∫ k=1
fN
,K δ(rK − r kABB (s)) ds
(0), T Z(nBh ,
(0),T nAB ,
VT , T ) =
(0), K (0), K min[nAB , nBh ]
∫0
VT
dVI VT
(10)
∫0
×
∫0
×
∫ +ρAK ∫ +ρBK ∫ +ξK exp(−H)
rxn, K dnBh
(0),I dnBh
fN
N
1, K 3/2b2 |drAB (s)/ds|2 ds −∫0 wAK (r1,ABK (s)) ds −∫ fN wBK (r1,ABK (s)) ds
∫ +r1,ABKe−3/2b
2
N
∫0 |dr1,ABK (s)/ds|2 ds
The normalized single chain partition functions for the ABB diblock copolymer and the B homopolmer are constructed analogously. We note that Q0,K AB is the denominator of eq 13. By discretizing the AB diblock copolymer chain into N monomer beads and integrating the N−1 where gM = (2πb2/ denominator of eq 13, we obtain Q0,K AB = VKgM αN−1 and Q0,K 3)3/2 is comparable to ν0 = 1/ρ0. Similarly, Q0,K Bh = VKgM ABB = . VKg(1+α)N−1 M We define the following dimensionless terms:
The expression for the B segment density in cell K is constructed analogously. Supramolecular assembly is incorporated into the model by placing a term for the total energy change due to reaction in each cell K in the Hamiltonian of the partition function given by eq 7. The total energy o o change from the reaction is Fbnrxn,K Bh where Fb = −ΔGrxn/kBT, ΔGrxn is the Gibbs free energy change associated with the reaction of an AB diblock copolymer with a B homopolymer, kB is Boltzmann’s constant, and T is temperature. The reactivity of our polymers provide nrxn,K Bh as additional degrees of freedom in our system. The sum over these degrees of freedom runs from no homopolymer reacted (nrxn,K Bh = 0) to (0),K (0),K all the limited reactant being consumed (nrxn,K Bh = min[nAB ,nBh ]). We convert the particle-based partition function in eq 7 to one based on statistical field theory.2,19,27,28 Placing 1 = ∫ +ρAK δ[ρAK − ρÂK ] into the partition function replaces the operator ρ̂KA with the function ρKA. We do the same for the operator ρ̂KB and substitute standard integral representations for the delta functionals. The standard integral representation of the delta functional that enforces incompressibility is used to introduce the Lagrange multiplier ξK(rK). The field-based partition function is (0),T nBh
∫ +r1,ABKe− ∫0
(13)
,K δ(rK − r lAB (s)) ds
0
(12)
with the application of the molar and volume balances and the 0,K incompressibility conditions (eq 1−eq 6). The terms Q0,K Bh , QAB , and are the unnormalized single chain partition functions in zero field Q0,K ABB for the B homopolymer and the AB and ABB diblock copolymers, respectively, and the normalized single chain partition functions for the AB diblock copolymer in external fields is
2
K
K nAB
drK {ν0χρAK (rK )ρBK (rK ) − wAK (rK )ρAK (rK )
K
K K K K K K ln Q Bh ln Q AB ln Q ABB − nBh − nAB − nABB
2
ds
= ν0χ
∫V
{
− wBK (rK )ρBK (rK ) − ξK (rK )(ρAK (rK ) + ρBK (rK ) − ρ0 )}
where b is the statistical segment length for all chains. Attractive interactions between segments of the same species are modeled using a quadratic interaction
U1K [r nBhBh , K ,
∑ K = I,II
2
,K dr kABB (s) ds
(1 + α)N
∫0
αN
H=
rK̅ =
rK , Rg
ϕUK (rK ) =
VK
VK̅ =
Rg 3
ρUK (rK ) ρ0
,
wU̅ K (rK ) = wUK (rK )N ,
VT̅ =
,
s̅ =
VT Rg 3
s N
ξK̅ (rK ) = ξK (rK )N
(14)
where U = {A,B} indicates the segment species and the radius of gyration of the AB diblock copolymer is Rg = b(N/6)1/2. We write the average unreacted parent concentrations of each polymer species in = NMn(0),K each box K as ϕ(0),K M M /(ρ0VK) where M = {Bh,AB} and the average concentrations upon reaction in each box K as ϕKJ = NJnKJ / (ρ0VK) where J = {Bh,AB,ABB}. We define the concentrations of rxn,K homopolymer reacted in each box K as ϕrxn,K Bh = NBhnBh /(ρ0VK). The polymerization indices of the polymer species in the melt are NBh = αN, NAB = N, and NABB = (1 + α)N. The reaction molar balances in cell K (eq 1−eq 3) in terms of dimensionless variables are K (0), K rxn, K ϕBh = ϕBh − ϕBh
∏ K = I,II
K (0), K ϕAB = ϕAB −
∫ +wAK ∫ +wBK K ϕABB =
(11)
(15)
rxn, K ϕBh
α
(16)
1 + α rxn, K ϕBh α
(17)
The molar and volume balances between simulation cells (eq 4−eq 5) in terms of dimensionless variables are
