Partition function zeros and finite size scaling for polymer adsorption.

Partition function zeros and finite size scaling for polymer adsorption Mark P. Taylor and Jutta Luettmer-Strathmann Citation: The Journal of Chemical Physics 141, 204906 (2014); doi: 10.1063/1.4902252 View online: http://dx.doi.org/10.1063/1.4902252 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/141/20?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Finite-size scaling relations for a four-dimensional Ising model on Creutz cellular automatons Low Temp. Phys. 37, 470 (2011); 10.1063/1.3610180 Exact partition function zeros and the collapse transition of a two-dimensional lattice polymer J. Chem. Phys. 133, 114106 (2010); 10.1063/1.3486176 Atomic hydrogen adsorption and incipient hydrogenation of the Mg(0001) surface: A density-functional theory study J. Chem. Phys. 131, 034706 (2009); 10.1063/1.3182851 Finite-size scaling analysis on the phase transition of a ferromagnetic polymer chain model J. Chem. Phys. 124, 034903 (2006); 10.1063/1.2161208 Crossover between strong- and weak-field critical adsorption and the determination of the universal exponent η J. Chem. Phys. 117, 902 (2002); 10.1063/1.1483066

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THE JOURNAL OF CHEMICAL PHYSICS 141, 204906 (2014)

Partition function zeros and finite size scaling for polymer adsorption Mark P. Taylor1,a) and Jutta Luettmer-Strathmann2,b) 1 2

Department of Physics, Hiram College, Hiram, Ohio 44234, USA Department of Physics and Department of Chemistry, University of Akron, Akron, Ohio 44325, USA

(Received 11 July 2014; accepted 11 November 2014; published online 26 November 2014) The zeros of the canonical partition functions for a flexible polymer chain tethered to an attractive flat surface are computed for chains up to length N = 1536. We use a bond-fluctuation model for the polymer and obtain the density of states for the tethered chain by Wang-Landau sampling. The partition function zeros in the complex eβ -plane are symmetric about the real axis and densest in a boundary region that has the shape of a nearly closed circle, centered at the origin, terminated by two flaring tails. This structure defines a root-free zone about the positive real axis and follows Yang-Lee theory. As the chain length increases, the base of each tail moves toward the real axis, converging on the phase-transition point in the thermodynamic limit. We apply finite-size scaling theory of partitionfunction zeros and show that the crossover exponent defined through the leading zero is identical to the standard polymer adsorption crossover exponent φ. Scaling analysis of the leading zeros locates the polymer adsorption transition in the thermodynamic (N → ∞) limit at reduced temperature Tc∗ = 1.027(3) [βc = 1/Tc∗ = 0.974(3)] with crossover exponent φ = 0.515(25). Critical exponents for = 0.97(5) and α = 0.03(4), respecthe order parameter and specific heat are determined to be β tively. A universal scaling function for the average number of surface contacts is also constructed. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4902252] tion temperature. The scaling variable

I. INTRODUCTION

Surfaces with end-grafted chain molecules are found in many natural and man-made systems.1–3 When the grafting density is so low that the chains do not interact with each other, the conformations of the chains are determined by the solvent quality and interactions with the surface.4–11 In good solvent conditions, where the chains are extended, a hard or repulsive surface leads to desorption of all but the tethered chain segment resulting in the so-called mushroom conformations of the chains. In contrast, an attractive surface adsorbs chain segments and leads to the so-called pancake conformations. The adsorption transition, which connects these states, is the focus of this work. Chain molecules tethered to an attractive surface undergo an adsorption transition as the temperature decreases. Since surface adsorption is controlled by a combination of the temperature T and the attractive bead-surface interaction energy − ( > 0) we define the reduced temperature T ∗ = kB T/, where kB is Boltzmann’s constant. The ratio of surface contacts and chain length, na /N, is the order parameter MN (T) for the transition; it is small for hard and slightly attractive surfaces, rises rapidly through the adsorption transition, and grows more slowly toward the maximum value, na /N = 1, as the temperature decreases further in the so-called strongcoupling region. The adsorption transition is continuous12 and accompanied by large fluctuations in the number of surface contacts. The critical point of adsorption is the dual limit N → ∞ and T ∗ → Tc∗ , where Tc∗ is the reduced critical adsorp-

a) [email protected] b) [email protected]

0021-9606/2014/141(20)/204906/10/$30.00

x = |κ|N φ with κ = (T − Tc )/T ,

(1)

measures the distance to the critical adsorption point and defines the exponent φ, the so-called crossover exponent. In the neighborhood of the transition, chain properties satisfy power laws with critical exponents. The number of surface contacts, for example, may be written as12 na (N, T ) = N φ f (x),

(2)

where f(x) is a universal scaling function, which is a positive constant at x = 0 (T = Tc ) and has the following behavior for large x:12 −1 for x 1 and T > Tc (desorbed), x f (x) ∝ (3) x −1+1/φ for x 1 and T < Tc (adsorbed). These scaling laws show, for example, that the number of surface contacts is independent of chain length for desorbed chains, proportional to N for adsorbed chains, and scales as Nφ at Tc . Similar scaling laws are available for the chain dimensions parallel and perpendicular to the surface.12 One prediction from these scaling relations is that the ratio of parallel and perpendicular contributions to the radius of gyration is independent of chain length at the critical adsorption point.13 This criterion has been used to determine the infinite-chain adsorption transition from finite-chain simulations. Unfortunately, the results suffer from significant statistical uncertainty13–15 and are not always consistent with other criteria that rely on the chain-length independence of observables at the critical adsorption point.13 While the scaling relations for the adsorption transition developed by Eisenriegler et al.12 have been confirmed in

