Thin films of diblock copolymer (BCP) with extended and defect-free lateral order can serve as ideal templates or scaffolds for fabricating nanoscale functional materials, as is desired in nanolithography and ultrafiltration membrane applications Albert2010 ; Sinturel2013 ; Li2015 . However, when a BCP thin film coats a solid substrate, it is usually kinetically trapped in a nonequilibrium and poorly ordered state with significant amount of defects, hindering the mainstream use of BCP thin films in nanotechnology Albert2010 ; Sinturel2013 ; Li2015 . Further treatments are thus needed to facilitate the fabrication of defect-free microdomains. As examples to techniques that have been developed to tailor the self-assembly behavior of BCP thin films we mentioned electric field alignment Russell1996 ; Russell2000 , shear alignment Marencic2007 ; Shelton2017 , microwave annealing Zhang2010 , and thermal annealing Albert2010 ; Li2015 ; Li2014 ; Hur2015 ; Hur2018 ; Muller2018 ; Song2018 ; Man2012 ; Man2016 . Besides these approaches, solvent vapor annealing (SVA) Albert2010 ; Sinturel2013 ; Baruth2014 ; Zhang2013 ; Zhang2014 ; Rudov2013 ; Gu2014a ; Gu2014b ; Gu2016 ; KimLibera1998 ; Cavicchi2007 ; Fredrickson2013 ; Fredrickson2016 ; Albert2011 ; Hur2014 ; Hannon2015 ; Hao2017 has been used to enhance the mobility of the polymer chains, and facilitates the annihilation of defects.
In a typical solvent vapor annealing process Albert2010 ; Sinturel2013 ; Hur2014 ; Zhang2014 ; Gu2014a ; Gu2014b , the BCP thin film is first exposed to vapors of one or more solvents for certain amount of annealing time. After a swollen and mobile polymer film is formed on the substrate, then it is dried by controlled solvent evaporation. During the film swelling in SVA, solvent molecules diffuse into the thin BCP film and screen (or dilute) the unfavorable interactions between polymer segments, leading to a lower effective Flory-Huggins parameter, , than its value for dry (no solvent) case Helfand1972 ; Lodge1995 ; Naughton2002 ; Gu2014a ; Gu2014b . Note that is a function of solvent volume fraction, . In addition, as the polymer mobility is greatly enhanced with increasing , the spacing of the microdomains also changes as a function Shibayama1983a ; Shibayama1983b ; Sinturel2013 ; Hur2014 ; Baruth2014 ; Zhang2013 ; Zhang2014 ; Rudov2013 ; Gu2014a ; Gu2014b ; Gu2016 .
Comparing to conventional thermal annealing (TA), SVA offers several unique advantages for defect annihilation Albert2010 ; Sinturel2013 ; Gu2014a : (i) SVA provides means to anneal BCPs that are sensitive to thermal degradation. (ii) SVA is generally more effective (with shorter annealing time) in removing defects for thin films of high-molecular-weight BCPs. (iii) SVA provides additional controlling parameters for tuning film morphology. For example, solvent molecules allow the segregation to vary continuously (by varying the swelling ratio or the solvent volume fraction) over a larger range of values than what could be achieved in TA by just changing temperature. In addition, selective solvent can be used to induce asymmetric swelling between BCP blocks, and thus change the morphology of the BCP film.
Despite the widespread use of SVA, a quantitative understanding of its effects in defect removal has not yet been established Albert2010 ; Sinturel2013 ; Gu2014a ; Gu2014b ; Hur2014 ; Baruth2014 ; Hannon2015 ; Hao2017 ; Fredrickson2013 ; Fredrickson2016 . Two principal reasons are that no standardized SVA process has been established, and the connection between the nanostructure formed in the swollen film and the final dried film has not been rigorously formulated. Therefore, it is of interest to perform detailed theoretical studies on the effects of the relevant system parameters during the swelling and removal of solvent in SVA Gu2014b ; Hur2014 ; Hannon2015 ; Hao2017 ; Fredrickson2013 . Such parameters include swelling ratio, annealing time, solvent evaporation rates, film thickness, temperature, and solvent selectivity. The understanding of their effects on the morphology of BCP thin films is necessary to finely control the SVA process in order to obtain the desired morphology.
