THE JOURNAL OF CHEMICAL PHYSICS 140, 044904 (2014)

Quantitative study of fluctuation effects by fast lattice Monte Carlo simulations: Compression of grafted homopolymers Pengfei Zhang and Qiang Wanga) Department of Chemical and Biological Engineering, Colorado State University, Fort Collins, Colorado 80523-1370, USA

(Received 31 August 2013; accepted 6 January 2014; published online 29 January 2014) Using fast lattice Monte Carlo (FLMC) simulations [Q. Wang, Soft Matter 5, 4564 (2009)] and the corresponding lattice self-consistent field (LSCF) calculations, we studied a model system of grafted homopolymers, in both the brush and mushroom regimes, in an explicit solvent compressed by an impenetrable surface. Direct comparisons between FLMC and LSCF results, both of which are based on the same Hamiltonian (thus without any parameter-fitting between them), unambiguously and quantitatively reveal the fluctuations/correlations neglected by the latter. We studied both the structure (including the canonical-ensemble averages of the height and the mean-square end-to-end distances of grafted polymers) and thermodynamics (including the ensemble-averaged reduced energy density and the related internal energy per chain, the differences in the Helmholtz free energy and entropy per chain from the uncompressed state, and the pressure due to compression) of the system. In particular, we generalized the method for calculating pressure in lattice Monte Carlo simulations proposed by Dickman [J. Chem. Phys. 87, 2246 (1987)], and combined it with the Wang-Landau– Optimized Ensemble sampling [S. Trebst, D. A. Huse, and M. Troyer, Phys. Rev. E 70, 046701 (2004)] to efficiently and accurately calculate the free energy difference and the pressure due to compression. While we mainly examined the effects of the degree of compression, the distance between the nearest-neighbor grafting points, the reduced number of chains grafted at each grafting point, and the system fluctuations/correlations in an athermal solvent, the θ -solvent is also considered in some cases. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4862498] I. INTRODUCTION 1–3

This is the fourth paper in our series of study, where fast lattice Monte Carlo (FLMC) simulations4 are directly compared with the corresponding polymer lattice field theories based on the same model system (Hamiltonian), thus without any parameter-fitting, to unambiguously and quantitatively reveal the effects of fluctuations and correlations either neglected or treated approximately in the theories. In the preceding two papers in this series (referred to as Papers II2 and III3 hereafter), we studied monodisperse, uncharged, and flexible homopolymer brushes on a planar, impenetrable, and nonadsorbing substrate in an implicit2 or explicit3 solvent, and found in both cases that the polymer segmental density profile in the direction perpendicular to the grafting substrate obtained from FLMC simulations is flatter than that from the corresponding lattice self-consistent field (LSCF) calculations neglecting the system fluctuations/correlations. This finding for uncompressed brushes implies some discrepancy between FLMC and LSCF results of the free energy and force (pressure) for compressed brushes. We therefore study in this work the fluctuation/correlation effects in such grafted homopolymers (in both the mushroom and brush regimes) compressed by a planar, impenetrable and nonada) E-mail: [email protected].

0021-9606/2014/140(4)/044904/16/$30.00

sorbing surface parallel to the grafting substrate. Here we only consider an explicit (either athermal or θ -) solvent, and refer readers to the Introduction of Papers II2 and III3 for the differences between implicit and explicit solvents, as well as various models for polymer brushes used in the literature. For homopolymer brushes on a planar substrate, two different cases of brush compression, by an impenetrable surface and by an opposing brush, have been widely studied with self-consistent field (SCF) theories. Dolan and Edwards first reported a continuum SCF study of both cases using an implicit and good solvent,5 and their work was later extended by Muthukumar and Ho.6 An explicit solvent was also used in both continuum7–9 and lattice10–14 SCF studies of brush compression. In addition, due to their simplicity, infinite-stretching theories (IST) have been most widely used in the literature,15–19 where the two cases of compression become identical as the brush interpenetration is neglected in the second case; we refer readers to the Introduction of Paper II2 for IST, and note here the recent comparison between IST and SCF results by Kim and Matsen for the compression of two opposing brushes,20 which unambiguously reveals the consequences of the further approximations in IST. Overall, these SCF studies closely follow those for uncompressed brushes summarized in the Introduction of Paper II.2

140, 044904-1

© 2014 AIP Publishing LLC

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Both cases of brush compression have also been studied with molecular dynamics,21–25 and continuum26, 27 and lattice28–31 Monte Carlo (MC) simulations; compression of two opposing brushes has further been studied with Brownian dynamics32 and dissipative particle dynamics33 simulations. In particular, Dickman proposed a method for calculating pressure in lattice MC simulations,28 which we generalize and combine with the Wang-Landau–Optimized Ensemble simulations34 here to efficiently and accurately calculate the Helmholtz free energy and the pressure for our model system of grafted homopolymers compressed by an impenetrable surface. While some of these simulation results have been compared with LSCF26, 30 or IST23, 24, 27, 29, 31, 33 predictions, none of these comparisons was done using exactly the same model system in both the theoretical and simulation studies (we refer readers to the Introduction of Paper II2 for the model differences). They therefore could not unambiguously quantify the effects of fluctuations/correlations neglected by the mean-field approximation inherent in all SCF (and IST) studies. In particular, Chakrabarti et al. briefly compared their lattice MC simulations with the LSCF calculations by Shim and Cates11 on the polymer segmental density profiles and the interpenetration of two opposing brushes in an athermal solvent, where the model systems are quite close except that the LSCF calculations were done in 1D while the MC simulations were done in 3D (i.e., they used different grafting patterns).30 On the other hand, grafted chains form mushrooms at low grafting densities. Compression of mushrooms by an impenetrable surface, however, has been much less studied in the literature. Dolan and Edwards first reported approximate analytical solutions to the SCF equation for ideal chains grafted on a flat substrate and compressed by an impenetrable surface.35 Li and Pincet, however, pointed out that the horizontal axis in Fig. 1 of Ref. √ 35 was mislabeled and should have been multiplied by 3, and showed that their experimental force profiles are well described by the theoretical prediction after this correction.36 Finally, in a series of papers,37–40 Arteca and coworkers have reported detailed characterization of molecular shape features of a single grafted chain with hard excludedvolume interactions compressed by an impenetrable surface; their focus, however, was on the simultaneous description of chain size and self-entanglement, as well as the relation between them. In this work, we perform both LSCF calculations and FLMC simulations based on exactly the same model system to unambiguously quantify the fluctuation/correlation effects in grafted flexible homopolymers (in both the mushroom and brush regimes) compressed by an impenetrable surface. We systematically vary the model parameters (including the degree of compression, the distance between the nearest-neighbor grafting points, the reduced number of chains grafted at each grafting point, the solvent quality, and the system fluctuations/correlations) to examine their effects on both structure and thermodynamics of the system. This work is also complementary to our recent study on the structural and phase transitions of uncompressed polymer mushrooms.41

J. Chem. Phys. 140, 044904 (2014)

II. MODEL AND METHODS A. Model system

We consider a grafted homopolymer system consisting of n chains each having N segments and nS solvent molecules on a simple cubic lattice having Lx layers in the x-direction and L layers in both the y- and z-directions. Each polymer segment or solvent molecule occupies one lattice site, and each lattice site is occupied by totally ρ0 ≡ (nN + nS )/V > 1 segments and solvent molecules, where V = L2 Lx is the total number of lattice sites. The first segment of all chains is grafted at x = 1 (i.e., in the first layer along the x direction), and the distance between two adjacent grafting points is Lg (thus a total of ng = L2 /L2g grafting points). Two impenetrable walls are placed at x = 0 and Lx + 1, respectively, which cannot be occupied by either polymer segments or solvent molecules, and the periodic boundary conditions are applied in both the y- and z-directions. The canonical-ensemble partition function of our model system is

