Introduction to polymer physics Lecture 4

Transcription

1 Introduction to polymer physics Lecture 4 Boulder Summer School July August 3, 2012 A.Grosberg New York University

2 Center for Soft Matter Research, New York University Jasna Brujic Paul Chaikin David Grier David Pine Matthieu Wyart

3 Ilya M.Lifshitz ( ) Evgenii M.Lifshitz ( ) Fermi-surfaces, metals, disordered systems (optimal fluctuations, Lifshitz tails), supersolids, polymers VdW interactions, Lifshitz point

4 Lecture 4: A bit more mathematics Course outline: Ideal chains (Grosberg) Real chains (Rubinstein) Solutions (Rubinstein) Methods (Grosberg) Closely connected: interactions (Pincus), polyelectrolytes (Rubinstein), networks (Rabin), brushes (Zhulina), semiflexible polymers (MacKintosh) This lecture outline: This lecture outline: Comment on Flory- Huggins Edwards Hamiltonian Polymers and classical mechanics (Semenov) Polymers and quantum mechanics (Lifshitz) Block-copolymers (Leibler)

5 Comment on Flory-Huggins (1) Flory-Huggins interpretation: Lifshitz (-like) interpretation: More phenomenological writing:

6 Comment on Flory-Huggins (2) Van der Waals equation of state: Van der Waals equation of state:

7 Comment on Flory-Huggins (2) FH free energy VdW equation of state Exercises: (1) Re-interpret common tangent construction in terms of equation of state; (2) Develop a competing theory to Flory-Huggins based on VdW equation, discuss its advantages and disadvantages.

8 Ideal chain To To begin with, suppose we we are interested in in only relatively few representative points: Position vectors of of representative points are {x {x i } i Suppose pieces between representative points are longer than Kuhn segment, then bond vectors {y {y i =x i =x i+1 i+1 -x -x i } i are (i) (i) independent, and (ii) Gauss distributed: g(y)=(2πa 2 /3) -3/2-3/2 exp[-3y 2 /2a 2 ]. ]. Here a is is not Kuhn segment, it it depends on on how many representatives we we have. x k x j y i

9 Ideal chain (cont d) Given independence and g(y)=(2πa 2 /3) -3/2-3/2 exp[-3y 2 /2a 2 ], ], probability distribution of the whole conformation on this level of description reads G(x 1,x,x 2,,x n )=g(y 1 )g(y 2 ) g(y n-1 n-1 ). ). This multiplicative structure is is the signature of ideal chain. x k x j y i

10 Generalization In general, {x i } i may include more than just coordinates of representative points, e.g., orientation/direction, internal state (helixcoil), etc. G(x 1,x,x 2,,x n )=g(x 1,x,x 2 )g(x 2,x,x 3 ) g(x n-1 n-1,x,x n ). ). This multiplicative structure is is still the signature of ideal chain. Here g(x,x ) is is conditional probability, it it satisfies g(x,x )dx =1 Note of care: bonds are NOT in in equilibrium

11 Comments: Although g(y)~exp[-3y 2 /2a 2 ] is is exponential, it it is is not right to to interpret g(y) as as equilibrium Boltzmann probability exp(-energy/k B T), because it it is is not energy, it it is is entropic spring. As As a result, there is is no no temperature. If If G and g are viewed as as TD probabilities, then normalization condition means that free unrestricted ideal chain is is taken as as zero level of of free energy. Imagine we we want to to switch to to a less detailed description, with fewer representative points {X {X i }; i }; then x k x j y i

12 Equation Green s function We can find it by integrating over all intermediate {x i }. i }. It satisfies equation

13 Operator If x is just coordinates and Then (think of of k-representation, exp[-k 2 2 a 2 2 /6]) /6]) Under proper conditions G n+1 n+1 G n + n G n and exp((a 2 /6)Δ) 1+(a 2 /6)Δ then n G n =(a 2 /6)ΔG n