Most previous theoretical and computational studies use either self-consistent field theory or particle-based simulation methods that includes a lot of molecular parameters Hur2014 ; Hannon2015 ; Fredrickson2013 ; Fredrickson2016 ; Hao2017 . Here, we propose a continuum two-fluid model for SVA processes in BCP thin films, and carry out detailed numerical analysis on two of the major (dimensionless) system parameters: (i) Swelling ratio , representing the ratio of the swollen film volume to its original volume , i.e., . (ii) Normalized solvent evaporation rate with representing the characteristic time for complete evaporation of solvent in the swollen film and characterizing the time for copolymer reorganization in the scale of microdomain period . To be specific, we investigate the removal of a typical defect occurring in symmetric BCP lamellae in their perpendicular orientation by SVA using non-selective solvent. In experiments of SVA for BCP thin films Gu2014a ; Gu2014b ; Baruth2014 , it has been well established that the swelling ratio and the solvent evaporation rate are the critical annealing parameters and should be specified explicitly. However, a quantitative understanding of their effects on the directed self-assembly using SVA and particularly on the efficiency of SVA in defect removal is still missing. Our main theoretical finding is that the increase of the lamellar spacing induced by added solvent facilitates a more efficient removal of defects in BCP films. Moreover, the dependence of the final morphology of the dried BCP film on and is summarized in a stability diagram, which can be verified in future experiments.
Ii Model and Methods
ii.1 Thermodynamics of BCP solutions: Generalized Landau-Brazovskii theory
When a BCP melt is exposed to the vapor of a good solvent, solvent molecules diffuse into the melt to form a swollen BCP solution, in which the chains gain some mobility. Here, specifically, we consider a BCP solution of monodisperse A/B diblocks dissolved in a neutral (non-selective) good solvent for which the polymer-solvent interactions follows . We restrict ourselves to symmetric A/B BCP where an A-block with monomers is covalently bonded to a B-block with monomers such that , and being the polymerization index. Furthermore, for simplicity, we assume incompressibility and zero volume change upon mixing in the BCP solution. In this case, the structure of the BCP solutions can be described by two independent order parameters: the solvent volume fraction with and the relative volume fraction of the two blocks
with . In addition, since the BCP solution is incompressible, we have
in which is the total volume fraction of A/B copolymer blocks.
It has been known that adding a neutral solvent to the BCP melt has a similar effect on the BCP melt as increasing the temperature Gu2014a ; Gu2016 . Namely, shifting the phase diagram of BCP melts towards higher temperature in the temperature-composition phase diagram Lodge1998 . Based on these understanding, we propose that the structure of a symmetric BCP solution, close to the order-disorder transition (ODT) between the disordered and lamellar phases, can be described by generalizing the Landau-Brazovskii free-energy Binder1994 ; Majewski2016 ; David1997 ; David2002 ; Ciach2013 ; Brazovskii1975 (in units of ) as
with being the copolymer closed packing chain density of an incompressible BCP melt, the monomer size, for the ODT, the gyration radius for Gaussian chains, and the monomer size. is the characteristic wavevector of ordered BCP microdomain, which gives the characteristic domain spacing . Note that for a swollen BCP solution with nonselective solvent, the effective interaction parameter in and the domain spacing are both functions of the solvent volume fraction, . Firstly, for , we follow the experimental measurements and take the following power-law form Helfand1972 ; Lodge1995 ; Naughton2002 ; Gu2014a ; Gu2014b ,
in which the exponent varies from to for a large range of Helfand1972 ; Lodge1995 ; Naughton2002 , representing the dilution effect of added solvent on the polymer chains by reducing the unfavorable A/B segmental interactions. Particularly, the case with can be obtained from the mean field dilution approximationHelfand1972 ; Lodge1995
, by assuming a uniform distribution of solvent throughout the BCP structure. The scaling ofwith the volume fraction of solvent is often approximated by another power law Shibayama1983a ; Shibayama1983b ; Sinturel2013 ; Hur2014 ; Baruth2014 ; Zhang2013 ; Zhang2014 ; Rudov2013 ; Gu2014a ; Gu2014b ; Gu2016
which represents the effects of added solvent on the microdomain spacing of the BCP solution. Here is a characteristic (either positive or negative) parameter of BCP solutions Shibayama1983a ; Shibayama1983b ; Sinturel2013 ; Hur2014 ; Baruth2014 ; Zhang2013 ; Zhang2014 ; Rudov2013 ; Gu2014a ; Gu2014b ; Gu2016 that depends on the degree of solvent selectivity, solvent volume fraction, and morphology of the ordered state. It is known from previous studies in both bulk BCP solutions Shibayama1983a ; Shibayama1983b and BCP films Sinturel2013 ; Gu2014a ; Gu2014b ; Gu2016
that the solvent-induced changes in lamellar spacing can be classified into two regimes according to the solvent volume fraction, which are separated by a crossover solvent volume fraction , another characteristic parameter of the BCP solution (e.g., for both polystyrene-polyisoprene (PS-PI) and polystyrene-poly(2-vinyl pyridine) (PS-P2VP) copolymers in solutions with non-selective solvent Shibayama1983a ; Gu2014a ; Gu2014b ):
Kinetically controlled regime with for dense polymer solutions with , which represents that the lamellar spacing increases with increasing solvent fraction. In this regime, is kinetically controlled Shibayama1983a ; Shibayama1983b ; Gu2016 in the sense that the copolymers does not gain much motility, the BCP structure reorganization cannot take place in a typical annealing time, and further addition of solvent therefore results in expansion in all directions. For example, it can be obtained from solvent volume conservation that for an isotropic swelling and for a uniaxial swelling of a (dense) BCP solution.