Z=

nS  n N 1  1   · · nS ! {g} mg ! k=1 s=1 k=1 r Rk,s

 · exp −β

n 

k

 hCk

k=1

− βH

E

·



δρ(r)+ ˆ ρˆS (r),ρ0 , (1)

r

grafting where mg is the number of chains grafted at the  point g with {g} being the set of grafting points (thus {g} mg = n), Rk,s denotes the lattice position of the sth segment on the kth chain, rk denotes the position of the kth solvent molecule, the summations are over all possible positions of all polymer segments and solvent molecules, respectively, “ · ” means that the products before it do not apply to the terms after it but the summations before it do, and β ≡ 1/kB T with kB being the Boltzmann constant and T the thermodynamic temperature. hCk is the Hamiltonian of the kth chain due to its connectivity; βhCk = 0 if the chain connectivity is maintained and ∞ otherwise. Finally, the Hamiltonian due to non-bonded interaction is given by ˆ ρˆS ] = βHE [ρ,

χ  ρ(r) ˆ ρˆS (r), ρ0 r

(2)

where χ is the Flory-Huggins interaction parameter between a polymer segment and a solvent molecule at the same lattice site r, the microscopic number density of polymer segments r are given by ρ(r) ˆ  and that of solvent molecules at nS ≡ nk=1 N δ and ρ ˆ (r) ≡ δ , respectively, S s=1 r,Rk,s k=1 r,rk and δ denotes the Kronecker δ-function. With the incompressibility constraint imposed at all ˆ r by the  last term in Eq. (1), we can rewrite βHE [ρ] ˆ − ρ(r)]. ˆ The partition function in = (χ /ρ0 ) r ρ(r)[ρ 0 Eq. (1) can then be rewritten as

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  n N n  1  1  nS ! χ  C · exp −β Z= · hk − ρ(r)[ρ ˆ ˆ 0 − ρ(r)] nS ! {g} mg ! k=1 s=1 ˆ ρ0 r r [ρ0 − ρ(r)]! k=1 ·



Rk,s

θ (ρ0 − ρ(r)) ˆ

r

  n  n N   1    χ  C · = · exp −β hk − ρ(r)[ρ ˆ ˆ − ln[ρ0 − ρ(r)]! ˆ 0 − ρ(r)] mg ! k=1 s=1 ρ0 r r {g} k=1 ·



Rk,s

θ (ρ0 − ρ(r)), ˆ

(3)

r

where θ (x) = 1 if x ≥ 0 and 0 otherwise. Here we have eliminated the dependence on the spatial positions of all solvent ˆ molecules {rk } in the summation, and nS ! r [ρ0 − ρ(r)]! is the number of different ways of arranging these solvent molecules for a given configuration {Rk,s } of all chains. We note that (1) different from conventional polymer lattice models using the self- and mutual-avoiding walk (i.e., ρ 0 = 1), our model uses multiple occupancy of lattice sites as in our previous work,1–4 which is a direct consequence of polymer lattice models being coarse-grained ones;42 (2) as a mean-field theory the LSCF theory for grafted homopolymers described in Sec. II C below gives exact results in the limit of ρ 0 → ∞ at finite mg /ρ 0 (thus mg → ∞), and the fluctuation/correlation effects we study refer to the differences between our fast lattice Monte Carlo (at finite ρ 0 and thus mg ) and LSCF results; and (3) readers are referred to Ref. 42 for quantitative mapping between experimental and our model systems, where the solvent entropy (i.e., the size ratio r between polymer segments and solvent molecules) is also quantitatively accounted for. To be consistent with Paper III,3 however, we set r = 1 in this work. B. Fast lattice Monte Carlo (FLMC) simulations

Our FLMC simulations are performed in a canonical ensemble, where n, N, nS , V , and χ are fixed. We use both the simple and the topological configurational bias trial moves as in Papers II and III.2, 3, 43 For both types of trial moves, a partial chain of l = 7 segments is cut and regrown. 1. Metropolis sampling

Here the acceptance criterion is44

Rn Wo E Pacc (o → n) = min 1, exp(−βH ) , Ro Wn

(4)

where o and n denote the old and trial configurations, re spectively, W ≡ r [ρ0 − ρ(r)]! ˆ is used to account for the entropy of solvent molecules, R is the Rosenbluth weight, which includes an additional factor to maintain the chain connectivity45 for the topological configurational bias trial moves, and HE denotes the difference in HE between the trial and old configurations. About (1 ∼ 20) × 106 Monte Carlo steps (MCS) are performed in each simulation, where

one MCS is defined as nN trial moves. The error bar of each ensemble-averaged quantity is estimated in the same way as in Paper I1 of our series. In FLMC simulations, the height of the grafted homopolymers along the x-direction is calculated as  ˆ y,z x ρ(r) hFLMC = 2  , (5) ˆ y,z ρ(r) where  denotes the canonical-ensemble average. The xcomponent of the mean-square chain end-to-end distance is calculated as  n 1 2 2 (Xk,N − Xk,1 ) , (6) Rex,FLMC = n k=1 where Xk, s denotes the x-component of Rk,s . The mean-square 2 chain end-to-end distance in the y- (or z-) direction Rey,FLMC is calculated similarly. Finally, the internal energy per chain of length N, uc,FLMC , is calculated as N E βH = −χ N eFLMC βuc,FLMC = (7) ρ0 V with the ensemble-averaged reduced energy density  1  eFLMC = − 2 ρ(r) ˆ ρˆS (r) . ρ0 V r

(8)

2. Wang-Landau–Optimized Ensemble (WL-OE) sampling

We also perform Wang-Landau–Optimized Ensemble (WL-OE) sampling to calculate the system pressure P caused by the compression of the impenetrable surface at x = Lx + 1 < N + 1, given by



1 ∂ ln Z(Lx ) ∂βF (Lx ) = 2 βP (Lx ) = − ∂V L ∂Lx n,N,χ,ρ0 n,N,χ,ρ0   1 Z(Lx + 1)  ≈ ln , (9) 2L2 Z(Lx − 1) n,N,χ,ρ0 where F(Lx ) is the Helmholtz free energy of the system with Lx lattice layers in the x-direction, and in the second line above the second-order central finite difference is used to approximate the derivative, which is more accurate than the first-order finite difference used in Ref. 28. To bridge

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the system described by Z(n, N, χ , ρ0 , Lx + 1) to that by Z(n, N, χ , ρ0 , Lx − 1), we introduce an intermediate system, which is the same as the former except that the polymer segments at x = Lx and Lx + 1 experience a repulsive potential U . The canonical-ensemble partition function of this intermediate system is Z  (Lx + 1, U  ) =

g(Nc ) → g(Nc )f,

n  N   1  · mg ! k=1 s=1 {g} Rk,s



χ  ρ(r)[ρ ˆ ˆ 0 − ρ(r)] ρ0 r k=1     − ln[ρ0 − ρ(r)]! ˆ − Nˆc U · θ (ρ0 − ρ(r)), ˆ

· exp −β

n 

hCk −

r

r

(10) where Nˆc is the number of segments in both x = Lx and Lx + 1 layers (following Ref. 46, we refer to these segments as “contacts”). Note that Z  (Lx + 1, U  = 0) = Z(Lx + 1) 2 and Z  (Lx + 1, U  → ∞) = Z(Lx − 1)(ρ0 !)−2L . Equation (9) then becomes βP (Lx ) ≈

1 Z  (Lx + 1, U  = 0) − ln ρ0 !. ln  2 2L Z (Lx + 1, U  → ∞)

(11)