where wKA(rK) and wKB (rK) are the fields acting on A and B segments, respectively, and ξK(rK) enforce incompressibility in each cell. The and the unreacted number of reacted homopolymer molecules nrxn,K Bh are treated as continuous variables. The parent compositions n(0),K Bh Hamiltonian of the partition function in eq 11 is the sum of the Hamiltonian in each box K (H = HI + HII). The Hamiltonian in the combined system is
νI =
(0),T (0),II ϕBh − ϕBh (0),I (0),II ϕBh − ϕBh
νII = 1 − νI 1867
(18) (19)
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where νK = V K/V T is the relative volume of cell K. The incompressibility conditions given by eq 6 in dimensionless variables are (0), K (0), K ϕAB = 1 − ϕBh
+
(20)
We substitute the dimensionless variables into the Hamiltonian of box K given by eq 12, make use of the balances in eqs 15−20, and apply Stirling’s approximation to derive the intensive Hamiltonian H̅ = HN/ (ρ0VK), which is given up to an additive constant (that is independent of phase volume fractions and relative volumes) by
H̅ = − −
⎧1 νK ⎨ ⎩ VK̅ K = I,II
∑
K ϕABB
1+α
α −
K ϕBh
ln
α
K ϕABB
1+α
−
−h
K ϕBh
− 1)}
α
K K K + ϕAB − ϕAB + ln ϕAB
rxn, K ⎫ ϕBh ⎬ α ⎭
K ϕABB
(21)
is a new dimensionless parameter for where h = Fb − reaction energy. The intensive Hamilonian can be expressed as H̅ = νIH̅ I + νIIH̅ II where H̅ K contains the terms corresponding to simulation cell K. The volume fractions of B homopolymer reacted ϕrxn,K Bh are obtained analytically from the reaction equilibrium statements in each cell K
K Q AB =
K K K Q AB Q Bh ϕABB K Q ABB
∂InZ ∂nJK
niK≠ J , VK , T
(23)
and substituting the results of eq 23 into the reaction constraints on the chemical potentials K K K μABB − μAB − μBh =0
σi K =
(24)
We complete the derivation of eq 22 by applying our dimensionless variables. The current work is restricted to the mean-field (self-consistent field) approximation made by finding the dominant (saddle-point) field configurations for each simulation cell K of the partition function. The mean-field equations that yield the dominant field configurations are
wA̅ K ( rK̅ ) = χNϕBK ( rK̅ ) − ξK̅ ( rK̅ ) w̅BK ( rK̅ ) = χNϕAK ( rK̅ ) − ξK̅ ( rK̅ )
1 = ϕAK ( rK̅ ) + ϕBK ( rK̅ )
K ϕAB
+
K Q AB
∫0
K ϕABB
(1 +
K α)Q ABB
f
(25)
F̅ =
(26) (27)
f
†,K K d s ̅ qABB ( rK̅ , s ̅ )qABB ( rK̅ , s ̅ )
K K d s ̅ qBh ( rK̅ , s ̅ )qBh ( rK̅ , α − s ̅ )
1 VK̅
∫V̅
K
(29)
†,K d rK̅ qAB ( rK̅ , s ̅ = 0)
(31)
K ⎫ ϕK ∂ ⎧ ϕBh K K K K ⎬ ⎨ + ϕAB + ABB ln Q ABB ln Q Bh ln Q AB K 1+α ∂ai ⎩ α ⎭ ⎪
⎪
⎪
⎪
⎧ − χN νK ⎨ ⎩ VK̅ K = I,II
∑
⎪
⎪
K K ln Q AB − ϕAB − K K ln ϕAB − ϕAB +
∫V̅
ϕAK ( rK̅ )ϕBK ( rK̅ )d rK̅ −
K ϕBh
α
K
K ϕABB
1+α K ϕABB
1+α
K ln Q ABB +
ln
K ϕABB
1+α
K ϕBh
α −
ln
K ϕBh
α
K ϕABB
1+α
−
−h
K ln Q Bh K ϕBh
α
K + ϕAB
rxn, K ⎫ ϕBh ⎬ α ⎭ ⎪
⎪
(33) The intensive free energy can be expressed as F̅ = νIF̅I + νIIF̅II where here F̅K contains the terms corresponding to simulation cell K. Chemical and mechanical equilibrium between simulation cells I and II are established when the reaction equilibrium conditions given by eq 22 are satisfied and the intensive Gibbs ensemble free energy (F̅)
K †,K d s ̅ qAB ( rK̅ , s ̅ )qAB ( rK̅ , s ̅ )
∫0
†,K K d s ̅ qABB ( rK̅ , s ̅ )qABB ( rK̅ , s ̅ )
(32) When all stresses are equal to zero (σKi = 0), the set of σKi are at their equilibrium values. The intensive free energy of a melt is obtained by applying the mean-field solutions in eqs 25−27 to the intensive Hamiltonian in eq 21. The intensive free energy is given by
The spatially dependent dimensionless densities of A and B segments in cell K are given by
ϕAK ( rK̅ ) =
α
1+α
The normalized single chain partition functions subject to fields for the B homopolymer QKBh and the ABB diblock copolymer QKABB are analogously expressed. The expressions for the spatially dependent dimensionless densities and single chain partition functions along with the mean-field equations given by eqs 25−27, the reaction molar balances given by eqs 15−17, and the reaction equilibrium statement given by eq 22 are sufficient to establish equilibrium within any cell K provided that the intensive free energy within the cell is minimized with respect to unit cell lattice constants aki . Finding these lattice constants relies on the calculation of stresses defined by