Thermodynamically controlled regime with for dilute polymer solutions with , which represents that the lamellar spacing decreases with increasing solvent fraction. This results from the decrease in segregation magnitude as seen in eq. (5) Sinturel2013 ; Gu2014a ; Gu2014b ; Gu2016 . For large , is thermodynamically controlled Shibayama1983a ; Shibayama1983b ; Gu2014a ; Gu2014b ; Gu2016 in the sense that the copolymers are sufficiently mobile and expand efficiently (by significant expansion of the average distance between the chemical junction points) along the lamellae interface for a given annealing time. Any addition of solvent can then result in a decrease of the domain spacing perpendicular to the interface. For example, it was found experimentally that in dilute BCP bulk solutions of PS-PI with nonselective solvents Shibayama1983a ; Shibayama1983b and in dilute BCP solution films of PS-P2VP with nonselective solvents Gu2014a ; Gu2014b .
The equilibrium structure of a swollen BCP solution is then given by the minimization of the Landau-Brazovskii free-energy with respect to and subject to the constraint of conserved solvent fraction and phase parameter :
with boundary conditions given at the neutral solid surface by
|and at the film-air interface|
is the normal unit vector of the film-air interface (with being the 2D velocity vector gradient operator in the -plane. Note that in eq. (7a) we have neglected the contribution from the concentration-dependence of and as shown in eq. (4a). This is equivalent to neglecting the influence of copolymers on the chemical equilibrium of solvent. The resulting equilibrium distribution is consistent with the assumption of uniform distribution of solvent through the BCP films. In addition, we would like to point out that the Landau-Brazovskii free-energy (3) can be further extended to include more physical effects by introducing more terms, e.g., solvent selectivity by coupling term (with representing representing B-preferred solvent), solvent accumulation at A/B interface by gradient term , and Marangoni effects by a concentration-dependent rigidity .
In addition, we would like to point out that, close to the ODT, the BCP system is described by two lengths: (i) the periodic spacing of BCP microdomain , and (ii) the correlation length , characterizing the decay of modulations and diverging at the ODT, . In particular, for BCP films below the ODT ( or ), a defect-free perpendicular lamellar state can be described in the single-mode approximation (in the limit of weak segregation, ) by
with being the amplitude obtained by minimizing free energy with respect to , and is the periodic spacing of the lamellae.
Finally, it is worthwhile to mention that Ohta and Ito OhtaIto1995 have constructed another Landau-type free energy to describe the microphase separation of general BCP-homopolymer mixtures. This, therefore, provides an alternative free energy model of BCP solutions by simplifying homopolymers to small solvent molecules.
ii.2 Dynamics of BCP solutions: Two-fluid model
With the generalized Landau-Brazovskii free-energy, we formulate a two-fluid model for the BCP solutions with non-selective solvent to explore the dynamic coupling between BCP copolymers and small solvent molecules during SVA. This is in analogy to a two-fluid model used to model polymer solutions, polymer blends, and diblock copolymer melts Onuki1990 ; DoiOnuki1992 ; Milner1993 ; Buxton2007 ; Vinals2012 ; Yabunaka2012 .
Let’s consider a BCP solution in which bulk flow and diffusion are taking place simultaneously, i.e., the velocity of the A/B blocks differ from each other and from the solvent velocity. First, from the conservation of solvent and A/B blocks we have
where represents the three components: solvent, A-block and B-block, respectively; and are the average velocities and volume fractions of each component, respectively. For simplicity, we assume incompressibility as given in eq. (2) and zero volume change upon mixing in the BCP solution, from which we obtain
with the volume averaged velocity defined by .
Note that and are both conserved order parameters of the BCP solution.