Instead of evaluating the above finite difference via numerical integration as in Ref. 28 (see Eq. (5) there), we note that the partition function in Eq. (10) can be rewritten as  (Nc ) exp(−Nc U  ), (12) Z  (Lx + 1, U  ) = Nc

where (Nc ) is the density of states having Nc contacts and is given by (Nc ) =

n  N   1  · ·δNc ,Nˆc mg ! k=1 s=1 R {g} k,s



× exp −β

n  k=1

−



hCk − 

χ  ρ(r)[ρ ˆ ˆ 0 − ρ(r)] ρ0 r

ln[ρ0 − ρ(r)]! ˆ ·

r



θ (ρ0 − ρ(r)). ˆ

(13)

r

Substituting Eq. (12) into Eq. (11) gives βP (Lx ) ≈ −

not improve P but only increases the simulation time. The trial moves are accepted according to the following criterion:44

g(Nc,o )Rn Wo WL E Pacc (o → n) = min 1, exp(−βH ) . g(Nc,n )Ro Wn (15) After each trial move, g(Nc ) and H(Nc ) of the current state (n if the trial move is accepted, and o otherwise) are updated as

1 (Nc = 0) ln  − ln ρ0 !. 2 2L Nc (Nc ) 34

(14)

In this work, we use WL-OE method to estimate (Nc ) for Nc ∈ [0, Nc,max ], which consists of two parts: first WL sampling is used to estimate (Nc ) crudely, and then OE sampling is used to estimate (Nc ) accurately. WL sampling used in WL-OE method34 is the same as that in the Wang-Landau– Transition Matrix method used in our previous work.1–3, 43 First an array g(Nc ) is set to unity and a histogram H(Nc ), which records the number of visits to the macrostate Nc in the simulation, is set to zero for all Nc ∈ [0, Nc,max ]. Nc,max is chosen to be large enough such that further increase of Nc,max does

H (Nc ) → H (Nc ) + 1,

(16)

where f is a modification factor with an initial value of f0 = e ≈ 2.718. The flatness of H(Nc ), i.e., H (Nc ) > N 0.8 Nc,max H (Nc,i )/(Nc,max + 1) for all Nc , is checked every c,i =0 1000 MCS. If H(Nc ) is flat, we reset H(N √ c ) = 0 for all Nc states and decrease f according to f → f . WL sampling is stopped when f < 1.00001, and we switch to OE sampling34 to improve the accuracy of g(Nc ). In OE sampling,34 we start the simulation with a set of weights w(Nc ) = 1/g(Nc ), where g(Nc ) is obtained from WL sampling; in addition, we use three arrays H(Nc ), H+ (Nc ), and H− (Nc ) with initial values of zero for all Nc . Here, H(Nc ) is used to record the number of visits to the macrostate Nc as above, H+ (Nc ) for the number of visits to the Nc -state with the most recently visited extreme state being Nc = 0, and H− (Nc ) for the number of visits to the Nc -state with the most recently visited extreme state being Nc,max . The trial moves are accepted according to

w(Nc,n )Rn Wo OE E exp(−βH ) . Pacc (o → n) = min 1, w(Nc,o )Ro Wn (17) After each trial move, the histogram of the current state is updated as H(Nc ) → H(Nc ) + 1. Furthermore, if the most recently visited extreme state is Nc,max , we update H− (Nc ) → H− (Nc ) + 1 while keeping H+ (Nc ) unchanged; otherwise, we update H+ (Nc ) → H+ (Nc ) + 1 while keeping H− (Nc ) unchanged. Note that H− (Nc ) + H+ (Nc ) = H(Nc ) for all Nc . After 130 000 MCS, we estimate the local diffusivity as D(Nc )∝[H(Nc )(dh/dNc )]−1 with h(Nc ) ≡ H+ (Nc )/H(Nc ) and update the weights as  1 dhi (Nc ) wi+1 (Nc ) = wi (Nc ) , (18) Hi (Nc ) dNc where i denotes the simulation step using wi (Nc ). We then increase the number of MCS in the simulation step by 1.3 times; reset H(Nc ), H+ (Nc ), and H− (Nc ) to 0 for all Nc ; and continue the simulation with wi+1 (Nc ). At the end of simulation step i, gi (Nc ) is estimated as ln gi (Nc ) = ln Hi (Nc ) − ln wi (Nc ).

(19)

OE sampling is stopped when the difference between two adjacent simulations in ln g(Nc )  Nc,max 2 Nc =0 [ ln gi+1 (Nc ) − ln gi (Nc ) ] /(Nc,max + 1) < 0.05, which usually requires six steps. (Nc ) is then estimated by g(Nc ) with a undetermined multiplicative constant, which is fixed by g(Nc = 0) = 1. Finally, the pressure P is estimated as βP (Lx ) = −

1 g(Nc = 0) − ln ρ0 !. ln  2L2 Nc g(Nc )

(20)

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J. Chem. Phys. 140, 044904 (2014)

Defining the reduced pressure as P∗ (Lx ) ≡ [βP(Lx ) + ln ρ 0 !]/ρ 0 , we have P ∗ (Lx ) = −

1 g(Nc = 0) . ln  2L2 ρ0 Nc g(Nc )

(21)

Similarly, we define the reduced Helmholtz free energy as F ∗ (Lx ) ≡ βF (Lx ) + (N − Lx )L2 ln ρ0 !,

(22)

where the last term is due to the difference in the translational entropy of solvent molecules between the compressed system with Lx lattice layers and the uncompressed one with N layers in the x-direction. Taking the uncompressed system with Lx = N as the reference, the difference in the reduced free energy per chain fc∗ (Lx ) ≡ (N/ρ0 V )[F ∗ (Lx ) − F ∗ (N )] can be calculated in a recursive way, Lx + 1 2N fc∗ (Lx + 1) + P ∗ (Lx ) Lx − 1 Lx − 1 (23) with the initial values fc∗ (N ) = 0 and P∗ (N) = 0. Similarly, the difference in the internal energy per chain is fc∗ (Lx − 1) =

uc (Lx ) = uc (Lx ) − uc (Lx = N ),

· exp(−βhC ) = zN−1 V , the lattice coordination number z = 6 for the simple cubic lattice usedhere, the single-particle partition function QS [iωS ] ≡ r exp[−iωS (r)/N]/V , the Helmholtz free energy per chain of length N for an ideal system of n ungrafted chains (but divided into ng groups according to the grafting pattern) and nS solvent molecules, which is non-interacting and not subject to the incompressibility constraint, is id Gn φ¯ 1 φ¯ ¯ ln V = − sc − ln −N (1 − φ) βfcid ≡ − ln  n n nS ! kB {g} mg ! (28) with φ¯ ≡ nN/ρ0 V denoting the average polymer volume fraction and scid the entropy per chain for the ideal system, and we have re-scaled variables according to N ω → ω, N ωS → ωS and N η → η. The LSCF equations are obtained by setting δβfc /δφ(r) = δβfc /δiω(r) = δβfc /δφS (r) = δβfc /δiωS (r) = δβfc /δiη(r) = 0 (i.e., the mean-field approximation47 ) and given by

(24)

and the difference in the reduced entropy per chain can be calculated as sc∗ = βuc − fc∗ .

iω(r) = χ N φS (r) + iη(r),

(29)

iωS (r) = χ N φ(r) + iη(r),

(30)

(25)

φ(r) =

N 

φg,s (r),

(31)