(22)
when used in conjunction with the reaction molar balances given by eqs 15−17. The reaction equilibrium statement in eq 22 is derived by calculating the chemical potentials for species J = {Bh,AB,ABB} in cell K from the partition function in eq 11 as
μJK =
∫0
K αQ Bh
∫f
(30) with initial conditions qKAB(rK̅ ,0) =1. The modified diffusion equation given by eq 30 multiplied by −1 on its right side is satisfied by the backward propagators for the AB diblock copolymer q†,K AB (rK ̅ ,s)̅ with the ,1) = 1. The forward qKABB(rK̅ ,s)̅ and backward conditions q†,K AB (rK ̅ q†,K ABB(rK ̅ ,s)̅ propagators for the ABB diblock copolymer and the forward propagators q†,K Bh (rK ̅ ,s)̅ for the B homopolymer satisfy analogous modified diffusion equations. The AB diblock copolymer normalized single chain partition functions subject to fields is expressed in terms of the propagators as
1+α
−1 ln(Nρ−1 0 gM )
α = eh K K ϕBh ϕAB 1 + α
K ϕBh
†,K K d s ̅ qAB ( rK̅ , s ̅ )qAB ( rK̅ , s ̅ )
∂s ̅ ⎧∇ 2 q K ( r , s ) − w K ( r )q K ( r , s ) for 0 < s < f A ⎪ rK̅ AB K̅ ̅ ̅ K̅ AB K̅ ̅ ̅ =⎨ ⎪∇r 2 q K ( rK̅ , s ̅ ) − w̅BK ( rK̅ )q K ( rK̅ , s ̅ ) for f < s ̅ < 1 ⎩ K̅ AB AB
⎪
⎪
K α)Q ABB
1
K ( rK̅ , s ̅ ) ∂qAB
1+α
K ϕBh
(1 +
∫f
The statistical weight, or propagator, for finding segment s ̅ of the AB diblock copolymer at position rK̅ in cell K is given by qKAB(rK̅ ,s). ̅ The propagators satisfy the modified diffusion equation
d rk̅ {χNϕAK ( rk̅ )ϕBK ( rk̅ ) − wA̅ K ( rk̅ )ϕAK ( rk̅ )
α
K Q AB K ϕABB
+
K
w̅BK ( rk̅ )ϕBK ( rk̅ )− ξK̅ ( rk̅ )(ϕAK ( rk̅ ) + ϕBK ( rk̅ ) K ϕBh ϕK K K K − ϕAB − ABB ln Q Bh ln Q AB
K + ln Q ABB
ln
∫V̅
K ϕAB
ϕBK ( rK̅ ) =
(28) 1868
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The exponential of the Hamiltonian matrix operators are evaluated by performing a matrix eigendecomposition. This results in the following solutions for the propagator amplitudes qKAB,i (s): ̅
of the overall system is minimized with respect to the unreacted parent and ϕ(0),II compositions of the homopolymer (ϕ(0),I Bh Bh ). The Gibbs ensemble free energy is minimized with respect to unreacted parent = 0. We calculate this derivative while compositions when ∂F̅/∂ϕ(0),K Bh enforcing the mean-field equations given by eqs 25−27 along with σKi = 0 and holding the spatially varying segment densities (ϕKA and ϕKB ), fields (w̅ KA and w̅ KB ), and average volume fractions of reacted B homopolymer (ϕrxn,K Bh ) constant. The solutions for the minimum of the free energy are restricted to those where the molar balances from reactions given by eqs 15−17, the molar and volumetric balances between cells given by eqs 18 and 19, and the incompressibility constraints given by eq 20 are satisfied. The resulting expressions for the partial derivatives are
K qAB, (0 < s ̅ < f ) = i
j K qAB, (f < s ̅ < 1) = i
∑ giK fiK ( rK̅ )
(34)
f Ki
(rK̅ ) possess a defined space group where are amplitudes. The symmetry, are ordered in increasing spatial frequency, and satisfy
∇rK̅ 2 fiK ( rK̅ ) = − λi K fiK ( rK̅ )
(36)
λKi
are eigenvalues of the Laplacian operator. The basis set f Ki where (rK̅ ) is generated from the symmetry operations of the space group according to Tyler and Morse.42 The spectral method provides an efficient algorithm for solving modified diffusion equations of the type given by eq 30. The modified diffusion equation in the symmetry adapted basis set is expressed as K ∂qAB, (s ) i ̅
∂s ̅
⎧∑ ( − λ K δ − i ij ⎪ ⎪ j =⎨ ⎪∑ (− λi K δij − ⎪ j ⎩
K K K ∑ wA,k ̅ Γ ijk)qAB, j( s ̅ )
for 0 < s ̅ < f
k K (s ) ∑ w̅B,Kk ΓijkK)qAB, j ̅
for f < s ̅ < 1
k
(37) The overlap integral is defined as K Γ ijk =
1 VK̅
∫V̅
K
d rK̅ fiK ( rK̅ )f jK ( rK̅ )fkK ( rK̅ )
(38)
The modified diffusion equation given by eq 37 can be solved exactly to a finite number of basis functions. The solutions for the K (s)̅ to an arbitrary number of basis propagator amplitudes qAB,i functions are A, K