To determine , , and , we employ Onsager’s variational principle Onuki1990 ; DoiOnuki1992 ; Milner1993 ; Buxton2007 ; Vinals2012 ; Xu2015 , in which the Rayleighian functional is given by with the change rate of free energy given by
The dissipation function is generally given by
where and the friction matrix must be symmetric. Furthermore, since has to be invariant under Galilean transformation, we get
and hence there are only three independent components in . Therefore, we can rewrite the Galilean-invariant dissipation function into the following form
where the resistance matrix (with ) is given by
Here we have assumed equilibrium conditions as given in eq. (7) at the boundaries of the BCP film. The first three terms in the energy dissipation eq. (17) are caused by the relative motion between the different polymer blocks and between polymer blocks and solvent. The fourth term represents the energy dissipation caused by solvent velocity gradients, where is the solution viscosity, and and are the friction constants per unit volume. Note that for neutral solvents, the friction constants for the relative motion between the A/B blocks. In addition, we would like to point out that the viscoelastic properties of the BCP solution can also be incorporated into the two-fluid model as discussed in Ref. DoiOnuki1992 for polymer solutions and blends.
Substituting eq. (II.2) into eqs. (14) and (17b), minimizing Rayleighian with respect to , , and and substituting the obtained results back into eqs. (II.2), we obtain the following set of dynamic equations Onuki1990 ; DoiOnuki1992 ; Milner1993 ; Buxton2007 ; Vinals2012 ; Xu2015
with the incompressibility condition (11). Here we have substituted the following constitutive relations resulted from the Rayleighian minimization
into conservation Eqs. (II.2). The motility coefficient matrix (with ) is the inverse of the resistance matrix given in eq. (II.2), which is also symmetric and its elements in the limit of dilute solution (i.e., small solvent fraction) are found to be
with the effective friction constant per volume depending on the relative values of and . It is worthwhile to point out that the copolymer diffusion usually depends on the morphology of the BCP solutions Hur2014 ; Vinals2012 ; Yokoyama2006 ; particularly for lamellar BCP with uniaxial symmetry, anisotropic diffusion is known Vinals2012 ; Yokoyama2006 to be able to enhance or reduce the rate of relaxation of long-wavelength perturbations of lamellae. But here for simplicity, we only consider the case of isotropic diffusion as a typical application of the two-fluid model.
is the viscous stress tensor. In addition, at, solvent evaporation occurs; from the evaporation flux defined by we can get kinematic boundary conditions as follows
in which is the solvent fraction of a swollen film at equilibrium with the vapor phase, is an adjustable velocity parameter, proportional to mass transfer coefficient, and relates the rate of evaporative loss from the free surface to inter-diffusion of the solvent through the film by with being the initial thickness of the BCP film. Note that here we have neglected the surface tension, which is usually very small in comparison to the bulk energy of the BCP film. In this case, the free surface of the BCP film stays flat during SVA.
Regardless of the behavior of the BCP properties, the mathematical problem is intrinsically non-linear due to the fact of a moving boundary. Numerical integration may be carried out following a scheme that was successfully employed in some other moving boundary problems Xu2015 ; Tu1990 ; Crank1987 . The method consists of the following steps:
Neglect the dynamic nature of , , and . The moving boundary is extrapolated to the forward time level by means of an explicit forward difference expression of Eq. (28).
Repeat the iteration schemes of the previous second and third steps.
ii.3 Two-dimensional model for thin BCP films
We consider the limit of no macroscopic convection DoiOnuki1992 ; Milner1993 and focus on thin films of concentrated BCP solutions (with relatively small ) where the relaxation dynamics throughout the film thickness (-direction) is much faster than the lateral (-plane) dynamics (as schematically shown in Fig. 1a). This limit is justified when the viscous time scale (with being the kinematic viscosity of the BCP solution) is much smaller than the evaporative time for an initially stationary film to evaporate to dry film, i.e., . Furthermore, we focus on the simple case that the evaporation of solvent is slow relative to vertical solvent/polymer diffusion, i.e., with and being the diffusion times through the film thickness of the solvent and the polymer segments, respectively, and and the corresponding collective diffusion constants. In this case, the variations of phase structure, measured by , and solvent distribution are negligible in the normal (-) direction, yielding and .
Under such assumptions, the evolution of the film thickness, the integrated solvent fraction, and the phase structure of BCP thin film is written as
where is the mobility coefficient, and is the chemical potential of the BCP solution.