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many simulations, the value of the exponent φ for threedimensional systems has been, until recently, a matter of some debate. Metzger et al.,13 in a 2002 review of the literature, reported a broad range of values within the theoretical bounds of 0.4 ≤ φ ≤ 0.67 (in three dimensions). In a typical analysis of simulation data, an estimate for the critical transition point is used to construct scaling plots of observables for several proposed values of the crossover-exponent φ. The graphs are inspected for data collapse and proper asymptotic behavior and the best value of φ is identified by visual inspection. Eisenriegler et al.12 estimated φ = 0.58(2) from a scaling analysis of simulations of a self-avoiding walk on a simple cubic lattice. Metzger et al.13 performed simulations of two off-lattice models with different interaction ranges and found φ = 0.50(2), suggesting that the exponent φ is “superuniversal,” having the same value, φ = 1/2, in two, three, and four dimensions. Descas et al.14, 15 performed simulations of the bond-fluctuation (BF) model16, 17 that is also employed in this work. Their scaling analysis for chains up to length N = 200 shows a strong correlation between estimates for the critical adsorption point and the exponent and they identified two sets of parameters that yield equally good scaling results: β c = 1.01 with φ = 0.59 and β c = 0.98 with φ = 0.50. Much higher precision studies, although also for short (N ≤ 200) chains, have been carried out by Janse van Rensburg and Rechnitzer18 who report φ = 0.500(4) and Hegger and Grassberger19 who find φ = 0.496(5), further supporting the idea that φ is super-universal. However, more recently, Grassberger carried out high precision studies on long chains (N up to 8000) using the pruned-enriched Rosenbluth method (PERM) to obtain the exponent φ = 0.484(2).20 Similar results, using the same method and with N up to 25 600, have been reported by Hsu and Binder21 and Klushin et al.22 In these most recent high precision studies, one must still identify the critical temperature in order to determine the exponent φ, so there remains some coupling between these two parameters in the analysis. An approach in which the determination of the exponent φ is independent of the determination of the transition temperature would clearly be desirable. In this work, we present such a method of analysis which avoids any correlation between the computation of the crossover exponent and the critical temperature. In particular, here we analyze the zeros of the canonical partition functions of tethered chains to obtain independent values for the critical point of adsorption and the exponent φ. The connection between the zeros of the partition function and phase transitions was first established by Yang and Lee, who proposed a general theory of phase transitions based on the behavior of the zeros of the grand canonical partition function in the complex fugacity plane.23, 24 In the YangLee theory, as the thermodynamic limit is taken, some zeros of the partition function move arbitrarily close to the positive real axis, giving rise to the non-analytic behavior in thermodynamic functions which characterizes phase transitions. Yang and Lee23, 24 demonstrated this behavior for the 2D Ising model in an external field. This approach was extended by Fisher to the canonical partition function where the zeros are defined in a complex temperature plane.25 In

J. Chem. Phys. 141, 204906 (2014)

the thermodynamic limit, the Yang-Lee zeros typically define smooth curves in the complex fugacity plane while the Fisher zeros tend to fill bounded regions in the complex temperature plane26 (with the isotropic Ising model studied by Fisher25 being an exception). For finite systems, Itzykson et al.27 have demonstrated that the leading partition function zeros (i.e., those closest to the positive real axis) follow scaling laws which allows one to extrapolate the locations of these leading zeros to the transition point in the thermodynamic limit. The distribution and behavior of the Yang-Lee and Fisher zeros have been studied for many systems and provide a means of computing phase transition properties including location, order, and critical exponents.28–33 For a recent review of the partition function zero formalism and applications see Ref. 34. The partition function zero method has been applied to the study of conformational phase transitions of model polymer chains. In particular, the approach has been used to study the collapse transition for lattice homopolymers,35–41 the protein-like folding transition of lattice heteropolymers,42, 43 and both the helix-coil44, 45 and collapse and freezing transitions46 of continuum polymer models. Most of these polymer applications study the scaling properties of the partition function zeros to obtain results in the long chain limit. However, with the exception of Ref. 46, these previous polymer studies have been restricted to rather modest chain lengths (typically N ≤ 36) for which the partition function zero “maps” are relatively sparse and accurate finite size scaling is difficult. In this work, we study the partition function zeros for flexible polymer chains, up to length N = 1536, endtethered to a flat attractive surface. We determine the density of states for a bond-fluctuation model of tethered chains via Monte Carlo (MC) simulation with Wang-Landau (WL) sampling.47, 48 This model is amenable to direct computation of the zeros of the canonical partition function since the latter is simply a polynomial whose coefficients are given by the density of states. We apply finite size scaling techniques to study the approach of the leading zeros toward the positive real axis and thereby determine details of the polymer adsorption transition in the thermodynamic limit.

II. MODEL AND METHODS A. Bond-fluctuation model and simulation method

In this work, we employ the BF model16, 17 to study individual flexible polymer chains tethered to an impenetrable surface in good solvent conditions. In the BF model, beads of a chain of length N occupy sites on a simple cubic lattice with a minimum distance of 2a between beads √ and allowed bond lengths in the range bmin = 2a to bmax = 10a, where a is the lattice constant. In the Cartesian coordinate system employed in this work, the surface spans the x − y plane at height z = −a, restricting monomers to the half-space z ≥ 0, and the tethered monomer is at z = 0. All monomers at z = 0 are considered to be in contact with the surface, each contributing an amount − to the energy. Thus, this surface adsorption model