K qAB, (0 < s ̅ < f ) = e sHi0 i K qAB, (f < s ̅ < 1) = i
(39) B, K
∑ e(s − f )Hij
A, K
e fHj0
j
(40)
The Hamiltonian operators are defined as HijU , K = − λi K δij −
∑ wU̅ K, k ΓijkK k
(43)
Matrices Y contain the eigenvectors in its columns and vectors tU,K contain the corresponding eigenconstants of the Hamiltonian operators HU,K. The analytical solutions for the propagator amplitudes are used to express the single chain partition functions for polymers in the fields as K †,K K K QKAB = q†,K AB,0(0), QABB = qABB,0(0), and QBh = qBh,0(α) for the AB and ABB diblock copolymers and B homopolymer, respectively. The single chain partition functions along with the spatially dependent dimensionless A and B segment densities given by eq 28 and eq 29 are used to construct amplitudes of basis-projected monomer densities ϕKA,i and ϕKB,i. The basis-projected monomer densities and fields acting on A and B segments are used to re-express the mean-field equations in eqs 25−27 in spectral space. To establish internal equilibrium within the phases, the Anderson mixing algorithm43 is employed to iteratively determine the fields that satisfy the mean-field equations for each candidate phase at the initial compositions and lattice constants αKi . The measure of the numerical inaccuracy in the field calculations are obtained from eq 21 in Matsen.43 The Anderson mixing algorithm is said to be converged when the numerical inaccuracy is within a tolerance of 1 × 10−10 in units of kBT. Using all the available histories up to 10 to construct the iterations of the fields yields reasonably good convergence speed and numerical stability. The stresses are eliminated by finding with a Newton−Raphson solver the lattice constants αKi where σKi = 0 within the specified tolerance (i.e., 1 × 10−6 in units of kBT). The terms proportional to stresses σKi are obtained by calculating the partial derivatives in eq 32 by finite differentiation. After each iteration of αKi , the fields are recalculated. A conjugate gradient algorithm is used to find the minimum in the intensive free energy of the overall system in terms of unreacted parent (0),I (0),II and ϕBh . Once the internal stresses are compositions ϕBh eliminated, we estimate the vector pointing to the minimum in the intensive free energy using the gradients calculated from ∂F̅/∂ϕ(0),I Bh and ∂F̅/∂ϕ(0),II Bh given by eq 34. Brent’s method is subsequently used to find the minimum along this vector. A new vector is calculated at the minimum using the Fletcher−Reeves algorithm. After each calculation of the new vector, we repeat the procedure for finding internal equilibrium in each phase. Convergence occurs when the convergence criteria of the conjugate gradient solver given by the magnitude of the gradient vector is below the required precision, which is 1 × 10−9 in free energy in units of kBT. The average volume fractions of reacted B are calculated analytically from the reaction homopolymer ϕrxn,k Bh equilibrium statements in eq 22 for every step of the Gibbs ensemble SCFT algorithm. It was previously demonstrated for polymer blends that Gibbs ensemble unit-cell SCFT enables rapid and automated calculation of phase coexistence regions that are bordered by single-phase domains of stability.41 The first step in phase coexistence calculations for reactive polymers is to enforce the desired symmetries in the simulation cells by calculating the overlap integrals in eq 38 from the appropriate set of f Ki (rK̅ ) basis functions. Equilibrium compositions of coexisting phases are obtained at given values of χN and h with our conjugate gradient algorithm while enforcing reaction equilibrium in each cell. We map out the phase diagram by incrementing χN or h and initializing the fields, lattice dimensions, and compositions from the results of the previous calculation. The size of the increment is selected based on the change in the equilibrium compositions with respect to the variable being incremented so that we properly resolve the features of the phase boundary.