We consider the limit of no macroscopic convection DoiOnuki1992 ; Milner1993 , i.e., and focus on dilute solution with small and small surface tension in comparison to the bulk energy. Hence, the film will stay flat with and from Eq. (21), we have , i.e., no cross-coupling between solvent and copolymer dynamics. This limit is justified when the viscous time (with being the solution kinematic viscosity) is much smaller than the evaporative time that is required for an initially stationary film to evaporate to dry film: . Then, the above set of equations reduces to the following 2D Cahn-Hilliard equation Goldenfeld1989 ; Muller2015 :
supplemented with boundary conditions at the bottom substrate (at )
and at the top free surface ,
with the evaporation flux given by
Moreover, the temporal evolution of the film thickness follows
In addition, note that for small volume fraction of solvent with and being the second virial coefficient, the dynamic equation for solvent in Eq. (II.3) reduces to simple diffusion equation and the collective diffusion constant of solvent is given by .
We will focus on thin BCP films where the relaxation dynamics through the film thickness (-direction) is much faster than lateral (-plane) dynamics. The variations of and are then negligible in the normal direction (-axis). This leads to an order parameter and solvent fraction . Particularly, the volume fraction of solvent (or the volume fraction of copolymers) is simplified to be an (spatially uniform) “external” control parameter (that can have time-variation depending on the rate of evaporation but no variation in space).
In this case, the integrated solvent fraction follows
from which and Eq. (35) we obtain
with representing the characteristic time for complete evaporation of solvent in the swollen film. The volume fraction of solvent is only a time-varying parameter following eq. (II.3) and the evolution of phase structure is determined by the following two dimensional dynamic equation
ii.4 Numerical Method and System Setup
Using the classical central finite-difference method Glotzer1995 ; Buxton2007 , we numerically integrate the dynamical equation (6) for the order parameter , and eq. (5) for the solvent fraction (in the -plane), as is schematically shown in Fig. 1a. The mesh size of our simulation box is chosen to be
from Eq. (4) we obtain . The simulation box size is taken to be . Periodic boundary conditions are employed in both directions. Time is measured in units of
which characterizes the time for copolymer reorganization in the scale of microdomain period . Note that the number of lamellar periods equals to an integer that is the closest to . It is five for and dry lamellar spacing , as can be seen in Fig. 1b.
We carry out a numerical study for the effects of solvent swelling and solvent evaporation on the morphology of a dry BCP film with a particular defect as shown in Fig. 1b. The uptake (swelling) rate of solvent usually strongly depends on the morphology of BCP film even for a non-selective solvent Zhang2014 . However, in experiments the exposure time to solvent vapor (the annealing time) is usually longer than that required for the film to swell and reach an equilibrium (solvent) volume fraction Gu2014a ; Zhang2014 . In our simulations, we ignore the effects of solvent uptake process and assume “instantaneous” saturation to equilibrium (solvent) volume fraction as schematically shown in Fig. 1c. After an annealing time , the solvent is removed by a controlled evaporation rate, , according to eq. (5).
Iii Results and Discussion
We present results about the effects of two key parameters on the defect removal during SVA: solvent-swelling ratio , that is a monotonic function of equilibrium volume fraction , and the normalized solvent-evaporation rate, .
iii.1 Effects of solvent-swelling ratio
The swelling of the film is one-dimensional along the normal -direction of the film, due to the lateral confinement of fixed area set by periodic boundary conditions, as shown in Fig. 1a. The swelling ratio is then given by
from the conservation of copolymers and eq. (28) with and being the film thicknesses of the original dry film and the swollen film, respectively.
As pointed out in the previous section, there are two major effects of a non-selective solvent in a swollen BCP film: diluting effect on , eq. (5), and changing the lamellar spacing , eq. (6). We first carry out simulations to study a simple case when the solvent-induced changes in lamellar spacing are negligible during solvent annealing. This corresponds to the limit of BCP solutions with in Eq. (6), which may result from the weak dependence of either on or on the incompatibility Gu2014a ; Gu2014b . The results show that the defect stability strongly depends on the solvent fraction (or swelling ratio given in Eq. (41)) in the swollen film.
For a relatively low swelling ratio (small ), the system is kinetically trapped (metastable). In the other limit of very large swelling ratio when (equivalently, or ), the BCP undergoes an order-disorder transition (ODT), and the BCP solutions evolve to a disordered state without microphase separated structure. As shown in Fig. 2, only in a narrow window of swelling ratio, (or equivalently ), the copolymer segments gain large enough mobility but still maintain an ordered structure such that the defect-free lamellar structure is obtained.