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has a discrete potential energy spectrum with allowed energy states En = −n where 1 ≤ n ≤ N. The density of states g(En ) describes the number of chain conformations with n surface contacts. To construct g(En ), we perform Monte Carlo simulations with WL sampling.47–49 The elementary moves for the simulations consist of displacement of individual beads to nearest-neighbor sites, pivot moves, and so-called cut-and-permute moves, where the chain is cut at a random bead, top and bottom are interchanged, and the chain is reassembled and tethered to the surface if the move is accepted.50 The acceptance criterion for the simulations is g(En ) ,1 , (4) p(n → m) = min g(Em ) where g(E) is the current estimate for the density of states. The density of states is updated according to the original Wang-Landau prescription48, 49 with refinement levels fk = exp (2−(k−1) ) for k = 4, 5, . . . , 30. We have performed simulations for chains of length N = 16–1536 and use flatness criteria of 95% for N ≤ 96, 90% for 64 ≤ N ≤ 512, and 80% for N > 512. For the longer chains (N > 512), we set the minimum energy in the simulations to EN/2 (or EN/3 for some of the N = 1536 runs). In all cases, we carry out multiple independent WL simulations (typically 10 or more), evaluate the densities of states individually, and calculate average and standard deviations for the resulting properties to estimate uncertainties.

written as βFN (T )/ = − ln ZN (T ) = − ln(g1 y) −

N−1

ln(y − wk )

k=1

= − ln(g1 y) −

N−1

1/2 ln (y − ak )2 + bk2 , (7)

k=1

where y is a real number such that 1 < y < ∞ corresponds to the physical temperature range ∞ > T ∗ > 0. In the second line, we have written the logarithm of the complex number y − wk in terms of its real and imaginary part and used the fact that the imaginary parts of the pairs of complex roots cancel. Since the WL simulations determine the density of states only up to an arbitrary multiplicative constant, taken to be g1 in Eq. (5), the free energy is only determined up to the N-dependent additive constant ln g1 . However, this arbitrary constant is absent from thermodynamic functions given by derivatives of the free energy, such as the internal energy U = ∂(βF)/∂β and heat capacity C = ∂U/∂T. The order parameter for the polymer adsorption transition, which is related to the internal energy per chain site through MN (T) = |UN (T)|/N, can be written in terms of the partition function zeros as

N−1 1 y y

∂(βFN /)

MN (T ) =

= N 1+ N

∂y (y − wk ) k=1 N−1 y(y − a ) 1 k = . 1+ N (y − ak )2 + bk2 k=1 (8)

B. Partition function zeros and thermodynamics

From the density of states one constructs the canonical partition function, given by a Boltzmann weighted sum over energy states En = −n, as11, 51 ZN (T ) =

N

g(En )e−En /kB T = g1 y

n=1

N−1

(gn /g1 )y n ,

(5)

n=0

where gn = g(En ) is the density of states for energy level En and y = eβ with β = /kB T. For our discrete energy state model, the canonical partition function is a polynomial of order N − 1 in the temperature dependent variable y. This polynomial can be rewritten in product form in terms of its N − 1 zeros or roots, {wk }, ZN (T ) = g1 y

N−1

(y − wk ).

(6)

k=1

The polynomial zeros are generally complex, coming in complex conjugate pairs, wk = ak ± ibk , and, since the gn are all positive, any real roots wk = ak are necessarily negative. The set of partition function zeros, {wk }, contains the same information as the original density of states {gn }. Thus, all thermodynamic functions derivable from ZN (T) can be expressed in terms of the N − 1 partition function zeros. For example, the single-chain Helmholtz free energy may be

Note that the sum in Eq. (8) approaches N − 1 in the low T (large y) limit, so that MN (T → 0) = 1 as expected. In general, for T < Tc (y > yc ) this sum must be O(N), whereas for T ≥ Tc (1 < y ≤ yc ) this sum must be O(Nμ ) with μ < 1 to give M∞ (T ≥ Tc ) = 0. Similarly, the specific heat is given in terms of the partition function zeros as N−1 β 2 ∂ 2 ln ZN y(ln y)2 −wk CN (T ) = = N kB N ∂β 2 N (y − wk )2 k=1

=

N−1 y(ln y)2 −ak (y − ak )2 + bk2 (2y − ak ) . (9) 2 N (y − ak )2 + bk2 k=1

(low T) limit Since CN (T) is a positive quantity, the large y of Eq. (9) allows us to establish the sum rule k ak ≤ 0 for the real parts of the partition function zeros. Notice that roots with positive real part ak and small imaginary part bk make a large contribution to CN (T) at y = ak (T ∗ = 1/ln ak ). In general, roots closest to the positive real axis, known as the leading roots, give the largest contribution to thermodynamic quantities. The Yang-Lee theory of phase transitions is based on the convergence of these leading roots onto the positive real axis in the thermodynamic limit (i.e., N → ∞).23, 24 This gives rise to non-analytic behavior of thermodynamic functions near the transition point. For example, the order parameter of the transition is described by a power as the transition is approached law with critical exponent β



J. Chem. Phys. 141, 204906 (2014)

M∞ (T ) ∼ |T − Tc |β .

(10)

Similarly, the specific heat behavior is described by a critical exponent α in the neighborhood of a phase transition, C∞ (T )/NkB ∼ |T − Tc |−α .