(35)
i
gki
∑ YijB, K exp[( s ̅ − f )t jB, K ]Y jlB, KYlmA, K
U,K
2.2. Numerical Methods. We perform calculations on unit cells of spatially periodic mesophases found in each box using the spectral method developed by Matsen.19 The spectral method allows for efficient, high resolution calculations because the only numerical approximation is the representation of all spatially dependent functions as a truncated series of symmetry adapted basis elements. Spatially dependent functions gK(rK̅ ) are expressed in each box K in terms of a symmetry adapted basis f Ki (rK̅ ) as gK ( rK̅ ) =
(42)
jlm
exp[ftmA, K ]YmA,0K
ν ν ∂F ̅ = − (0),I K (0),II FI̅ + (0),I K (0),II FII̅ (0),K ∂ϕBh ϕBh − ϕBh ϕBh − ϕBh ⎛ 1 ⎞ ϕK 1 K K K⎟ + νK ⎜⎜ − ln Q Bh + ln Q AB + ln Bh − ln ϕAB ⎟ α α ⎝ α ⎠
∑ YijA, K exp[ s ̅ t jA, K ]Y jA,0 K
(41)
where U = {A,B}. 1869
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3. RESULTS AND DISCUSSION The utility of our new method for calculating coexisting phases for reactive polymers as described in section 2 is demonstrated on an incompressible melt blend of AB diblock copolymers with composition f = 0.6 and B homopolymers that are the same length as the overall diblock copolymer (α = 1). The blend components reversibly bond to form ABB diblock copolymers of length 1 + α and composition f = 0.3. A schematic of the reversible reaction is shown in Figure 1. For Figure 3. Phase diagram at χN = 9 in terms of reaction favorability h versus average B segment volume fraction ϕB of a melt of AB diblock copolymers with f = 0.6 reacting with B homopolymers with α = 1 to form ABB diblock copolymers. Candidate morphologies for calculating the phase diagram were disordered (D), lamellar (L), hexagonal (H), gyroid (G), and spherical (S). The S phase is represented by body centered cubic spheres. Morphologies where the minority domains are primarily A segments are indicated by a prime symbol (e.g., G′).
Figure 1. Schematic illustrating the reversible reaction of AB diblock copolymer and B homopolymer to form ABB diblock copolymer. For the current study, f = 0.6 and α = 1.