It is interesting to note that the mechanism of defect removal by adding solvent to reduce is very similar to a thermal annealing (TA) by increasing temperature also reducing Li2015 ; Li2014 ; Hur2015 ; Muller2018 . Firstly, the presence of a threshold value of incompatibility (given by a threshold of solvent volume fraction or swelling ratio) in SVA has also been seen in TA ( in TA) Li2015 ; Li2014 ; Hur2015 ; Hur2018 ; Song2018 ; Muller2018 . For small (large swelling ratio), defects are not even metastable and removes spontaneously without any barriers to overcome. For large (small swelling ratio), defects are metastable and some free-energy barriers needs to be overcome. However using our method, we can not quantify the dependence of the energy barrier of the defect on or . We can only check when the energy barrier of the defect removal vanishes as is increased (as discussed above) and examine the dynamic processes of the defect removal.
We found that the dynamic processes of the defect removal in SVA is symmetric in the sense that the two horizontal arms break simultaneously, as seen in Fig. 2a. This is in contrast to the kinetic pathway of defect removal that is predicted in TA, in which the defect removal pathway is asymmetric Li2014 in the sense that the two horizontal arms in the defect shown in Fig. 1b break sequentially, one by one by crossing two consecutive free energy barriers Hur2018 ; Muller2018 ; Song2018 .
More generally, the change in lamellar spacing during solvent annealing of BCP films is visible and cannot be neglected Sinturel2013 ; Gu2014a ; Gu2016 . At small below some critical value Shibayama1983a ; Shibayama1983b , the spacing increases with increasing solvent volume fraction. In Fig. 3 we take which is smaller than , and , the equilibrium lamellar spacing increases from for the dry state to for the swollen state. Correspondingly, the energy-preferred number of lamellar periods changes from five to three. Fig. 3 shows that such an increase in lamellar spacing tends to drive long-wavelength structural evolution. Note that after defect removal the system evolves to a metastable defect-free state with four lamellar periods instead of energy-preferred three periods. This again indicates the addition of solvent alters the kinetic pathway of the microstructure evolution of the BCP film.
In addition, comparing these results with spacing changes to those without spacing changes (for example, results shown in Fig. 2), we find that, firstly in both cases, the defect-free state can be obtained only in a narrow window of solvent volume fraction: large enough to make defect state unstable and smaller than ODT to be still in ordered lamellar states. This observation agrees with experiments Baruth2014 . Secondly, and more interestingly, the defect structure of BCP solutions with is still metastable in the latter case without spacing changes, but becomes unstable in the former case with spacing increases (as shown in Fig. 3). We, therefore, conclude that the solvent-induced increase in lamellar spacing facilitates a more efficient removal of defects.
iii.2 Effects of solvent-evaporation rate
It is well-known that the final morphology of the dried BCP film depends, not only on the swelling ratio and the morphology in the swollen state, but also on the evaporation rate of solvent, Sinturel2013 ; Gu2014a ; Zhang2014 ; Gu2016 . To examine this dependence, we carry out simulations for swollen films with different swelling ratio (i.e., with different ) and for solvent evaporation at different rates . As schematically shown in Fig. 1c, the solvent is removed by evaporation following eq. (5) after time of solvent annealing.
For small swelling ratio (i.e., ), the amount of solvent in the swollen film is not sufficient to enhance polymer chain mobility such that the original defect is still metastable and kinetically trapped. In the dried film after solvent evaporation, the original defect persists and is found to be independent of the evaporation rate.
In contrast, at large swelling ratio (i.e., ) where , the system passes through the ODT and the swollen BCP film is disordered. After the solvent evaporates, the film re-enters the ordered phase but the final dry film is usually trapped in a poorly ordered state with significant amount of defects and the detailed phase structures strongly depend on the evaporation rates. To obtain a defect-free state, one has to direct the film morphology by well controlling the solvent evaporation rate Albert2011 ; Fredrickson2013 ; Fredrickson2016 , or applying shear flows Marencic2007 ; Shelton2017 .
At intermediate swelling ratio (i.e., ), the copolymer chains have sufficient mobility and the defect is removed spontaneously if the annealing time is large enough, as shown in Fig. 3. In SVA experiments, the solvent is usually evaporated after some small annealing time, . In our simulations, we choose an annealing time , after which the solvent is evaporated with controlled rates, . We find that the phase structure of the dried film strongly depends on the evaporation rate as shown in Fig. 4. The initial defect Fig. 4a can be removed only at slow evaporation with in Fig. 4c (a snapshot before the defect is removed is also shown in Fig. 4b). At fast evaporation, , the defect persists as shown in Fig. 4e, while at intermediate evaporation rates with , the initial defect becomes unstable but is re-trapped to another defective structure (but with larger spacing) as shown in Fig. 4d.