{g} s=1

φg,s (r) =

C. Lattice self-consistent field (LSCF) theory

Here we start with the same Hamiltonian as in our FLMC simulations, and insert in Eq. (1) the identity 1 = (ρ0 /2π )2V  ˆ exp[iωS ·(ρ0 φ S − ρˆ S )], dφdωdφ S dωS exp [iω·(ρ0 φ − ρ)] where vectors ρˆ ≡ {ρ(r)} ˆ and ρˆ S ≡ {ρˆS (r)} have V elements; φ ≡ {φ(r)} and φ S ≡ {φS (r)} are the normalized density (volume fraction) fields of polymer segments and solvent ˆ 0 and ρˆ S /ρ0 ; molecules, respectively, constrained to ρ/ρ and ω ≡ {ω(r)} and ωS ≡ {ωS (r)} are the conjugate fields imposing these constraints, respectively. We then substitute the following integral form of the δ-function into Eq. (1),   1 δρ(r)+ dη exp[−ρ0 iη·(φ + φ S − 1)], ˆ ρˆS (r),ρ0 = (2π )V r (26) where the vector η ≡ {η(r)} is the conjugate field imposing the incompressibility constraint, and the vector 1 2V 3V has  V elements of 1. We finally have Z = [ρ0 /(2π ) ] dφdωdφ S dωS dη exp{−(ρ0 V /N)βfc [φ, iω, φS , iωS , iη]} with 1 βfc = [χ Nφ · φ S − φ·iω − φ S ·iωS + iη·(φ + φ S − 1)] V  mg N ¯ ln QS [iωS ] + βfcid , ln Qg [iω] − N (1 − φ) − ρ V 0 {g} (27) where the single-chain partition function of chains with   their first segment grafted at g is Qg [iω] ≡ N s=1  N Rs · exp[−βhC − i N ω(R )/N]/G, G ≡ s Rs s=1 s=1

mg exp[iω(r)/N] ∗ qg,s (r)qN+1−s (r) ρ0 V Qg

φ¯ exp[iω(r)/N ] ∗ qg,s (r)qN+1−s (r), ng N Qg

iωS (r) 1 − φ¯ exp − , φS (r) = QS [iωS ] N =

φ(r) + φS (r) = 1,

(32) (33) (34)

where φg,s (r) is the volume fraction at r of the sth segment of those chains with their first segment grafted at g; in the second line of Eq. (32), we have assumed mg = m and used ng = n/m. qg,s (r) is the probability of finding a partial chain of s segments that starts from g and ends at r, and qt∗ (r) is the probability of finding a partial chain of t ≡ N + 1 − s segments that starts from anywhere in the system and ends at r. According to the chain connectivity, we have the recursive relations exp[−iω(r)/N]  ∗ ∗ (r) = qt (rn ), qt+1 z r n

∗ qt=1 (r)

= exp[−iω(r)/N ]

qg,s+1 (r) =

(35)

exp[−iω(r)/N]  qg,s (rn ), z r n

qg,s=1 (r) = exp[−iω(r)/N ]δr,g

(36) qt∗ (r)

with the boundary conditions of qg,s (r) = 0 and = 0 at x = 0 and Lx + 1. The single-chain partition function Qg can

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be rewritten as 1  ∗ Qg = qg,s (r)qN+1−s (r) exp[iω(r)/N] = qN∗ (g)/V . V r (37) It is worth noting that, if {iω, iωS , iη} is a solution of the LSCF equations, {iω + c, iωS + c, iη + c} is also a solution with c being an arbitrary constant; the corresponding {φ, φ S } and βfc, LSCF remain unchanged. We therefore set QS = 1 − φ¯ to obtain a unique solution. Then from Eqs. (29), (30), (33), and (34) we obtain iω(r) = −N ln[1 − φ(r)] + χ N[1 − 2φ(r)],

(38)

which, along with Eq. (31), is solved using the Anderson mixing method48, 49 with the convergence criterion of max{r} |iω(r) + N ln[1 − φ(r)] − χ N[1 − 2φ(r)]| < 10−12 . Once the LSCF equations are solved, we calculate the free energy per chain as β(fc,LSCF − fcid ) =

1  {χ N φ 2 (r) + N ln[1 − φ(r)]} V r −

φ¯  ¯ ln(1 − φ), ¯ ln Qg − N(1 − φ) ng g

III. RESULTS AND DISCUSSION

Our model system of compressed grafted homopolymers in an explicit solvent is specified by a few physical parameters: the chain length N, the Flory-Huggins interaction parameter χ between polymer segments and solvent molecules, the distance between the nearest-neighbor grafting points Lg , ¯ x L2g /N which is directly related to the reduced m/ρ0 = φL chain-grafting density σ/ρ0 = (N − 1)m/ρ0 L2g with m being the number of chains grafted at each grafting point and ρ 0 the total number of polymer segments and solvent molecules at each lattice site, the degree of compression Lx , and ρ 0 ; the first five completely determine the LSCF results, which are exact in the limit of ρ 0 → ∞ (thus m → ∞), and the last one controls the system fluctuations in FLMC simulations, the results of which should approach the LSCF predictions with increasing ρ 0 (thus m). Fixing N = 40, we mainly investigate the effects of Lg and m/ρ 0 in an athermal solvent (χ = 0) using both FLMC simulations and LSCF calculations. The effects of ρ 0 are also investigated at given Lg and m/ρ 0 using FLMC simulations. To investigate the effects of solvent quality, we further consider the θ -solvent (χ = 0.5) in a few cases. In all the cases, Lx is varied from the smallest integer value max{3, (m/ρ0 )(N/L2g ) + 2}, where a denotes the integer part of a, to N.

(39) and the internal energy per chain as βuc,LSCF = −χ N eLSCF

A. LSCF results

(40)

with eLSCF = −

1  φ(r)[1 − φ(r)]. V r

(41)

The height of the grafted homopolymers hLSCF and the mean2 in the x-direction square chain end-to-end distance Rex,LSCF are calculated in the same way as in Papers II and III.2, 3 To calculate the mean-square chain end-to-end distance in the y2 , first the calculation box is dupli(or z-) direction Rey,LSCF cated in both the y- and z-directions such that the number of layers in these two directions in the new box L ≥ 2N − 1 to avoid the effect of periodic boundary conditions, then the propagators qg, s (r) and qt∗ (r) are calculated in this new box with appropriate initial conditions4 and further are used to cal2 2 in a way similar to that of calculating Rex,LSCF culate Rey,LSCF 2 from Eq. (18) in Paper II. Finally, using the second-order central finite difference, the pressure can be calculated as βPLSCF (Lx ) ρ0 ≈−

(Lx + 1)βfc,LSCF (Lx + 1) − (Lx − 1)βfc,LSCF (Lx − 1) 2N (42)

with βfc, LSCF given by Eqs. (28) and (39). As in FLMC simu∗ , the reduced lations, we calculate the reduced pressure PLSCF ∗ free energy difference per chain fc,LSCF , and the reduced en∗ in LSCF calculations. tropy difference per chain sc,LSCF

We first report the structure and thermodynamics of grafted homopolymers obtained from LSCF calculations, which provide the mean-field reference for subsequent analysis of the fluctuation effects reported in Sec. III B below.