affects the selection of morphology. This is due to the depletion of the limiting polymer reactant at each composition. The limiting reactant volume fractions vary between 0 and 0.054 at h = 5. At low ϕB the limiting reagent is B homopolymer, and at high ϕB the limiting reagent is AB diblock copolymer. Therefore, as h is increased the population of polymer species saturates to AB + ABB at ϕB < 0.7, and saturates to ABB + Bh at ϕB > 0.7. At matched stoichiometry (ϕB = 0.7) the system progresses from a AB + Bh to ABB as h is increased. The system transitions from behavior that is consistent with the presence of a blend to that of a pure cylindrical forming diblock copolymer.19,20 Because of the presence of up to three species in the melt, we observe coexistence regions that are bordered by single phase regions. For cases where Bh is the limiting reagent (ϕB < 0.7) and h is sufficiently high to produce ordered phases, we illustrate the composition of coexisting phases with Figure 4 by plotting the relative composition of polymer species along the borders of the L+G coexistence region. Figure 4 shows that coexistence occurs between higher B segment composition phases (L) that are richer in ABB and Bh and lower B segment composition phases (G) that are richer in AB. We also observe that overall the equilibrium is shifted toward high AB + ABB concentration for a large range of h values for both phases. ABB and Bh become equal in concentration once the reaction energy is lowered to approximately h = 0. Lowering the reaction energy h below zero results in blends that are primarily AB + Bh with enough ABB remaining to stabilize ordered structures. Alternatively, when AB is the limiting reagent (ϕB > 0.7) and h is sufficiently high to produce ordered states, coexistence occurs between high B segment composition
convenience, we limit our search of phase morphologies to disordered (D), lamellar (L), hexagonal (H), gyroid (G), and spherical (S), which are found on the phase diagram of the AB diblock incompressible melt calculated by Matsen and coworkers.19,21 The S phase is represented by body-centeredcubic spheres. Close-packed spherical morphologies are also observed on the phase diagram, such as face-centered-cubic and hexagonally close-packed spheres. However, we do not search for them since the free energy differences between various spherical phases are generally very small. We likewise neglect the more recently predicted23 and observed44 Fddd network morphology since the domain of stability is small compared to other morphologies in diblock copolymer melts. The lowest free energy ordered phase morphologies at h = 5 and χN = 9 for different values of average B segment volume fraction ϕB = ϕ(0),T + (1 − f) (1 − ϕ(0),T Bh Bh ) are displayed in Figure 2. Morphologies where the minority domains are primarily A segments are indicated with a prime symbol (e.g., G′). Isosurfaces in Figure 2 are located at spatially r varying dimensionless B segment densities of ϕB(r) = 0.5 except for G, where the iso-surface was selected to be ϕB(r) = 0.3 to better illustrate the structure. The phase diagram in Figure 3 was calculated by varying the reaction energy h at a fixed χN = 9 for the classical diblock copolymer morphologies.19,20 The AB diblock copolymer and B homopolymer (Bh) blend is disordered at χN = 9. As reaction becomes more favorable as h is increased, the product ABB diblock copolymer stabilizes ordered phases due to the increased segregation strength stemming from its larger size. At high h, the phase boundaries become parallel with the h-axis indicating that the reaction equilibrium in the system no longer
Figure 2. Lowest free energy ordered phase morphologies at h = 5 and χN = 9 with respect to the total B segment volume fraction ϕB in the melt. Each illustration consists of eight unit cells. The phases are disordered (D), lamellar (L), hexagonal (H), gyroid (G), and spherical (S). The S phase is represented by body centered cubic spheres. Morphologies where the minority domains are primarily A segments are indicated by a prime symbol (e.g., G′). Isosurfaces are located at ϕB(r) = 0.5 except in G where the isosurface was selected to be ϕB(r) = 0.3 to better illustrate the structure. 1870
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The concentration of AB does not increase significantly because there is a strong shift of phase boundaries to higher cumulative B segment concentrations with decreased h. As can be seen from the phase diagram in Figure 3, phase boundaries shift toward higher B composition with decreasing h. Examination of the segment distribution within an L unit cell reveals the reasons underlying the h dependence of the phase boundaries. Figure 6 shows the distribution of the A and B blocks on the various polymer species in the L unit cell located on the boundary of the L + H region at h = 5. Under these conditions, only AB and ABB are present in appreciable amounts. The identical A blocks on both AB and ABB fill the A-rich microdomain equivalently. However, the differently sized B blocks exhibit a partitioning within the unit cell by size. This effect is similar to that seen in polydisperse block copolymer systems.45,46 The smaller B block localizes near the interface, and the longer B block on ABB partitions near the center of the B domain. A decrease in h releases untethered Bh and AB into the unit cell. The partitioning of Bh to the center of the B microdomains mitigates the entropic penalty of filling space in the B microdomain with AB, and increases configurational entropy in the system. This softening of the B-domain favors shifting phase boundaries to higher composition in the B component toward morphologies with increased interfacial curvature toward the B-domain containing size-disperse B-species. The distribution of the A and B blocks on the various polymer species in the L unit cell on the boundary of the L + H region at h = −1 (see Figure 6) demonstrates that the addition of the Bh to the system both through the decrease in h and increase in ϕB allows AB to qualitatively maintain the distribution in the unit cell seen at h = 5, and, hence, remain in the L morphology. In order to validate the quality of our results, we compare the phase diagram in our current study to the results in Mester et al.41 In the region of Figure 3 where AB is the limiting reagent (ϕB > 0.7) and h is high, the equilibrium melt blend approximately matches the diblock copolymer and homopolymer system at a segregation strength of χN = 18 in Figure 4 of Mester et al.41 Comparing the equilibrium compositions on the S′ + H′ boundaries in the current studies to the S + H boundaries in Mester et al.,41 we see that there is good agreement in the results. The difference in terms of
Figure 4. To illustrate trends in the compositions of coexisting phases when Bh is the limiting reactant (ϕB < 0.7), the average compositions of phases along the borders of the L + G coexistence region are plotted in terms reaction energy h. The average compositions are in terms of volume fractions of AB diblock (ϕAB) and ABB diblock (ϕABB) copolymers and B homopolymer (ϕBh).