(11)

A positive value of the exponent, α > 0, corresponds to power-law divergence of C∞ (T) at the transition temperature Tc , while α = 0 indicates logarithmic divergence or a discontinuity of C∞ at Tc , and α < 0 describes a heat capacity that remains finite and continuous as Tc is approached. C. Finite size scaling

For a finite-size system, the thermodynamic functions remain analytic at Tc but, with increasing size, these functions approach their asymptotic power-law behavior as the leading partition function zero approaches the thermodynamic-limit transition point on the real positive axis. In the finite-size scaling theory of partition function zeros, a crossover exponent φ is defined to measure how the distance between the leading zero and the transition point scales with system size.27 Here, we apply finite-size scaling theory to our system, show that the exponent φ is identical to the crossover exponent for the adsorption transition defined in Eq. (1), and derive scaling relations. In the complex w-plane, the difference between the leading zero wl and the transition point on the real axis, yc = eβc , is expected to scale as27 wl − yc ∼ DN −φ ,

(12)

where D = d1 + id2 is a complex constant. (This scaling form actually applies to all zeros in the neighborhood of the critical point, but in the present analysis we will focus on the leading zero only.) An asymptotically equivalent expression can be written for the leading zero wl = al + ibl in the complex inverse temperature plane ln(wl ) − βc ∼ (D/yc )N −φ ,

(13)

where ln(wl ) = ln(al + ibl ) = βl + iτl with βl = + bl2 )1/2 and τ l = tan −1 (bl /al ). The real part of Eq. (13) gives β l − β c ∼ N−φ , which implies a scaling relation for the temperature of the leading root in terms of the chain length, |1 − Tc /Tl | ∼ N−φ . Since the behavior of the order parameter for temperatures near Tc is dominated by the leading root, Eqs. (8) and (12) and the real part of (13) yield MN (T) ∼ Nφ − 1 ∼ |T − Tc |1 − 1/φ for T near Tc . This allows us to write the order parameter in a finite size scaling form at temperature Tc and identify the critical of Eq. (10) exponent β ln(al2

MN (Tc ) ∼ N

φ −β

= −1 + 1/φ. with β

(14)

For the average number of adsorbed chain sites na (N, T), this implies na (N, Tc ) = NMN (Tc ) ∼ Nφ , which reproduces the scaling relation of Eq. (2) (at T = Tc ) and shows that the finite size scaling crossover exponent introduced in Eq. (12) is identical to the traditional polymer crossover exponent. (More

directly, inserting Eq. (12) into Eq. (8) at T = Tc yields the relation na (N, Tc ) − 1 ∼ Nφ .) Similarly, for the specific heat near Tc , Eq. (9) gives CN /N kB ∼ 1/N bl2 which, when combined with bl ∼ N−φ from the imaginary part of Eq. (12) and the temperature scaling from the real part, yields CN /NkB ∼ N2φ−1 ∼ |T − Tc |2−1/φ . Recalling the definition of the specific heat exponent in Eq. (11), the scaling of the specific heat at the transition temperature may be written as CN (Tc )/N kB ∼ N αφ with α = 2 − 1/φ.

(15)

The relation between the exponents α and φ is analogous to the Josephson hyperscaling relation between the specific heat exponent and the exponent of the correlation length in the Ising model.52 Since the specific heat is given by a temperature derivative of the internal energy, the order parameter must also follow the scaling form MN (Tc ) ∼ Nαφ−φ which, on com= 1 − α parison with Eq. (14), gives the exponent identity β for polymer adsorption. III. RESULTS A. Partition function zero maps

In Fig. 1, we show density of states functions, in the form ln[gn /g1 ], for tethered flexible chains of lengths N = 64, 128, 256, 512, 1024, and 1536. In each case, the plotted function is an average over results from multiple independent WL simulations. For N = 512, the density of states function is seen to span more than 370 orders of magnitude. The normalized density of states, Pn = gn / m gm represents the probability that in an athermal system (i.e., chain tethered to a hard wall) exactly n chains sites are in simultaneous contact with the wall. For small contact numbers n, we find Pn to be 0

N = 64 128 256

-500

512

0.0 -0.5

-1000 -1.0

1024

ln(Pn)/N

from below (T < Tc ),

ln[gn/g1]

204906-4

-1.5

-1500 0.0

0.0

(n-1)/N 0.5

0.2

1536

1.0

0.4

0.6

0.8

1.0

n/N FIG. 1. Logarithm of the density of states gn , relative to the value for n = 1, vs scaled contact number n/N for an end-tethered chain of length N = 64–1536, as indicated. Results shown for each N are an average over multiple independent WL simulations, although the maximum variation between simulations is smaller than

the line thickness. Inset: Same data as in the main panel, with Pn = gn / m gm , plotted as ln (Pn )/N vs (n − 1)/N. In this form, the density of states data for all but the smallest n/N are seen to collapse onto a master curve. This scaling is used to extend the densities of state for chains with N > 512 over the full energy range, as shown by the dashed lines in the main plot.


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20

Im(w)

10

0

-10

-20 20

2

2

10

Im(w)

0 2

0 2

4

4

0

-10

-20 -20

-10

0

10

Re(w)

20 -20

-10

0

10

20

Re(w)

FIG. 2. Partition function zeros in the complex w-plane for an end-tethered chain of length N = 128 (a), 256 (b), 512 (c), and 1024 (d). Positive physical ∗ temperatures correspond to the real axis with Re(w) > 1 where y = Re(w) = e1/T . The gn shown in Fig. 1 are the coefficients for the ZN (T) polynomials whose roots are shown here. The number of roots in each case is N − 1 and in (a), (b), (c), and (d) the number of roots falling outside the plot range is 6, 15, 29, and 55, respectively. The insets in (c) and (d) show the root structure in the neighborhood of the positive real axis. Leading zeros are colored red.

