Partition function zeros and finite size scaling for polymer adsorption

Transcription

1 CompPhys13: New Developments in Computational Physics, Leipzig, Nov , 2013 Partition function zeros and finite size scaling for polymer adsorption Mark P. Taylor 1 and Jutta Luettmer-Strathmann 2 1 Dept. of Physics, Hiram College, Hiram, OH USA 2 Dept. of Physics, University of Akron, Akron, OH USA 20 N = 128 Im(w) 10 0 desorbed Re(w) adsorbed

2 Polymer adsorp.on transi.on: n s (T c ) ~ N φ tethered chain (par.ally adsorbed: T~T c ) cool system tethered chain (desorbed: T>>T c ) tethered site cool system tethered chain (fully adsorbed: T<<T c ) a9rac.ve surface (side view) = desorbed site = adsorbed site Eisenriegler, Kremer, Binder, JCP 77, 6296 (1982) [φ = 0.58] Metzger, Muller, Binder, Bashnagel, Macromol Theory Sim 11, 985 (2002) [lit. review: φ = ; φ=0.50] Grassberger, J Phys A: Math Gen 38, 323 (2005) [φ = 0.48] Decas, Sommer, Blumen, Macromol Theory Sim 17, 429 (2008) [φ = ] Klushin, Polotsky, Hsu, Markelov, Binder, Skvortsov, PRE (2013) [φ = 0.48]

3 Bond fluctua.on model for tethered chains Simple cubic lacce (a = 1) Bond lengths: Bead 1 fixed at (1,1,1) 108 bond vectors Interac.ons between beads i and j with distance r ij : hard core repulsion for r 2 ij < 4 Total energy: Interac.ons with the surface: contribu.on to the internal energy of ε s for z i = 1 E = n s ε n s = number of surface contacts T = k B T/ε = reduced temperature

4 Density of States and Wang-Landau Sampling I Density of States: g(e n ) = volume of configurational phase space for energy state E n Thermodynamics: microcanonical entropy: S(E) = k B lng(e) canonical partition function: Z(T) = g(e)exp(-e/k B T) iterate m levels m=20 is standard we need m>25 Wang-Landau algorithm*... an iterative simulation method to compute g(e n ): Starting w/ g(e n )=1, H(E n )=0 n, ƒ 0 =e Generate sequence of chain conformations using acceptance criteria: P acc (a b) = min 1, g(e a) g(e b ) Update DOS: g(e n ) ƒ m g(e n ) Update visitation histogram: H(E n ) H(E n )+1 When histogram ~flat... reduce modification factor: ƒ m+1 = (ƒ m ) 1/2 reset histogram to zero: H(E n ) = 0 n *Wang & Landau, PRL 86, 2050 (2001); PRE 64, (2001).

5 Wang-Landau Sampling II Success of the WL methods depends on underlying MC move set These "standard" moves easily sample most of configuration space: Single bead displacement Pivot But for good sampling of chain configurations near the grafting point we also use: Cut and Permute Causo, J. Stat. Phys. 108, 247 (2002)

6 Polymer Adsorption: Single Chain DOS and Specific Heat 0 N = 32 Canonical Analysis Partition Function: Z = g(e) e E/kT Probability: P(E,T) = g(e) e E/kT / Z ln[g n /g 1 ] Average Energy: E(T) = EP(E,T) Specific Heat: C(T) = d E(T) /dt n s /N n s = number of surface contacts C N (T) / Nk B N = 32 expected behavior: C inf /Nk B ~ T - T c -α T *

7 Polymer Adsorption: Single Chain DOS and Fraction Adsorbed ln[g n /g 1 ] N = n s /N n s = number of surface contacts expected behavior: M inf (T) ~ T - T c β 0.0 Canonical Analysis Partition Function: Z = g(e) e E/kT Probability: P(E,T) = g(e) e E/kT / Z Average Energy: E(T) = EP(E,T) Order Parameter: M(T) = n s (T) /N = E(T) /Nε M N (T) = n s (T) /N <n s (T)>/N T N =

8 Polymer Adsorption: Partition Function Zeros Model has discrete energy spectrum: -ε, -2ε,..., -Nε Partition function is a polynomial in y=e ε/kt : Z(T) = g(e) e E/kT = n g n y n or Z(T) = Π k (y-w k ) where w k = a k + ib k are the complex zeros of Z(T) Properties: complex zeros come in pairs a ± ib any real zeros must be negative sum of Re(w k ) is negative, i.e., k a k < 0 All thermodynamics can be expressed in terms of the zeros {w k } Example: Heat Capacity (physical temp. range: y > 1) C(y) k B 2 k ln Z max yw = β 2 = (ln y) 2 k β 2 k= 0(y w k ) 2 Zeros near real axis contribute most

9 Partition Function Zeros: Yang-Lee theory In 1952 C. N. Yang and T. D. Lee proposed a very general theory for phase transitions based on the distribution of the zeros of the grand canonical partition function in the complex plane. Non-analytic behavior of thermodynamic functions arises when some zeros move onto the positive real axis in the thermodynamic limit. In 1965 M. Fisher extended the approach to the canonical partition function and zeros in the complex temperature plane. Fisher, in Lecture Notes in Theoretical Physics (U. of Colorado Press, 1965). Yang & Lee, Phys. Rev. 87, 404, 410 (1952).