phases that are richer in Bh, and lower B segment composition phases that are richer in ABB and AB (see Figure 5). On the
Figure 5. To illustrate trends in the compositions of coexisting phases when AB is the limiting reactant (ϕB > 0.7), the average compositions of phases along the borders of the S′ + H′ coexistence region are plotted in terms reaction energy h. The average compositions are in terms of volume fractions of AB diblock (ϕAB) and ABB diblock (ϕABB) copolymers and B homopolymer (ϕBh).
boundaries of the S′ + H′ coexistence region, the equilibrium is shifted toward high ABB concentration for a large range of h values for both phases. As the reaction energy is decreased, the concentration of Bh increases and ABB decreases. The concentrations become equal around h = 1.5 in both phases.
Figure 6. Contribution of the AB and ABB polymer species to the cumulative A and B segment concentration profiles in the unit cell at h = 5 and h = −1 on the L phase boundary of the L + G coexistence region. Distances x are expressed in terms of the radius of gyration of the AB diblock Rg = b(N/6)1/2. 1871
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homopolymer composition is 0.016 in the hexagonal phase and 0.017 in the spherical phase. The difference is even lower when applying the same comparison to the D′+S′ phase boundaries due to the higher degree of conversion of the limiting reactant (97.3% versus 98.8% on the respective spherical phase boundary). Figure 3 shows that as h is decreased, the widths of the coexistence regions increase due to lower compatibility between the reactants than the reactants with the product. Eventually, the borders of different coexistence regions intersect, which leads to the elimination of a nonmacrophase separated morphology below the value of the h of the intersection. This creates coexistence between new phases. Below h = −1.376, L and G phases are the only ordered mesophases that remain on the phase diagram. Stabilizing phases and calculating coexistence at these low values of h is difficult because the unreacted homopolymer swells the B rich microdomains of the ordered phases substantially, which increases the number of basis functions required to resolve the phase and flattens the free energy landscape with respect to the system variables. Therefore, while the trend in the borders of the remaining coexistence regions suggest the elimination of the G phase, it becomes prohibitively expensive to calculate ordered phases at lower values of h. Instead, the trend is merely extrapolated, and is indicated by dashed lines. Moreover, the stability of ordered phases at those low values of h is likely an artifact of the mean field assumption, and fluctuation effects would cause the system to be disordered.2,47 While the L and H′ phases do not trend toward elimination as h is decreased, we argue that their apparent stability against unbinding at low h in Figure 3 is also an artifact due to the mean-field assumption. The dashed vertical borders of the L and H′ phases with the D′ phase in Figure 3 indicate the highest B segment composition L and H′ phases that we were able to stabilize with the mean-field calculations at h = −1 and h = 0, respectively. However, a full analysis beyond mean-field theory would very likely terminate these phases at lower B compositions due to fluctuationinduced destruction of long-range order. We note that the phase diagram in Figure 3 was constructed with the assumption that coexistence is only present in our system between two compositionally distinct phases. This assumption is valid because all three polymer species need to be in the blend in substantial amounts for coexistence to occur between three compositionally distinct phases. The region where this can occur is small due to the restrictions of reaction energy h and stoichiometry. Furthermore, since ABB is derived from AB and Bh, the ABB product is highly miscible with its building blocks making it unlikely that it would form its own pure phase. Moreover, the miscibility of homopolymer and diblock copolymer blends has been established in previous theoretical studies,27,28,41 and the ABB + Bh blend system falls within this established miscible regime. However, the method we present is readily extended to 3+ coexisting phases for situations that may require doing so.41 In order to explore the interplay between dynamic bonding and incompatibility, Figure 7 shows phase behavior as a function of segregation strength χN at h = 3. The phase coexistence regions narrow with decreased χN due to increasing compatibility between A and B polymer segments. Eventually, the coexistence regions narrow enough that they are much smaller than the line thickness in Figure 7. When this occurs, boundaries between pairs of phases are calculated by locating the blend composition where the free energies of the candidate
Figure 7. Phase diagram at h = 3 in terms of immiscibility χN versus average B monomer volume fraction ϕB of a melt of AB diblock copolymers with f = 0.6 bonding with B homopolymers with α = 1 to form ABB diblock copolymers. We construct the phase diagrams for disordered (D), lamellar (L), hexagonal (H), gyroid (G), and spherical (S) phases. The S phase is represented by body centered cubic spheres. Morphologies where the minority domains are primarily A are indicated by a prime symbol (e.g., G′). The mean-field critical point where the RPA spinodal is equal to the binodal is located at (ϕB,χN) = (0.579,6.80) is indicated by a point (•).