1. Effects of Lg

Here, we fix m/ρ 0 = 0.1 in an athermal solvent (χ = 0); varying Lg is then equivalent to varying the reduced chaingrafting density σ /ρ 0 . For uncompressed systems (i.e., Lx = N), Fig. 1(a) shows the dependence of the height of grafted homopolymers h0LSCF and the mean-square chain end-to-end distance in the x-direction (perpendicular to the 2 . At small σ /ρ 0 (i.e., large Lg ), grafting substrate) Rex,LSCF 0 2 both hLSCF and Rex,LSCF are independent of σ /ρ 0 , indicating that the system is in the mushroom regime. At large 2 increase with increasing σ /ρ 0 σ /ρ 0 , both h0LSCF and Rex,LSCF due to the repulsion (excluded-volume interaction) among chains grafted at adjacent grafting points (referred to as the “crowding” effect), indicating that the system enters the brush regime. Fig. 1(b) shows that the scaling exponents k1 ≡ 2 /2d ln(σ/ρ0 ) are d ln h0LSCF /d ln(σ/ρ0 ) and k2 ≡ d ln Rex,LSCF almost the same and increase monotonically with increasing σ /ρ 0 at large σ /ρ 0 . There is no abrupt transition between the mushroom and brush regimes, and at our highest chain-grafting density (i.e., uniform grafting with Lg = 1) k1 ≈ 0.37 is slightly larger than the value of 1/3 predicted by both the scaling theory50, 51 and the infinite-stretching theory (IST)15, 16, 52–54 for homopolymer brushes in an implicit, good solvent.55

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044904-7

P. Zhang and Q. Wang

(a)

J. Chem. Phys. 140, 044904 (2014)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

FIG. 1. Parts (a) and (b) show how LSCF predictions of (a) the height h0LSCF and the mean-square chain end-to-end distances in the x- and y-direction 2 2 2 Rex,LSCF and Rey,LSCF , respectively, and (b) the exponents k1 ≡ d ln h0LSCF /d ln(σ/ρ0 ) and k2 ≡ d ln Rex,LSCF /2d ln(σ/ρ0 ) estimated using the second-order finite difference method, of grafted homopolymers in the uncompressed state vary with the reduced chain-grafting density σ /ρ 0 ; the corresponding distance Lg between the nearest-neighbor grafting points is shown as the upper horizontal axis. Parts (c)–(g) show how LSCF predictions  of (c) the height hLSCF ,  2 ¯ − φ) ¯ with φ¯ ≡ r φ(r)/V denoting the , (e) the ensemble-averaged reduced energy density eLSCF ≡ − r φ(r)(1 − φ(r))/V normalized by φ(1 (d) Rey,LSCF ∗ from the uncompressed state (where Lx = N), average polymer volume fraction, (f) the difference in the reduced Helmholtz free energy per chain fc,LSCF ∗ due to compression, of grafted homopolymers vary with the distance Lx between the grafting substrate and the compressing and (g) the reduced pressure PLSCF ∗ ∗ of the bulk solution of ungrafted homopolymers at the same φ¯ surface. Part (h) compares PLSCF of grafted homopolymers with the osmotic pressure PFH 2 ¯ obtained from the Flory-Huggins theory. The chain length N = 40, m/ρ0 = φLx Lg /N = 0.1 with m being the number of chains grafted at each grafting point and ρ 0 the total number of polymer segments and solvent molecules at each lattice site, and the Flory-Huggins interaction parameter between polymer segments and solvent molecules χ = 0 (i.e., an athermal solvent) are used here. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

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044904-8

P. Zhang and Q. Wang

Fig. 1(a) also shows the mean-square chain end-to-end 2 as distance in the y- (or equivalently the z-) direction Rey,LSCF 2 a function of σ /ρ 0 . We see that Rey,LSCF remains constant at small σ /ρ 0 and decreases monotonically with increasing σ /ρ 0 at large σ /ρ 0 ; the latter is again due to the “crowding” effect, which makes chains less extended in the directions parallel to the grafting substrate. Based on Fig. 1(a), we can consider systems with Lg  20 as in the mushroom regime. For compressed systems, Fig. 1(c) shows the height of grafted chains hLSCF as a function of Lx , both normalized by h0LSCF , at various Lg . At given Lg , hLSCF / h0LSCF ≈ 1 for small compression (i.e., Lx / h0LSCF  2.5) and decreases monotonically with decreasing Lx for larger compression. At given Lx / h0LSCF , hLSCF / h0LSCF decreases monotonically with increasing Lg for small Lg  10, and is independent of Lg 2 for larger Lg . Rex,LSCF exhibits the same behavior as hLSCF and is not shown. On the other hand, Fig. 1(d) shows that, 2 at given Lg , Rey,LSCF remains constant for Lx / h0LSCF  2.5 and increases monotonically with decreasing Lx for smaller Lx / h0LSCF , indicating that chains become more and more extended in the directions parallel to the grafting substrate with increasing compression in the latter case. At given Lx / h0LSCF , 2 increases monotonically with increasing Lg for small Rey,LSCF Lg  20 (the “crowding” effect) and is independent of Lg for larger Lg (i.e., in the mushroom regime). Based on Figs. 1(c) and 1(d), we can consider systems with Lx / h0LSCF  2.5 as uncompressed. The internal energy per chain uc is given by βuc = −χ Ne, with  the ensemble-averaged reduced energy density e = − r φ(r)[1 − φ(r)]/V . For an athermal (χ = 0) solvent, we have the trivial result of βuc = 0. Fig. 1(e) there¯ − φ) ¯ as a function of Lx / h0 fore shows −eLSCF /φ(1 LSCF at various Lg , where the average polymer volume fraction φ¯ ¯ − φ) ¯ = 1 for a homogeneous = N m/ρ0 L2g Lx . Since −e/φ(1 system, its deviation (decrease) from 1 thus quantifies the system inhomogeneity (within its volume). We see that, for uncompressed systems (i.e., Lx / h0LSCF  2.5), mushrooms (i.e., Lg  20) are more homogeneous than brushes due to their much smaller polymer volume fraction φ(r). On the other hand, while compression makes brushes more homogeneous, it slightly increases the inhomogeneity of mushrooms due to the increased φ(r). Fig. 1(f) shows how the difference in the reduced free energy per chain from the uncompressed state (where Lx ∗ = N) fc,LSCF varies with Lx / h0LSCF at various Lg ; note that ∗ 2 fc,LSCF Lg is half of the reduced work per grafting point required to move the compressing surface from Lx + 1 to Lx ∗ − 1. We see that fc,LSCF L2g monotonically increases with decreasing Lx / h0LSCF as expected. For Lg  10, however, ∗ L2g increases with increasing Lg at given Lx / h0LSCF fc,LSCF  1.2. This is because we divide Lx by h0LSCF here, which increases with increasing Lg as shown in Fig. 1(a); if we plot ∗ ∗ L2g vs. Lx (not shown), one then sees that fc,LSCF L2g fc,LSCF increases with decreasing Lg ( 10) at all Lx as expected. With Lx / h0LSCF as the horizontal axis, our data (except the Lg = 1 case) collapse onto the same curve for large compression (i.e., ∗ L2g is independent Lx / h0LSCF  1). We also see that fc,LSCF ∗ of Lg for Lg  10, as expected. Our behavior of fc,LSCF L2g

J. Chem. Phys. 140, 044904 (2014)

in the brush regime is qualitatively consistent with previous SCF studies.8, 9, 12, 13 For an athermal solvent, the difference in the reduced entropy per chain from the uncompressed state sc∗ = −fc∗ and is not shown. ∗ varies Fig. 1(g) shows how the reduced pressure PLSCF 0 ∗ 2 with Lx / hLSCF at various Lg ; note that PLSCF Lg is the force exerted on the compressing surface by chains grafted at the ∗ ≥ 0 in all the cases and same grafting point. We see that PLSCF ∗ . Fig. 1(h) compares exhibits the same behavior as fc,LSCF ∗ of grafted homopolymers with the osmotic pressure of PLSCF the bulk solution of ungrafted homopolymers at the same φ¯ obtained from the Flory-Huggins theory (which is equivalent ∗ = (1/N to our LSCF theory for a homogeneous system), PFH ∗ 2 ¯ ¯ ¯ is − 1)φ − ln(1 − φ) − χ φ . We see that, at given Lg , PLSCF ∗ ¯ much smaller than PFH for small φ (i.e., small compression for grafted homopolymers) due to the chain-grafting, and ap∗ ¯ indicating the dominance of with increasing φ, proaches PFH the solution osmotic pressure over the chain-grafting. The latter explains the collapse of our data for large compression shown in Fig. 1(g). For Lg = 4 and 79, however, we see ∗ ∗ > PFH at large φ¯ (i.e., large compression for grafted hoPLSCF mopolymers) due to the lateral inhomogeneity of the grafted systems. 2. Effects of m/ρ 0