10 Yang-Lee theory of phase transitions System with two phase transi.ons Collapse and freezing of a SW chain imaginary axis complex par..on func.on zeros Region 1 expanded Region 2 collapsed Region 3 crystal real axis Yang & Lee, Phys. Rev. 87, 404, 410 (1952). real axis Taylor, Aung, Paul, PRE 88, (2013). high T low T

11 Roots maps for polymer adsorp.on follow Yang Lee behavior... leading zeros pinch down onto posi.ve real axis: Im(w) N = 64 imaginary axis (a) real axis N = 128 (b) Approach of the leading zeros towards the real axis: 1.6 N = Im(w) N = 256 (c) Re(w) N = desorbed (d) adsorbed Re(w) Im(w l ) Re(w l )

12 Finite size scaling of the leading zeros cri.cal point (real: y=e β =e 1/T ) crossover exponent Expected scaling form: w 1 y c ~ DN φ leading root (complex) complex constant (D=d a +id b ) Itzykson, Pearson, Zuber, Nucl. Phys. B 220, 415 (1983) Scaling of Im[w 1 ] gives φ: Im[w 1 ] ~ d b N φ Linear correla.on between Im[w 1 ] and Re[w 1 ] gives y c : Im[w 1 ] ~ (Re[w 1 ] y c )d b /d a Alternately, linear correla.on between Im[w 1 ] and β 1 gives β c : Im[w 1 ] ~ (β 1 β c )y c d b /d a Complex inverse temperature: ln(w 1 ) = β 1 + iτ 1

13 Adsorption transition: The crossover exponent φ 1 Imaginary part of leading zeros vs chain length scaling behavior: Im(w l ) ~ N!" Im[w l (N)] N!1/2 " = 0.535(11) N

14 Adsorption transition: The transition temperature T c 1.6 For long chains the leading zeros are expected to show a linear approach the Re(w) axis (or the β axis) N = N = 32 y = Re(w) = e 1/T β = 1/T 1.2 Im(w l ) 0.8 linear fit to N! y c = 2.68(1)! 0.0 c = 0.986(3) Re(w l ) 1.2 Im(w l ) linear fit to N! ! c = 0.984(2) ! l Result for transi.on temperature: T c = 1.016(2)

15 Finite size scaling for thermodynamics PFZ scaling form in complex temperature plane: ln(w 1 ) β c ~ (D/ y c )N φ leading root (complex: w=β+iτ) cri.cal point (real) complex constant (D=d a +id b ) crossover exponent Real part gives: β 1 β c ~ N φ 1 T c /T 1 ~ N φ Leads to scaling rela.ons: M N (T c ) ~ N β φ n s (T c ) = NM N (T c ) ~ N φ C N (T c ) /Nk B ~ N αφ β = 1+1/φ α = 2 1/φ And exponent iden..es: order parameter number adsorbed specific heat

16 Finite size scaling at T c : Number absorbed and order parameter n a (N,T c ) M N (T c ) scaling behavior: n a (N,T c ) ~ N!! = 0.529(12) scaling behavior: M N (T c ) ~ N #!" N N!" = 0.468(20) M N (T) 1.0 Agrees with Result from Im(w 1 ): φ = 0.535(11) M N (T) = n a (T)/N N = T Result for order parameter exponent: β = 0.875(41) Agrees with: β = 1 + 1/φ

17 Finite size scaling at T c : Specific Heat C N (T c )/Nk B 1 scaling behavior: C N (T c )/Nk ~ N!"!" = 0.062(7) C N (T) / Nk B N = N 1000 Result for specific heat exponent: α = 0.116(13) Agrees with exponent iden.ty: α = 2 1/φ T *

18 Polymer Adsorption: Summary of Results Transi.on temperature: T c =1.016(2) Crossover exponent: φ = 0.535(11) Specific heat exponent: α = 0.116(13) Order Parameter exponent: β = 0.875(41) Sa.sfy exponent iden..es: α = 2 1/φ β = 1 + 1/φ Caveat: Grassberger and Hsu et al. find φ = 0.48 studying very long lacce chains via the PERM algorithm. They suggest larger values of φ are caused by strong correc.ons to scaling. Note that φ = 0.48 gives a nega.ve α, indica.ng a non diverging specific heat. Grassberger, J Phys A: Math Gen 38, 323 (2005). Klushin, Polotsky, Hsu, Markelov, Binder, Skvortsov, PRE (2013).

19 Summary and Outlook Adsorption transition for a tethered polymer chain Findings: Partition function zeros display Lee-Yang behavior. Finite size scaling (FSS) of zeros locates T c and determines φ. FSS determines specific heat and order parameter exponents. To do: Include correction to scaling terms in this analysis. Carry out same analysis for a continuum chain model. Funding: NSF (DMR ) Special thanks to Wolfgang Paul's group for their hospitality! Happy "American" Thanksgiving