phases are equal. At a B monomer concentration of ϕB = 0.4 which contains no contribution due to Bh, the transitions between phases maps to the phase diagram of the pure AB diblock copolymer with f = 0.6.19,20 Since the homopolymer reactant Bh is scarce near ϕB = 0.4, phase coexistence between compositionally distinct phases disappears. All the phases on the diagram have a mean-field order-to-disorder transition at a single critical point except the G and G′ phases which join with the H and H′ phases at slightly higher values of χN. We define the critical point as the point where the spinodal predicted by the random phase approximation (RPA) and the binodal as calculated by SCFT are equal. This behavior is also qualitatively demonstrated in the pure diblock phase diagram.19,20 This point located at (ϕB,χN) = (0.579,6.80) is indicated by a dot (•) on the phase diagram. Figure 8 shows that the conversion of B homopolymer, ε = ϕrxn,T /ϕ(0),T , in the L region of Figure 7 at ϕB = 0.57925 B B
(0),T Figure 8. Conversion of B homopolymer, ε = ϕrxn,T , versus χN B /ϕB in the L region of Figure 7 at ϕB = 0.57925. The higher concentration of B functional ends in the B-rich microdomains that results from stronger segregation between A and B segments at high χN causes a small increase in ε with increased χN.
slightly increases as the segregation strength χN increases. This is due to the higher concentration of B functional ends in the Brich microdomains that results from stronger segregation between A and B segments at high χN. This higher conversion slightly counteracts the tendency toward the widening of phase coexistence regions with increased χN in the same way that 1872
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increasing h serves to stabilize wide windows of single phase coexistence.
4. CONCLUSIONS We incorporated our newly derived model for supramolecular assembly in the canonical ensemble into the Gibbs ensemble unit cell SCFT technique to take make use of its greater efficiency and stability41 compared to the grand canonical method for studying coexistence of complex phases produced in self-assembling systems with reaction. The effectiveness of our method was demonstrated by calculating phase diagrams for a simple melt blend of AB diblock copolymers with A block volume fraction f that reversibly react at their B termini with B homopolymers of relative length α to produce longer ABB diblock copolymers. The phase behavior depended on the final composition of the melt (in terms of Bh, AB, and ABB), which was derived from the chemical equilibrium of end-group binding. The reaction energy h parameter drove conversion of precursor polymers to ABB, where a higher h indicated a greater probability for an AB + Bh reaction to take place. The high degree of compatibility of ABB with the reactant polymers (AB + Bh) created a large variety of wide single-phase regions. The coexisting phase regions that occur are relatively narrow with respect to the single-phase regions. The coexistence regions widen with decreased h due to the greater incompatibility of the reactants (AB + Bh) with each other than with the product (ABB). When the formation of ABB diblock copolymer was unfavorable, two disordered phases were likely to coexist, one rich in AB diblock copolymer and one rich in B homopolymer. As demonstrated by the current study, reversibly bonding polymers have a large parameter space in terms of reaction energy, stoichiometry, and compositions and sizes of polymers in the blend, which increases significantly with additional reaction components or pathways. The possibility of a wide variety of phase behaviors makes Gibbs ensemble unit cell SCFT for reversibly bonding polymers a powerful tool for the design of responsive materials.
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AUTHOR INFORMATION
Corresponding Author
*E-mail: (G.H.F.)
[email protected]. Notes
The authors declare no competing financial interest.
■
ACKNOWLEDGMENTS This work was supported by the CMMT Program of the National Science Foundation under Award No. DMR 1160895. We acknowledge support from the Center for Scientific Computing from the CNSI, MRL, an NSF MRSEC (DMR1121053), and NSF CNS-0960316. Z.M. thanks CSP Technologies, Inc., for a graduate fellowship.
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REFERENCES
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