14 Diffusion equation Under proper conditions G n+1 n+1 G n + n G n and exp((a 2 /6)Δ) 1+(a 2 /6)Δ then n G n =(a 2 /6)ΔG n Diffusion equation returns the well known solution G n (x,x )~exp[-(3/2na 2 )(x-x ) 2 ] no new information: what we put in in - we got out Challenging exercise Challenging exercise: Find operator g for worm-like chain. Hint: in this case, x={x,n}

15 Continuum limit Given g(y)~exp[-3y 2 /2a 2 ], ], probability of conformation G(x 1,x 2,,x n )=g(y 1 )g(y 2 ) g(y n-1 n-1 ) ~exp[-3[(x n-1 1 -x 2 ) 2 +(x 2 -x 3 ) 2 + +(x n-1 -x n ) 2 ]/2a 2 ] =exp(- [3( x/ s) 2 /2a 2 ]ds) Looks very much like exp[(i/ħ) [kinetic energy]dt]

16 Green s function in continuum limit When we integrate over all intermediate {x i } in G(x 1,x 2,,x n )~exp[-3[(x 1 -x 2 ) 2 +(x 2 -x 3 ) 2 + +(x n-1 -x n ) 2 ]/2a 2 ] =exp(- [3( x/ s) 2 /2a 2 ]ds) we arrive at path integral: G n (x,x )= exp(- [3( x/ s) 2 /2a 2 ]ds) Dx(s) This is the partition sum of our polymer the sum over all conformations. In this continuum form, it satisfies n G n =(a 2 /6)ΔG n Thus, we can do cont. limit via G n+1 G n + n G n and exp((a 2 /6)Δ) 1+(a 2 /6)Δ or via path integral

17 A bit more general: φ(x) Still assume ideal polymer, but suppose each monomer (really -- representative point) carries energy φ(x) which depends on position (or in general on x). Instead of G(x 1,x 2,,x n )=g(x 1,x 2 )g(x 2,x 3 ) g(x n-1,x n ). we now have (assuming k B T=1) G(x 1,x 2,,x n )=g(x 1,x 2 )e -φ(x 2 ) g(x 2,x 3 )e -φ(x 3 ) e -φ(x n-1 ) g(x n-1,x n ). exp[-φ(x 1 )] exp[-φ(x n )]

18 Continuum limit with φ(x) We had G(x 11,x,x 2,,x n n )~exp[-3[(x 11 -x -x 2 ) 2 +(x 2 -x -x 3 ) 2 + +(x n-1 n-1 -x -x n n ) 2 ]/2a 2 ] =exp(- [3( x/ s) 2 /2a 2 ]ds) Now we we insert φ(x) in in the exponential:

19 Continuum limit with φ(x) φ(x We had 1 ) φ(x 2 ) φ(x 3 ) φ(x n ) G(x 11,x,x 2,,x n n )~exp[-3[(x 11 -x -x 2 ) 2 +(x 2 -x -x 3 ) 2 + +(x n-1 n-1 -x -x n n ) 2 ]/2a 2 ] =exp(- [3( x/ s) 2 /2a 2 ]ds) Now we we insert φ(x) in in every point in in the exponential:

20 Continuum limit with φ(x) φ(x We had 1 ) φ(x 2 ) φ(x 3 ) φ(x n ) G(x 11,x,x 2,,x n n )~exp[-3[(x 11 -x -x 2 ) 2 +(x 2 -x -x 3 ) 2 + +(x n-1 n-1 -x -x n n ) 2 ]/2a 2 ] =exp(- [3( x/ s) 2 /2a 2 ]ds) Now we we insert φ(x) in in every point in in the exponential and arrive at at G(x 11,x,x 2,,x n n )~exp(- [3( x/ s) 2 /2a 2 +φ(x(s))]ds) Looks very much like exp[(i/ħ) [kinetic energy potential energy]dt] except for disturbing sign difference!

21 Green s function with φ(x) Green s function is is the sum over all all conformations with fixed ends path integral: G n n (x,x )= exp(- [3( x/ s) 2 /2a 2 +φ(x(s))]ds)dx(s) This is is partition sum of of the chain with fixed ends Looks very much like, but simpler than exp[(i/ħ) [kinetic energy potential energy]dt] Dx(t) except for disturbing sign difference!