independent of chain length, in agreement with the fact that the average number of surface contacts is independent of N for desorbed chains. For large contact numbers, the chain behaves like a set of independent beads. In the inset to Fig. 1, we show the logarithm of the probability scaled by chain length N, ln (Pn )/N, as a function of (n − 1)/N. In this form, the density of states functions for different N are found to fall onto a single master curve for large contact numbers. Deviations from this scaling behavior become apparent when the contact numbers approach the adsorption transition values (which is near N1/2 ). This scaling behavior allows us to extend the partial density of states functions computed for N > 512 to the full energy range (as shown by the dashed lines in the plot). As shown in Eq. (5), the canonical partition function for the N-mer chain ZN (T) can be written as a polynomial of order N − 1 with polynomial coefficients gn /g1 . Due to the large order of the polynomials for long chains, accurate methods are required to compute the roots, which leads us to use MATHEMATICA53 for these calculations. The zeros of the ZN (T) polynomials, obtained from Fig. 1 density of state functions, are shown in the complex w-plane in Fig. 2. The N − 1 roots making up each of these “root maps” are exactly symmetric with respect to the Re(w)-axis and the mean of the real parts ak = Re(wk ) is negative as expected from the above noted sum rule.

The root maps shown in Fig. 2 exhibit a characteristic structure that becomes more defined with increasing chain length. An approximately circular ring of roots surrounds a root-free region near the origin and is terminated near the real axis by intersections with two flaring lines of roots. These lines of roots define a fan shaped boundary to a root free zone for Re(w) > Re(wl ), where wl (N ) is the leading root (i.e., the root closest to the positive real axis). With increasing chain length the leading roots move closer to the Re(w)-axis and the root density along the boundaries increases. This is consistent with the Yang-Lee model for phase transitions, where roots approach the positive real axis with increasing system size and, in the thermodynamic limit, divide the real axis into single phase regions separated by phase transitions. Although the root maps obtained from our multiple independent WL simulations can show significant variation in the location of some individual roots, the nearly circular and linear boundaries delineating a root free region about the Re(w)-axis, and in particular the leading roots, are highly reproducible. We give these leading roots (and the number of independent WL simulations performed) for each chain length N in Table I. We note that the zeros are a “local” property of the partition function polynomial and thus, different regions of the root map can be computed accurately using a truncated polynomial.46 In particular, the leading zeros we compute using truncated density of states functions for N > 512 are identical to the



J. Chem. Phys. 141, 204906 (2014)

TABLE I. Real and imaginary parts of the leading partition function zeros for tethered BF model N-mer chains obtained from nrun independent WL simulations. The numbers in parentheses are uncertainty estimates (standard error) in the last digit shown. Im(wl )

nrun

16 20 24 32 40 48 64 80 96 128 160 192 256 320 384 512 640 768 1024 1280 1536

2.582(1) 2.733(1) 2.816(1) 2.895(3) 2.926(2) 2.940(2) 2.940(1) 2.933(1) 2.926(3) 2.907(3) 2.884(2) 2.867(3) 2.850(3) 2.833(2) 2.816(2) 2.800(2) 2.787(5) 2.776(2) 2.764(6) 2.750(3) 2.743(5)

2.215(2) 1.948(1) 1.745(1) 1.469(3) 1.286(2) 1.156(2) 0.971(2) 0.850(2) 0.765(2) 0.651(2) 0.572(2) 0.514(3) 0.443(4) 0.387(2) 0.354(3) 0.305(3) 0.271(4) 0.242(3) 0.210(3) 0.181(3) 0.169(5)

12 12 12 12 12 12 12 12 12 18 10 16 10 10 10 16 10 12 10 10 8

leading roots obtained when we extend the density of states across the full energy range. We have investigated this issue in some detail for N = 512 and computing the density of states out to energy level N/2 (or even N/3) is more than sufficient to obtain the correct leading partition function zeros for long chains. As one check on our numerics, we compare the average energy (order parameter) and specific heat computed from the partition function zeros, via Eqs. (8) and (9), with the same quantities computed directly from the density of states as done in Refs. 11 and 51. In all cases the results agree. In Fig. 3, we show the adsorption transition order parameter and specific heat functions for a range of chain lengths N. The evolution of the order parameter from MN = 1 (complete adsorption) to MN ≈ 0 (desorbed state) defines the transition region which is characterized by a very broad specific heat peak. With increasing N the specific heat peaks grow and move to higher temperature which correlates with the movement of the leading partition function zeros wl (N) toward both smaller Im(w) and Re(w) values. The asymmetry of the specific heat peaks is a reflection of the asymmetric root structure in the vicinity of the wl (N).

B. Analysis of the leading zeros

To determine the location and nature of the polymer adsorption transition in the N → ∞ limit, we examine the approach of the leading partition function zeros toward the positive real axis using Eqs. (12) and (13). Each of these equations involves two unknowns, the transition temperature Tc∗ = 1/ ln(yc ) = 1/βc and the crossover exponent φ,

MN(T)

Re(wl )

0.3

(a) N = 64 –––– 128 –––– 256 –––– 512 –––– 1024 –––– 1536 ––––

0.5

0.2 0.1 0.0 0.8

1.0 T* 1.2

0.0 1.5

CN(T) / NkB

N

1.0 MN(T)

204906-6

512

(b)

1.0

N = 32

0.5

0.0 0.0

0.5

T*

1.0

1.5

FIG. 3. (a) Polymer adsorption transition order parameter MN (T) and (b) specific heat per monomer CN (T)/NkB vs reduced temperature T ∗ for an endtethered chain of length N = 64–1536 in (a) and N = 32–512 in (b), as indicated. These results are computed from partition function zeros via Eqs. (8) and (9) and are identical to results computed from the density of states. The inset in (a) shows the approach of the order parameter to zero. The vertical dashed lines in the (a) inset and in (b) locate the N → ∞ adsorption transition temperature Tc∗ found in this work.