All these results can be understood in terms of the annealing time and the evaporation time . We understand that it is that gives the time allowing the copolymers to re-organize during the SVA process (with characteristic time ). For a given small , when the solvent is removed rapidly, the copolymers lose mobility almost instantly, and are kinetically trapped into the defective swollen morphology. When the solvent is removed gradually, the domain spacing changes and the defects are destabilized; moreover the BCP microdomains have time to relax such that the defects can be removed just as in the case of very long annealing times (without solvent evaporation). In the annealing process of thin BCP films on neutral substrates by non-selective solvents, the most efficient pathway to remove defects is first to swell the film to a sufficiently large solvent fraction where defects become unstable, and only then to evaporate the solvent rapidly when the defect is almost “healed” spontaneously.
In addition, we note that the dependence of the final morphology of dried BCP films on solvent-swelling ratio and solvent-evaporation rate can be summarized in a stability diagram of the defect as shown in Fig. 5. Three regions are identified: (I). Stable-defect region at small and large , in which the original defect is still stable or metastable and persists in the dried film. (II). Defect-free region at large and small , in which the defect becomes unstable and is removed spontaneously. (III). Intermediate-defect region, in which the original defect is unstable but is re-trapped in a new defect state. Such a trapping or transition to a new defect lamellar structure has also been reported in TA of lateral in-plane defect pairs Hur2018 ; Muller2018 and cross-sectional out-of-plane defect pairs Song2018 in BCP films.
When the solvent-induced changes in lamellar spacing are negligible (see the inset of Fig. 5), there are only two regions: (I). Stable-defect region, and (II). Defect-free region; the intermediate-defect region (III) does not appear. Moreover, the defect-free region (II) is much smaller when the spacing changes. Note that numerically, the accuracy of the obtained stability diagram depends on the choice of the mesh size; only a small enough mesh size can fully resolve the “shape” of the free-energy landscape and gives accurate stability. Here, we have chosen the mesh size to be as defined in Eq. (39) which is checked to give the same diagram as a smaller mesh size, . Furthermore physically, the stability of the defect also depends generally on the ratio of the lamellar spacing between swollen state and dry state, which is determined by the swelling ratio and the exponent defined in Eq. (6), which will be discussed in the next section. Our computational prediction agrees, at least qualitatively, with experimental observations Gu2014a ; Gu2014b that solvent-swelling ratio and the solvent-evaporation rate not only affect the domain spacing of BCPs but also govern the removal of defects or lateral ordering of the BCP microdomains. But the predicted stability diagram should be verified quantitatively in future experiments.
iii.3 Effects of spacing decrease induced by added solvent at large solvent fraction
In this section, we present some more numerical results. As discussed in Sec. II.1, the effects of added solvent on the changes of lamellar spacing can be classified into two regimes separated by a crossover solvent volume fraction, . In previous sections, we have all taken to be very large such that and lamellar spacing increases with added solvent. Here, we choose small and explore the effects of spacing changes in the two regimes, particularly the effects of the spacing decrease with increasing solvent fraction.
We find that, in both regimes, the change in lamellar spacing destabilize the defect, as shown in Figs. 6 and 7. However, the kinetics of defect instability is very different in the two regimes. The solvent-induced spacing increases at small and tends to drive long wavelength structural evolution, as shown in Fig. 3 in the main text. In contrast, the solvent-induced spacing decreases at large and introduces small wavelength undulations, as shown in Fig. 6. Moreover, the kinetically-controlled spacing-increase (when ) during solvent annealing is essential for removing BCP defects efficiently, as shown in Fig. 7. Note that the obtained defect-free lamellae adopt a tilted orientation in the -plane to adjust to the smaller domain size under the constraints imposed by a fixed size of the simulation box.
Finally, we would like to point out that at large solvent fraction (over some crossover value ), the addition of solvent can induce decrease of lamellar spacing in both bulk BCP solutions Shibayama1983a ; Shibayama1983b and BCP films Sinturel2013 ; Gu2014a ; Gu2016 . For example, for polystyrene-polyisoprene (PS-PI) copolymer in solutions with non-selective solvent Shibayama1983a ). To explore the effects of solvent induced decrease in lamellar spacing, we have also carried out more simulations. We find that the solvent-induced spacing decreases at large induces small-wavelength undulations in contrast to the long wavelength structural evolution, as shown in Fig. 3. These small-wavelength undulations are usually not enough to remove defects. Efficient defect removal occurs only for cases with significant spacing-increase during solvent annealing at small solvent fraction (when ).