22 Applicability of cont. limit Nothing changes significantly over the length scale of of Kuhn segment. Let φ(x) change over length scale λ>>b. Then we can choose equivalent chain with λ>>a>>b Because a>>b, we we can approximate g(y) as as Gaussian Because λ>>a, we we have a 2 Δ~a 2 /λ /λ 2 <<1, and we we can expand exp((a 2 2 /6)Δ) 1+(a 2 2 /6)Δ /6)Δ

23 Edwards Hamiltonian Green s function is is the sum over all all conformations with fixed ends path integral: G n n (x,x )= exp(- [3( x/ s) 2 /2a 2 +φ(x(s))]ds)dx(s) External or or self-consistent field Two body interaction Very short ranged potential

24 Example 1 Imagine a polymer pinned down by one end at the top of a potential bump (e.g., Flagstaff mountain), and hanging down from there. If the potential is strong and temperature low, the most probable conformation will correspond to the minimum of the expression in the exponent G(x 1,x 2,,x n )~exp(- [3( x/ s) 2 /2a 2 +φ(x(s))]ds) i.e. x(s) corresponds to min [3( x/ s) 2 /2a 2 +φ(x(s))]ds A.N. Semenov, Sov. Phys. JETP, 61, p. 733, 1985

25 Example 1 (cont d) i.e. x(s) corresponds to min [3( x/ s) 2 /2a 2 +φ(x(s))]ds This looks exactly like: if classical mechanics applies, then particle chooses trajectory x(t) such as to minimize action min [m( x/ t) 2 /2-U(x(t))]dt Dictionary: s->t, 3/a 2 -> m, x/ s ->v, φ(x)->-u(x). It is surprising, but has physical meaning that φ(x) is MINUS potential.

26 Example 1 (cont d further) Optimal trajectory in mechanics satisfies F=ma or mv 2 /2+U=const Optimal conformation of a polymer satisfies Per little chain piece δs: i.e., chain stretching energy is large at large potential and small where potential is small, thus φ(x) is analogous to U(x). Free energy F=-k B TlnG reads

27 First serious application Consider a brush, and look at at one chain there. It It cannot wiggle sideways because of of peer pressure. There is is some sort of of a (selfconsistent) field φ(x), where x is is height above the plane, which favors chain rushing up, to to better life. What is is this field? Every chain is is a particle that starts at at x=0 and reaches some height x=h, different for different chains, in in one and the same time t=n φ x

28 Isochronous pendulum Reverse: the time (period) does not depend on on height (amplitude). Isochronous pendulum!!! For a harmonic oscillator, ω=(k/m) 1/2 1/2 independently of of amplitude; this is is true for harmonic oscillator ONLY. For real pendulum, period becomes longer at at very large amplitudes. As candelabra swings more, does it also swing faster? φ x

29 Potential in the brush Thus, brush is is like a harmonic oscillator, U(x)~x 2 or or φ(x)~-x 2 Famous parabolic profile It It is is very clever that φ(x)->-u(x), because φ(x) in in the brush must be be a bump We can even establish coefficient in in φ(x)=-kx 2 /2, because quarter-period is is N: N: k=(3π 2 /4)(k B T/N 2 a 2 ) In this form, the agument is due to S.T.Milner, T.A.Witten, and M.E.Cates Europhys. Lett., 5 (5), pp (1988) A Parabolic Density Profile for Grafted Polymers. E.Zhulina in her lecture is likely to present another independently developed view on the same topic

30 Potential and density Given the harmonic oscillator potential the particle energy conservation reads Total energy is is established from boundary condition that particle arrives to to the amplitude position at at zero velocity, or or chain is is not stretched at at the end: Density: #monomers δs δsper height interval δx: δs/δx, i.e., inverted velocity:

31 Density Inverted velocity is is density of of one chain whose end is is located at at given height h: h: Different chains end at at different heights h. h. Given σ chains per unit area, total density is is the sum: where H is is total brush height (maximal h), h), and ψ(h) is is the fraction of of chains ending at at h both to to be be found from the self-consistency condition which relates field φ(x) to to the total density ρ(x): φ(x)-φ(h) = k B Tv Tvρ(x) Exercise: Show that for a chain with free end at the height h, monomer number s is located on average at the height x=h sin(πs/2n)

32 Self-consistency Self-consistency condition relates field φ(x) to to the total density ρ(x): φ(x)-φ(h) = k B Tv Tvρ(x): Since we we know both φ(x) and ρ 11 (x,h), we we get equation: The rest is is a few lines of of algebra. Integrate both sides over h from 0 to to H to to get This completes the mean field solution.