however, since the equations are complex we can use the real and imaginary parts to construct independent equations for these two variables. For example, eliminating φ between the real and imaginary parts of Eq. (12), gives Im(wl ) = (d2 /d1 )(Re(wl ) − yc ). Similarly, Eq. (13) can be written as τ l = (d2 /d1 )(β l − β c ). Thus, we expect a linear correlation between the real and imaginary parts of the wl and β l + iτ l leading zeros with identical slopes and intercepts locating the N → ∞ transition point yc or β c , respectively. Such plots of the wl and β l + iτ l leading zeros are shown in Figs. 4(a) and 4(b). In both cases, there is pronounced curvature in the data, however, for the longest chain lengths (N > 512) we do see the expected linear correlation associated with the asymptotic scaling regime. Linear fits to the large N data in Figs. 4(a) and 4(b) give slopes of 2.1(3) and 1.9(2) and intercepts of yc = 2.66(2) and β c = 0.978(6), respectively. Agreement of these slopes within uncertainty supports our construction, however, the large error bars on the intercepts (due, in part, to the small data set being fit) lead to an imprecise determination of the transition temperature. We can get an improved constraint on the transition point by noting that the complete set of leading roots is very well described by a curve quadratic in the imaginary part (i.e., the Fig. 4(a) data set is well fit by Re(wl ) = yc + c1 Im(wl ) + c2 Im(wl )2 with an analogous form for the Fig. 4(b) data set). We comment on the meaning of this particular quadratic form at the end of this section. These quadratic fits, shown by the dashed lines in Fig. 4, give intercept values of yc = 2.657(10) and β c = 0.974(3) which correspond to transition temperature estimates of Tc∗ = 1.024(4) and 1.027(3). Since the quadratic


204906-7


J. Chem. Phys. 141, 204906 (2014)

0.5

1.5 N =32

(a)

quadratic fit

1.0

scaling behavior:

N =32

(b)

1

quadratic fit

0.4

Im(wl),

= 0.535(15)

l

~N

64

64 l

Im(wl)

0.3 128

128

0.1

0.2

0.0 2.6

2.8

3.0

512

0.1

yc = 2.657(10) c = 0.977(4)

1536 c

3.2

0.0 0.9

1.0

Re(wl)

1.1

1.2

FIG. 4. Leading partition function zeros for end-tethered chains of lengths N = 32–1536, as indicated. (a) Results in the complex w-plane. (b) Results in the complex inverse temperature plane ln(w) = β + iτ . In each case, the intercepts of the linear and quadratic fits (solid and dashed lines, respectively) with the real axis provide estimates for the N → ∞ transition temperature Tc∗ = 1/βc .

correction is smaller for the β l + iτ l zeros, we use the value of Tc∗ = 1.027(3) obtained from that data set as our best estimate of the transition temperature. To obtain the crossover exponent, we use the imaginary parts of Eqs. (12) and (13) which read Im(wl ) = d2 N −φ and τ l = (d2 /yc )N−φ , respectively. We plot Im(wl ) and τ l vs N in Fig. 5(a) and show power law fits to the N ≥ 320 data. The resulting exponent estimates are φ = 0.535(15) and φ = 0.510(15) which just agree within the error bars. In the scaling regime, we expect the ratio Im(wl )/τl to equal yc and for our longest chains we find this ratio to be roughly 2.75(10) in agreement with the yc value obtained above. An alternate means of determining the scaling exponent for power law behavior q(N) ∼ Nφ (where q(N) is any quantity of interest), and also identifying the onset of the scaling regime, is to define an N-dependent effective exponent given by20–22

( > 0) the effective exponent given by Eq. (16) will have the asymptotic behavior c , N

0.56

0.52

(b) 0.48 30

100

1000

N

0.56

0.52

(c) 0.48 0.00

0.04

–1/2

0.08

N

FIG. 5. (a) Scaling plot showing the imaginary part of the leading partition function zeros Im(wl ) (circles) and τ l (squares) vs chain length N. The solid lines are power law fits to these data for N ≥ 320 yielding values for the crossover exponent φ as indicated. (b) Effective crossover exponent φ eff (N), computed from Im(wl ) (circles) and τ l (squares) with Eq. (16), vs chain length N. The dashed lines are meant as a guide for the eye. (c) φ eff (N) (as in (b)) plotted vs N−1/2 for N ≥ 192. The lines are linear fits to the data shown.

(16)

We construct effective exponents φ eff (N) for Im(wl ) and τ l in Fig. 5(b). For short chains strong corrections to scaling are evident, however, for N > 300 the exponent constructed from τ l appears to reach a plateau of φ eff (N) ∼ 0.515 with the exponent from Im(wl ) showing a slow decay toward this value. This onset of scaling behavior near N ∼ 300 was used to set the N-range for the power law fits in Fig. 5(a). The increase in the error bars for φ eff (N) with increasing N is due to the increased fractional uncertainty in the zeros, whose values decrease with increasing N. The slow decay of φ eff (N) obtained from the Im(wl ) suggests that corrections to scaling need to be considered here. If we assume the corrections to scaling are of the form54, 55 c (17) q(N ) ∼ N φ 1 + N

φeff (N ) = φ +

1000

N

1.3

l

φeff (N ) = ln[q(2N)/q(N/2)]/ ln(4).