In summary, we have constructed a two-fluid model to study the microphase separation dynamics of block copolymer (BCP) thin films and applied it to solvent vapor annealing (SVA) processes. We focus on the effects of added (non-selective) solvent on the defect annihilation in ordered BCP thin films on neutral substrates. More specifically, we investigate the solvent-facilitated removal of a typical defect occurring in symmetric lamellar BCP thin films with perpendicular (to the substrate) lamellae.
One of our key findings is that the added solvent not only dilutes the unfavorable interactions between polymer segments, but also changes the characteristic spacing of the BCP lamellae. We find that the solvent-induced increase of the lamellar spacing in dense polymer solution (that is at small solvent volume fraction) facilitates an efficient removal of defects. The kinetic pathways of defect removal during SVA are quite different from that of thermal annealing (TA) and show sensitive dependence on the volume fraction and dynamics of added solvent. The final morphology of the dried BCP film after SVA depend not only on solvent-swelling ratio (or solvent volume fraction) but also solvent-evaporation rate . Such a dependence is summarized in a stability diagram (Fig. 5), which can be verified in future experiments.
Based on these results, we are able to make an important observation about the annealing of thin BCP films on neutral substrates by non-selective solvents. Namely, the most efficient pathway to remove the defects is to first swell the film to a sufficiently large solvent fraction where defects become unstable, and only then to evaporate the solvent rapidly when the defect almost “heals” spontaneously. We believe that the insight provided in this theoretical work on defect removal from perpendicular lamellar BCP thin films by SVA has the potential to greatly expand and diversify the range of applications of solvent-assisted, directed self-assembly of BCP thin films.
We would like to conclude with a few remarks.
(i) Effects of the dynamics of swelling (solvent uptake). The swelling dynamics of BCP films strongly depends on the BCP morphology Gu2014a ; Zhang2014 . For example, films with perpendicular lamellae swell faster than parallel lamellae because a higher amount of solvent penetrates through the A/B interfaces that are exposed to the free surface. This effect is associated with the inhomogeneous distribution of solvent within microphase separated film. In this work, we have neglected the coupling of the swelling dynamics with BCP morphology (or phase) dynamics completely. Nonetheless, the inhomogeneous solvent distribution and morphology-dependent swelling rate can have significant effects on the defect removal during solvent annealing process.
(ii) Structure and dynamics through the film thickness. In our simulations, we have assumed that the relaxation dynamics through the film thickness is much faster than their lateral dynamics. Based on this assumption, we have neglected any variations of the structure and dynamics through the film thickness completely. However, it has been shown in many experiments that the structure and dynamics though the film thickness can be of importance during SVA processes Albert2010 ; Sinturel2013 ; Gu2014a ; Zhang2014 . For example, as solvent is introduced or removed, there must be a re-distribution of BCP chains in the film and some tilting of the lamellae is expected. Moreover, film thickness is an important parameter, use as a means allowing a direct access to the dynamics of the structural rearrangements KimLibera1998 ; Cavicchi2007 ; Albert2011 ; Hao2017 . By controlling the film thickness in the fully swollen state, Cavicchi et al. Cavicchi2007 found that either parallel or perpendicular cylinder orientation could be obtained. In addition, Kim and Libera KimLibera1998 suggested that the solvent diffusion to the free surface and the solvent concentration gradients within the film imposed by different solvent evaporation rates are responsible for the different microdomains orientations. More recently, experiments Albert2011 indicated that for a swollen film of parallel cylinder morphology, if the solvent is removed instantaneously, well-ordered parallel cylinders occur; however, if the solvent is removed very slowly, the parallel cylinders reorient and became perpendicular to the substrate. Note that it has been known that surface tension we have neglected also play some important roles in determining the orientation of the lamellae in BCP film Sinturel2013 ; Hao2017 .
We believe that the insights provided in this theoretical work on defect removal from perpendicular thin BCP films by SVA have the potential to greatly expand and diversify the range of applications of solvent-assisted, directed self-assembly of BCP thin films.
We would like to thank S. Komura and T. Qian for discussions and helpful suggestions. This work was supported in part by Grant No. 21822302 and 21434001 of the National Natural Science Foundation of China (NSFC), the joint NSFC-ISF Research Program, jointly funded by the NSFC under Grant No. 51561145002 and the Israel Science Foundation (ISF) under Grant No. 885/15. X. X. was supported by Guangdong Technion – Israel Institute of Technology. D.A. acknowledges the hospitality of the ITP (CAS) and Beihang University, Beijing, China, and a CAS President International Fellowship Initiative (PIFI).
Conflict of Interest
The authors declare no conflict of interest.
block copolymer, solvent vapor annealing, defect removal, two-fluid model, Onsager’s variational principle
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