33 Results for brush: Figure from Milner- Witten-Cates, EPL 1988 ρ(x)/ρ(0) σa 2 =0.04 σa 2 =0.08 ψ(h) x/h,h/h

34 Applicability of mean field Mean field applies if thermal blob g T (the distance needed to develop selfavoidance correlations) is larger than the distance g to the neighboring chains. The former is estimated from vg 1/2 /a 3 ~1, the latter from a 2 g~1/σ Thus the conditions: v 2 /a 6 <<σa 2 <<1

35 Green s function with φ(x) (repeated) Green s function is is the sum over all all conformations with fixed ends path integral: G n n (x,x )= exp(- [3( x/ s) 2 /2a 2 +φ(x(s))]ds)dx(s) This is is partition sum of of the chain with fixed ends Looks very much like, but simpler than exp[(i/ħ) [kinetic energy potential energy]dt] Dx(t) except for disturbing sign difference! In In QM, it it satisfies Schroedinger equation. By By analogy, G G n n / n = (a (a 2 /6)ΔG nn -φ(x)g nn

36 Bilinear expansion Since it it looks like Schroedinger equation, let us us proceed as as we we are taught in in QM: expand x-dependence over eigenmodes Then

37 Ground state dominance If If there is is discrete energy level, then largest l l (smallest energy) dominates and where λ and ψ are ground state energy and wave function of of Reminder from QM: wave function of of ground state can be be always chosen positive (has no no zeros). Examples: chain in in a box, in in a tube, in in a slit, etc

38 Phase transition Remember that φ(x) -> -> φ(x)/k B T. T. Presence or or absence of of discrete level is is determined by by temperature: Challenging exercises: (1) (1) Suppose field f(x) is is some potential well in in 3D, but φ(x)=0 outside a certain finite volume (or rapidly goes to to zero). What is is the order of of phase transition? (2) Suppose field φ(x) depends only on on one coordinate, say z, z, and is is equal to to zero outside finite interval of of z; z; is is there phase transition? (3) Suppose field φ(x) depends only on on one coordinate, say z, z, and is is equal to to infinity at at z<0, zero at at z>a, and some negative value between z=0 and z=a; is is there phase transition?

39 Free energy, density Given that free energy is isf=tnλ To To find density, consider Thus, n(x)=ψ 2 (x) (assuming ψ is is properly normalized)

40 Entropy We know free energy is is F=TNλ and density n(x)=ψ 2 (x) Entropy: Notes: (1) (1) ψ is is an an order parameter; (2) S~Na 2 /R /R 2 is is recovered.

41 Self-consistency Particle which digs well for itself. Free energy: Minimize this with respect to to ψ, ψ, with Lagrange multiplier λ ensuring that ψ is is properly normalized: Equilibrium free energy:

42 Virial expansion for μ* μ* Virial expansion

43 Solution for large globule In In the interior, ψ=const: Equilibrium free energy: (1/2)Bn 2 +(1/3)Cn 3,, has minimum at at B<0 at at n~-b/c: exactly the same answer as as before. In In the surface layer, full non-linear PDE has to to be be addressed.

44 Challenging problem Consider polymer with excluded volume, but no no attractions (B>0,C>0) collapsed in in a potential box with infinite walls. What is is the profile of of the self-consistent potential μ*?

45 Summary Chain connectivity is one thing and interactions is the other: this is a possible and fruitful point of view. Quasi-classical limit for stretched chains. Ultra-quantum limit for collapse. One has to watch for physics not only doing magic estimates, but also performing routine calculations.