= 0.510(15)

100 0.60

= 0.974(3)

eff(N)

512 1536

(a)

256

256

eff(N)

0.5

(18)

where c and c are non-universal constants. Thus, a plot of φ eff (N) vs N− should, for large N, yield a straight line whose intercept gives the true (N → ∞) exponent φ. We show such an analysis in Fig. 5(c), where we assume the corrections to scaling go approximately at = 1/2.20 The linear fits shown in this plot give intercept values of φ = 0.522(20) and φ = 0.515(25) for the Im(wl ) and τ l results, respectively. These intercept values show some variation with the number of data points included in the analysis and the assumed value of , but the result φ = 0.515(25) covers most of this variation and we take this value as our best estimate of the crossover exponent given by the leading partition function zeros. We have only included the data for N ≥ 192 in Fig. 5(c) since the shorter chain results show a distinctly different scaling behavior than the long chain data, as might be expected for higher order correction-to-scaling terms. If we include the full data set in the Fig. 5(c) analysis, the intercept values are φ ≈ 0.52 and 0.49 for the Im(wl ) and τ l results, respectively. Our determination of the transition temperature Tc∗ and crossover exponent φ for the scaling of the leading partition



C. Critical exponents

Having obtained the critical temperature and crossover exponent for the polymer adsorption transition from the leading partition function zeros, we can proceed to carry out finite size scaling to determine the order parameter and specific heat critical exponents. We will first check the self-consistency of our results by examining the scaling of the number of surface contacts na (N, T) which, according to Eq. (2), at Tc should follow na (N, Tc ) ∼ Nφ . A standard technique for determining both Tc and φ is to construct φ eff (N, T), using q = na (N, T) in Eq. (16), for a range of temperatures and to study the large N behavior of these functions.20–22 At Tc we expect φ eff (N, Tc ) → φ true (i.e., a constant) for large N whereas for T < Tc or T > Tc we expect to see deviations from φ true which increase with N. We show this construction in Fig. 6(a) for a range of temperatures spanning our partition function zero result of Tc = 1.027(3). The variation of φ eff (N, T) across the temperatures consistent with our Tc result is not insignificant and in Fig. 6(b) we show a correction to scaling plot for these temperatures. The asymptotic exponent values extrapolated in Fig. 6(b) range from ∼0.47 → 0.53 and, for our range of chain lengths, there is no clear means to distinguish which value in this range is more correct. We note that Hsu has carried out this type of analysis for the BF model using PERM for chains up to N = 104 and concludes that Tc = 1.0298(3) and φ true = 0.487(5).57 Hsu’s57 high precision results depend on very long chain data which are currently out of reach for our simulation methods. Clearly, for our data set, extracting Tc and φ from plots like Fig. 6 would be a delicate task. However, Fig. 6 does show that our partition function results of Tc = 1.027(3), φ = 0.515(25) are consistent with a more standard analysis.

T = 1.020

na ~ N

0.55

1.025 eff(N)

function zeros is similar to, but slightly different from the standard approach found in the literature.37, 39–41, 44, 46, 56 The usual procedure is to first use the imaginary part of Eq. (12) to obtain the exponent φ, and then insert this exponent into the real part of Eq. (13) to determine β c . This method recognizes that the real part of the leading zeros obeys better scaling in the β-τ plane than in the complex w-plane (as is clearly seen in Fig. 4). Our results for φ eff (N) suggest that the imaginary part of the leading zeros also displays better scaling behavior in the β-τ plane so that τ l may be preferable to Im(wl ) for the determination of φ. Finally, the particular quadratic form followed by both sets of complex leading zeros is consistent with a correction to scaling of the type shown in Eq. (17) for Eq. (12), with ≈ φ. If the amplitude of the correction term for the imaginary part is sufficiently small, combining the real and imaginary parts of Eq. (12) will yield the quadratic form Re(wl ) = yc + c1 Im(wl ) + c2 Im(wl )2 shown in Fig. 4(a). The mapping from the w to the ln(w)=β + iτ complex plane mixes the real and imaginary parts of the wplane zeros thus introducing order N−2φ terms into both β l and τ l . Since expansion of the logarithm in this mapping also generates a leading correction term of order N−2φ the two sameorder corrections must (almost) completely cancel for τ l and partially cancel for β l .

J. Chem. Phys. 141, 204906 (2014)

0.50

1.027 1.030

0.45

(a) 30 0.53

1.035

100

N

1000

T = 1.025 eff(N)

204906-8

1.027

0.50

1.030

(b)

0.47

0.00

0.05

–1/2

N

0.10

FIG. 6. Effective crossover exponent φ eff (N), computed from na (N, T) with Eq. (16), for a range of temperatures, as indicated, vs chain length N in (a) and vs N−1/2 in (b). The dashed lines in (a) are a guide for the eye only while the solid lines in (b) are linear fits to the data shown.

We now use our critical temperature result of Tc = 1.027(3) to study the scaling of MN (T) and C(T)/NkB at T = Tc . Plots of these function vs chain length N are shown in Figs. 7(a) and 7(b). In Fig. 7(a), the power law fit to φ = 0.498(7). MN (Tc ) vs N yields a crossover exponent of β Using the value φ = 0.515(25), we obtain an order parameter = 0.97(5) in agreement with the exponent exponent of β = 1/φ − 1 = 0.94(5). In Fig. 7(b), the identity result of β power law fit to C(Tc )/NkB vs N yields an exponent of αφ = 0.014(13). Again, using the result φ = 0.515(25) this

scaling behavior: 0.06

MN(Tc) ~ N

(a)

= 0.498(7) [ =0.97(5)]

0.02 scaling behavior: 0.60

0.54

CN(Tc)/Nk ~ N

(b)

300

= 0.014(13) [ =0.03(3)] 600

N

1000

2000

FIG. 7. Log-log scaling plots at the transition temperature Tc∗ = 1.027 showing (a) the adsorption order parameter MN (Tc ) and (b) the specific heat CN (Tc )/NkB vs chain length N. The solid lines are power law fits to the data shown. Using φ = 0.515(25) the associated critical exponents are found to be β = 0.97(5) and α = 0.03(3) consistent with the exponent identities discussed in the text.