With the relation σ/ρ0 = (N − 1)m/ρ0 L2g , only two of the three variables (σ /ρ 0 , m/ρ 0 , and Lg ) are independent. Instead of holding σ /ρ 0 constant to examine the effects of Lg (i.e., the chain-grafting pattern), which is difficult to control in experiments and known to have rather small effects on both brushes and mushrooms (as supported by our LSCF results), here we hold Lg constant and vary m/ρ 0 to examine its effects on brushes (with Lg = 1) and mushrooms (with Lg = 79) in an athermal solvent (χ = 0). While this is also equivalent to varying σ /ρ 0 , different effects, sometimes opposite to those shown in Sec. III A 1, are found for mushrooms as shown below. First of all, Table I lists h0LSCF for both brushes and mushrooms at various m/ρ 0 , as well as the scaling exponent k ≡ d ln h0LSCF /d ln(m/ρ0 ). We again see that, for brushes at large m/ρ 0 (= 0.1 and 0.2), k is larger than the value of 1/3 predicted by both the scaling theory50, 51 and IST15, 16, 52–54 for homopolymer brushes in an implicit, good solvent.55 Fig. 2(a) shows hLSCF / h0LSCF as a function of Lx / h0LSCF at m/ρ 0 = 0.1 and 0.2, which is similar to Fig. 1(c). In addition, we see that hLSCF / h0LSCF increases with increasing m/ρ 0 for TABLE I. The height h0LSCF (in units of lattice spacing) of grafted homopolymers in the uncompressed state predicted by LSCF theory. The numbers in the parentheses are the corresponding scaling exponent k ≡ d ln(h0LSCF )/d ln(m/ρ0 ) estimated by the second-order finite difference method. h0LSCF (k)

Lg

1 79

m/ρ 0 (χ = 0)

m/ρ 0 = 0.1,

0.05

0.1

0.2

χ = 0.5

8.714 (0.305) 5.072 (0.013)

11.006 (0.369) 5.137 (0.024)

14.538 (0.434) 5.246 (0.036)

8.161 4.999

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044904-9

P. Zhang and Q. Wang

J. Chem. Phys. 140, 044904 (2014)

(a)

(b)

(c) 2 FIG. 2. LSCF predictions of (a) the height hLSCF , (b) the mean-square chain end-to-end distances in the y-direction Rey,LSCF , and (c) the difference in the ∗ reduced Helmholtz free energy per chain fc,LSCF from the uncompressed state. N = 40 and χ = 0 are used here. See the caption of Fig. 1 for more details.

compressed brushes (i.e., Lx / h0LSCF  1.5) but is independent 2 exhibits the same behavior of m/ρ 0 for mushrooms. Rex,LSCF as hLSCF and is not shown. 2 as a function of Lx / h0LSCF Fig. 2(b) shows Rey,LSCF at m/ρ 0 = 0.1 and 0.2, which is similar to Fig. 1(d) and can be understood accordingly. In particular, we see that, 2 decreases with increasing m/ρ 0 at given Lx / h0LSCF , Rey,LSCF (thus σ /ρ 0 ) in the brush regime due to the “crowding” effect (i.e., the excluded-volume interaction among chains grafted at adjacent grafting points). The opposite, however, occurs in the mushroom regime due to the excluded-volume interaction among chains grafted at the same point; this is not seen in 2 is indeFig. 1(d), where m/ρ 0 is kept constant and Rey,LSCF pendent of Lg (thus σ /ρ 0 ). ¯ − φ) ¯ exhibits similar behavAt m/ρ 0 = 0.2, −eLSCF /φ(1 ior to Fig. 1(e) and is not shown. In addition, for brushes the same effect of decreasing system homogeneity with increasing σ /ρ 0 is found in both cases (i.e., by either increasing m/ρ 0 or decreasing Lg ). For mushrooms, however, increasing m/ρ 0 (thus σ /ρ 0 ) at constant Lg makes the system less homogeneous, opposite to the effect of increasing σ /ρ 0 (by decreasing Lg at constant m/ρ 0 ) shown in Fig. 1(e). ∗ L2g as a function of Lx / h0LSCF Fig. 2(c) shows fc,LSCF at m/ρ 0 = 0.1 and 0.2, which is similar to Fig. 1(f). While ∗ L2g varies with σ /ρ 0 in the same trend for brushes fc,LSCF ∗ in both cases, for mushrooms fc,LSCF L2g increases with increasing m/ρ 0 (thus σ /ρ 0 ) at constant Lg but is independent of Lg (thus σ /ρ 0 ) at constant m/ρ 0 . The dependence ∗ L2g on m/ρ 0 is also stronger for brushes than for of fc,LSCF

mushrooms, due to the “crowding” effect in brushes. Fi∗ ∗ exhibits the same behavior as fc,LSCF and is not nally, PLSCF shown. 3. Effects of χ

Here we fix m/ρ 0 = 0.1, and examine the effects of solvent quality on both brush (with Lg = 1) and mushroom (with Lg = 79). We use χ = 0 and 0.5, corresponding to a good (athermal) and θ -solvent, respectively. Fig. 3(a) shows hLSCF / h0LSCF as a function of Lx / h0LSCF in the two solvents, which is similar to Fig. 2(a). For mushroom, our data in different solvents collapse onto the same curve. For brush, comparing to Fig. 2(a) we see that decreasing solvent quality (i.e., increasing χ ) has the same effects as 2 of both mushdecreasing m/ρ 0 ; this is also found for Rey,LSCF room and brush (data not shown). The opposite, however, is ¯ − φ) ¯ (data not shown); that is, increasfound for −eLSCF /φ(1 ing χ makes the system less homogeneous. ∗ as a function of Lx / h0LSCF in Fig. 3(b) shows fc,LSCF the two solvents. Comparing it with Figs. 1(f) and 2(c), we see that for brush decreasing solvent quality has the same effects as decreasing σ /ρ 0 , which makes chains less stretched. For mushroom, our data in different solvents collapse onto the same curve. Note that, at the same Lx (not the same ∗ is larger in the athermal solvent than in Lx / h0LSCF ), fc,LSCF ∗ ≥ 0 in both the θ -solvent, as expected. We also find that PLSCF ∗ (data not solvents and exhibits the same behavior as fc,LSCF shown).

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044904-10

P. Zhang and Q. Wang

J. Chem. Phys. 140, 044904 (2014)

(a)

(b)

(c) ∗ FIG. 3. LSCF predictions of (a) the height hLSCF , (b) the difference in the reduced Helmholtz free energy per chain fc,LSCF from the uncompressed state, and ∗ (c) the differences in the internal energy per chain uc, LSCF and in the reduced entropy per chain sc,LSCF from the uncompressed state in the θ -solvent (i.e., χ = 0.5). N = 40 and m/ρ 0 = 0.1 are used here. See the caption of Fig. 1 for more details.

∗ For an athermal solvent, βuc,LSCF = 0 and sc,LSCF ∗ = −fc,LSCF . Fig. 3(c) therefore shows how these two quantities vary with Lx / h0LSCF in the θ -solvent. For nearly un∗ L2g is compressed systems (i.e., Lx / h0LSCF  1.5), fc,LSCF ∗ small ( 0 in all the cases; that is, the system fluctuations/correlations make grafted chains more stretched in the x-direction (perpendicular to the grafting substrate), consistent with Papers II and III on uncompressed homopolymer brushes.2, 3 For uncompressed systems (i.e., Lx / h0LSCF  2.5), we find ρ 0 h/hLSCF ≈ 0.05 for brushes and 0.5 for mushrooms. For smaller Lx / h0LSCF , h/hLSCF decreases monotonically with increasing compression (i.e., decreasing Lx ), which may be explained by the increased φ¯ (the system inhomogeneity, however, needs to be taken into account). This may also explain the larger fluctuation/correlation effects in mushrooms than in brushes at given Lx / h0LSCF and ρ 0 . Finally, our ρ 0 h/hLSCF data with different ρ 0 -values collapse, indicating that the fluctuation/correlation effects are inversely 2 2 /Rex,LSCF exproportional to ρ 0 within our data range. Rex hibits qualitatively the same behavior as h/hLSCF and is not 2 2 /Rex,LSCF ≈ 0.2 for unshown. Note, however, that ρ0 Rex compressed brushes and 1.1 for uncompressed mushrooms, larger than the corresponding ρ 0 h/hLSCF -value given above. 2 2 /Rey,LSCF as a function of Fig. 5(b) shows Rey 0 2 2 /Rey,LSCF Lx / hLSCF at ρ 0 = 10 and 20. We see that ρ0 Rey ≈ 0.5 for uncompressed brushes and 0.85 for uncompressed 2 2 /Rey,LSCF increases mushrooms. More interestingly, Rey 0 with increasing compression for Lx / hLSCF  2.5, which is 2 shown in Fig. 1(d). similar to the behavior of Rey,LSCF Fig. 5(c) shows −e/eLSCF as a function of Lx / h0LSCF at ρ 0 = 10 and 20. We see that LSCF theory underestimates −e (thus the system homogeneity) in all the cases, consistent with the above result that chains are less stretched in LSCF cal-

culations. In particular, −ρ 0 e/eLSCF ≈ 1 for uncompressed brushes and 1.4 for uncompressed mushrooms. More interestingly, with increasing compression, −e/eLSCF decreases for compressed brushes (i.e., Lx / h0LSCF  1.5) but increases for compressed mushrooms (i.e., Lx / h0LSCF  2.5). In addition, −ρ 0 e/eLSCF for mushrooms with different ρ 0 do not collapse, although their differences are rather small (less than 0.05). ∗ ∗ /fc,LSCF − 1 as a function Fig. 5(d) shows fc,FLMC 0 of Lx / hLSCF at ρ 0 = 10 and 20. For brushes, we see that LSCF theory overestimates fc∗ at large compression (i.e., Lx / h0LSCF  1.1) but underestimates it at smaller compression. For mushrooms, LSCF theory underestimates fc∗ in all the cases. The inset of Fig. 5(d) shows that ∗ ∗ − fc,LSCF approaches 0 with increasing Lx ; that fc,FLMC ∗ ∗ fc,FLMC /fc,LSCF − 1 increases with increasing Lx is there∗ towards 0 as shown fore due to the quick decrease of fc,LSCF in Fig. 1(f), which also amplifies the relative uncertainty of ∗ . Consistent with the above, the flucour estimated fc,FLMC ∗ ∗ − fc,LSCF | or tuation effects, measured by either |fc,FLMC ∗ ∗ |fc,FLMC /fc,LSCF − 1|, are larger in mushrooms than in brushes at given Lx / h0LSCF and ρ 0 . For an athermal solvent, fluctuations of the difference in the reduced entropy per chain from the uncompressed ∗ ∗ ∗ − sc,LSCF = −(fc,FLMC state (where Lx = N) sc,FLMC ∗ − fc,LSCF ) and is not shown. Finally, Fig. 5(e) shows ∗ P ∗ /PLSCF − 1 as a function of Lx / h0LSCF at ρ 0 = 10 and 20, which is similar to Fig. 5(d); the only difference is that LSCF theory overestimates P∗ for brushes at Lx / h0LSCF  1.3.

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044904-13

P. Zhang and Q. Wang

J. Chem. Phys. 140, 044904 (2014)

(a)

(b)

(c)

(d)

(e)

(f)

2 , (c) and (d) the ensembleFIG. 7. Fluctuation/correlation effects on (a) the height h, (b) the mean-square chain end-to-end distance in the y-direction Rey averaged reduced energy density e, (e) the difference in the reduced Helmholtz free energy per chain fc∗ from the uncompressed state, and (f) the difference in the reduced entropy per chain sc∗ from the uncompressed state in the θ -solvent (i.e., χ = 0.5). N = 40, ρ 0 = 10, and m/ρ 0 = 0.1 are used here.

3. Effects of m/ρ 0

Here we fix ρ 0 = 10 and χ = 0, and examine the effects of m/ρ 0 (equivalently σ /ρ 0 ) on the fluctuations in both brushes and mushrooms. Fig. 6(a) shows h/hLSCF as a function of Lx / h0LSCF at m/ρ 0 = 0.1 and 0.2, which is similar to Fig. 5(a). In addition, for compressed brushes (i.e., Lx / h0LSCF  1.5), we see that h/hLSCF (and h) decreases faster with decreasing Lx / h0LSCF at larger m/ρ 0 . On the other hand, h/hLSCF decreases with increasing m/ρ 0 at given Lx / h0LSCF , which may be explained by the increased 2 2 ¯ Rex /Rex,LSCF exhibits qualitatively the same behavior as φ. h/hLSCF and is not shown. That increasing m/ρ 0 decreases the system fluctuations is 2 2 /Rey,LSCF (data not shown), which is simalso found for Rey ilar to Fig. 5(b), and for −e/eLSCF shown in Fig. 6(b), which is similar to Fig. 5(c). In addition, for the compressed brush at

2 2 /Rey,LSCF increases faster with m/ρ 0 = 0.2, we find that Rey 0 decreasing Lx / hLSCF than at m/ρ 0 = 0.1, which is similar to 2 the behavior of Rey,LSCF shown in Fig. 2(b). For this case, we also see that −e/eLSCF exhibits a shallow minimum around Lx / h0LSCF ≈ 0.83; note that both −eLSCF and e (not shown) exhibit a maximum around Lx / h0LSCF = 1 (where φ¯ ≈ 0.5), as explained at the end of Sec. III A 3 above. ∗ ∗ /fc,LSCF − 1 as a function Fig. 6(c) shows fc,FLMC 0 of Lx / hLSCF at m/ρ 0 = 0.1 and 0.2, which is similar to ∗ ∗ Fig. 5(d). In addition, fc,FLMC /fc,LSCF − 1 with different m/ρ 0 nearly collapse, which is different from the above finding that increasing m/ρ 0 decreases the system fluctuations. The inset of Fig. 6(c) shows that, for both compressed brushes at Lx / h0LSCF  1.1 and compressed mushrooms at ∗ ∗ deviates more from fc,LSCF with Lx / h0LSCF  2, fc,FLMC ∗ increasing m/ρ 0 (note the corresponding increase of fc,LSCF

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044904-14

P. Zhang and Q. Wang

in these cases as shown in Fig. 2(c)). Finally, the fluctuation effects on P∗ exhibit qualitatively the same behavior as those on fc∗ and are not shown here. 4. Effects of χ