204906-9


1.6

10

Tc = 1.027

T < Tc 1.1

ƒ(x)

T > Tc 0.6 0.0

0.5

x

= 0.515 T < Tc 1.0

1

0.1

T > Tc

N = 1536 –––– 1024 –––– 512 –––– 320 ––––

0.1

x 1

1

10

FIG. 8. Contact number scaling function f(x) = na (N, T)/Nφ vs the scaling variable x = |κ|Nφ , where κ = (T − Tc )/T, for chain lengths N = 320–1536, as indicated. The dashed lines show the expected large x scaling forms, given in Eq. (3), for both the T < Tc and T > Tc branches. The inset shows, in linear scale, an expanded view of f(x) for x ≤ 1.

gives a specific heat exponent of α = 0.03(3). This value of α is consistent with the exponent identity α = 2 − 1/φ and = 0.06(5). Our actual uncertainties in the exponents β α are somewhat larger than found in the Fig. 7 analysis due to the uncertainty in Tc . Our best estimates for these critical = 0.97(5) and α = 0.03(4). exponents are β

D. Surface contact scaling function

The values of Tc∗ and φ, which characterize the polymer adsorption transition, can be used to investigate the more general scaling behavior of the number of surface contacts na (N, T) in the neighborhood of the transition as given in Eq. (2). In Fig. 8, we plot the average surface contact number scaling functions f(x) = na (N, T)/Nφ vs the scaling variable x = |κ|Nφ , where κ = (T − Tc )/T, for chains of lengths N = 320–1536. The data are presented in log-log format to assess the expected large x power-law scaling forms given in Eq. (3). In this plot, the data separate into distinct branches for temperatures above and below the transition and we see very good data collapse for all N values on both branches up to x ≈ 3. For larger x, the results for the different chain lengths begin to show systematic variation, as the shorter chain results sequentially “peel off” from the scaling curve. For x > 10, our longest chain results begin to deviate from the expected N → ∞ asymptotic behavior. In the inset to Fig. 8, we show this scaling function near the transition and, again, good data collapse is observed in this region. This data collapse and large x scaling is further evidence for the accuracy and consistency of our transition temperature and crossover exponent values.12, 14

IV. DISCUSSION

In this work, we have studied the adsorption transition of an isolated flexible polymer chain end-tethered to an attractive surface. We performed Monte Carlo simulations with Wang-Landau sampling to determine the density of states of a tethered chain in the bond-fluctuation model, constructed the canonical partition function, which is a polynomial in the

J. Chem. Phys. 141, 204906 (2014)

temperature variable y = eβ , and computed the zeros of this polynomial. Near the infinite-chain adsorption transition temperature Tc , the behavior of thermodynamic functions is dominated by the contribution of the leading partition function zeros, which approach the real axis in the thermodynamic limit, N → ∞. Our scaling analysis for the leading zeros leads to the well known polymer adsorption result na (N, Tc ) ∼ Nφ , which allows us to identify the crossover exponent used in the finite size scaling of the leading roots with the polymer adsorption crossover exponent. We apply finite-size scaling to study the movement of the leading zeros toward the positive real axis as the chain length increases and to determine details of the polymer adsorption transition in the thermodynamic limit. In particular, we obtain values for both the transition temperature Tc∗ and the crossover exponent φ by independent analyses of the scaling behavior of the real and imaginary parts of the leading zeros. Our result for the transition temperature of the BF model is Tc∗ = 1.027(3) which agrees within error bars with the result obtained by both Descas et al.14 and by Hsu57 for the same model. As noted by Descas et al.,14 φ and Tc values tend to be correlated in the analysis of MC data and more than one pair of values may result in good scaling plots. This helps explain the persistence in the literature of different values for the crossover exponent φ, with the most popular being φ ≈ 0.5 and 0.58. The approach presented here overcomes the difficulties associated with correlated Tc and φ values since the exponent φ is obtained directly from the scaling of the imaginary part of the leading complex zeros wl (N) and τ l (N) shown in Fig. 5. Our result of φ = 0.515(25) is consistent with Hsu’s recent (unpublished) value of φ = 0.487(5),57 but is just outside the uncertainty range of the current most precise estimate of φ = 0.484(2) obtained by Grassberger and others.20–22 We note that a φ value smaller than 0.5 yields > 1.0 and a specific heat an order parameter exponent of β exponent of α < 0. This implies that, in the limit N → ∞, CN (T)/NkB does not diverge and both M(T) and CN (T)/NkB go smoothly to zero as T → Tc from below. Since the set of long chain, high precision results for φ in the literature were all obtained using the same techniques we believe it is useful to investigate the adsorption transition for long chains using alternate methods that evaluate different quantities. This is part of the motivation for this study, although better statistics are clearly needed to make strong statements about exponents. In conclusion, since modern simulation techniques provide direct access to the density of states, and thus the partition function of many body systems, analysis of the partition function zeros has become a practical calculation tool. As shown here, when combined with finite size scaling methods, this first principles approach can provide much detailed information about phase transitions and should find many applications to polymer systems. ACKNOWLEDGMENTS

M.P.T. thanks Hsiao-Ping Hsu for helpful discussions and for generously sharing unpublished data. Financial support


204906-10


from Hiram College and the National Science Foundation (NSF) (DMR-1204747) are gratefully acknowledged. 1 T.

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