Here we fix m/ρ 0 = 0.1 and ρ 0 = 10, and examine the effects of solvent quality χ on the fluctuations in both brush and mushroom. Fig. 7(a) shows how h/hLSCF varies with Lx / h0LSCF at χ = 0 and 0.5, which is similar to Fig. 5(a). In addition, for brush, h/hLSCF is larger in the θ -solvent than in the athermal solvent, consistent with Paper III on uncompressed homopolymer brushes;3 for mushroom, however, the opposite is found. We also see that, for brush, h/hLSCF decreases faster with decreasing Lx / h0LSCF in the θ -solvent than in the athermal solvent. On the other hand, in the athermal solvent, h/hLSCF is larger for mushroom than for brush; in the θ -solvent, however, the opposite is found (for Lx / h0LSCF 2 2  0.8). Rex /Rex,LSCF exhibits qualitatively the same behavior and is not shown. 2 2 /Rey,LSCF as a function of Fig. 7(b) shows Rey 0 Lx / hLSCF in both solvents, which is similar to Fig. 5(b). 2 2 /Rey,LSCF is larger in In addition, at given Lx / h0LSCF , Rey the athermal solvent than in the θ -solvent. We also see that, 2 2 /Rey,LSCF increases faster with decreasing for brush, Rey 0 Lx / hLSCF in the θ -solvent than in the athermal solvent. On the 2 2 /Rey,LSCF is larger other hand, in the athermal solvent Rey for mushroom than for brush, while the opposite is found in the θ -solvent. Fig. 7(c) shows −e/eLSCF in both solvents, which is similar to Fig. 5(c) except that −e/eLSCF for brush in the θ solvent exhibits a maximum around Lx / h0LSCF = 1. To understand it, Fig. 7(d) shows eL2g , which is approximately proportional to L−1 x for mushroom at all Lx and for brush at large Lx  1.5h0LSCF ; for strongly compressed brush, e exhibits a maximum. Similar behavior is also found for −e LSCF in both solvents (data not shown). This indicates that r φ(r)[1 − φ(r)] is nearly constant for mushroom at all Lx and for brush at large Lx  1.5h0LSCF , and that the maximum for brush is caused by φ¯ ≈ 0.5 (see our above discussion on βuc,LSCF shown in Fig. 3(c)). When we divide e by −eLSCF , the maximum disappears in the athermal solvent but remains in the θ -solvent. We also see from Fig. 7(c) that, at given Lx / h0LSCF , −e/eLSCF is larger in the θ -solvent than in the athermal solvent. ∗ ∗ /fc,LSCF − 1 in both solFig. 7(e) shows fc,FLMC vents, which is similar to Fig. 5(d). Clearly, LSCF theory underestimates fc∗ in the θ -solvent. At given Lx / h0LSCF , ∗ ∗ /fc,LSCF − 1 is larger in the θ -solvent than in fc,FLMC the athermal solvent for brush, while the opposite is found for mushroom. On the other hand, in the athermal sol∗ ∗ /fc,LSCF − 1 is larger for mushroom than for vent fc,FLMC brush, while the opposite is found in the θ -solvent. The in∗ − set of Fig. 7(e) shows that, for Lx / h0LSCF  1.3, (fc,FLMC ∗ 2 fc,LSCF )Lg increases with increasing χ for brush, while the ∗ exhibits qualiopposite is found for mushroom. P ∗ /PLSCF ∗ ∗ − 1 and is tatively the same behavior as fc,FLMC /fc,LSCF ∗ ∗ − sc,LSCF not shown. For the athermal solvent, sc,FLMC

J. Chem. Phys. 140, 044904 (2014) ∗ ∗ = −(fc,FLMC − fc,LSCF ). The inset of Fig. 7(f) therefore ∗ ∗ = −χ N [e − e(Lx = N )] − shows sc,FLMC − sc,LSCF ∗ ∗ (fc,FLMC − fc,LSCF ) as a function of Lx / h0LSCF in the θ solvent, where LSCF theory overestimates sc∗ . Fig. 7(f) ∗ ∗ /sc,LSCF − 1 as a function of Lx / h0LSCF in shows sc,FLMC the θ -solvent. We note that, for nearly uncompressed systems (i.e., Lx / h0LSCF  1.5), fc∗ L2g is small ( 0 and −e > 0 in all the cases; that is, the system fluctuations/correlations make grafted chains more stretched, thus leading to a more homogeneous system. Within our ρ 0 -range (i.e., ρ 0  10), the fluctuation/correlation effects are approximately proportional to 1/ρ 0 . For uncompressed systems (i.e., Lx / h0LSCF  2.5) at m/ρ 0 = 0.1 and χ = 0, we find ρ 0 h/hLSCF ≈ 0.05 for brushes (with Lg = 1) and 2 2 /Rex,LSCF ≈ 0.5 for mushrooms (with Lg = 79), ρ0 Rex 2 2 0.2 for brushes and 1.1 for mushrooms, ρ0 Rey /Rey,LSCF ≈ 0.5 for brushes and 0.85 for mushrooms, and −ρ 0 e/eLSCF ≈ 1 for brushes and 1.4 for mushrooms. We also find that fluctuation/correlation effects (measured by h/hLSCF , 2 2 /Rey,LSCF , and −e/eLSCF ) are smaller in brushes than Rey in mushrooms and decrease with increasing m/ρ 0 (at constant Lx / h0LSCF ). In addition, h/hLSCF decreases monotonically with increasing compression (for compressed systems). 2 2 ¯ Rey /Rey,LSCF , These may be explained by the increased φ. however, increases with increasing compression. We also find that, with increasing compression, −e/eLSCF decreases for compressed brushes (i.e., Lx / h0LSCF  1.5) but increases for compressed mushrooms (i.e., Lx / h0LSCF  2.5). Furthermore, LSCF theory overestimates fc∗ at large compression (i.e., Lx / h0LSCF  1.1) but underestimates it at smaller compression for brushes, and underestimates it for ∗ ∗ /fc,LSCF − 1 is mushrooms. At given Lx / h0LSCF , fc,FLMC ∗ ∗ | nearly independent of m/ρ 0 , and both |fc,FLMC − fc,LSCF ∗ ∗ and |fc,FLMC /fc,LSCF − 1| are larger in mushrooms than in brushes. For both compressed brushes at Lx / h0LSCF  1.1 and ∗ deviates compressed mushrooms at Lx / h0LSCF  2, fc,FLMC ∗ more from fc,LSCF with increasing m/ρ 0 . The above results are for the athermal solvent. In the θ solvent (where m/ρ 0 = 0.1 is used), however, different results are found; that is, LSCF theory underestimates fc∗ and overestimates sc∗ , and the fluctuation effects (measured 2 ∗ ∗ 2 /Rey,LSCF , and fc,FLMC /fc,LSCF −1 by h/hLSCF , Rey ∗ ∗ or fc,FLMC − fc,LSCF ) are smaller for mushroom than for brush. h/hLSCF is larger in the θ -solvent than in the athermal solvent for brush, while the opposite is found for 2 2 /Rey,LSCF is smaller mushroom. At given Lx / h0LSCF , Rey while −e/eLSCF is larger in the θ -solvent than in the

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∗ ∗ athermal solvent, and both fc,FLMC /fc,LSCF − 1 and ∗ ∗ fc,FLMC − fc,LSCF are larger in the θ -solvent than in the athermal solvent for brush with the opposite found for mushroom. 2 exhibits the Finally, we note that, in all the cases, Rex ∗ 2 same behavior as h, and P Lg (the force exerted on the compressing surface by chains grafted at the same grafting point) is greater than or equal to 0 and  exhibits the same behavior as fc∗ L2g . Furthermore, −e = r φ(r)[1 − φ(r)]/V provides a link between the structure and thermodynamics of the system. In particular, −e is approximately proportional to L−1 x for mushrooms (with Lg = 79) at all Lx and brushes (with Lg = 1) at large Lx  1.5h0LSCF , and exhibits a maximum for compressed brushes around Lx / h0LSCF = 1, where φ¯ ≈ 0.5. The latter in some cases leads to non-monotonic behavior of −e/eLSCF and βuc with varying Lx / h0LSCF . Last but not least, for nearly uncompressed systems (i.e., Lx / h0LSCF  1.5) in the θ -solvent, fc∗ is small compared to sc∗ ≈ βuc .

ACKNOWLEDGMENTS

We thank Mr. Sadanand Singh in Professor Juan de Pablo’s group for introducing the optimized-ensemble sampling to us, and Q.W. thanks Professor Juan de Pablo for his hospitality provided during a visit to his group in December 2011. Financial support for this work was provided by NSF CAREER Award No. CBET-0847016, which is gratefully acknowledged. 1 P.

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