1. Universität Mainz, Germany
IRTG Lecture: Introduction into Self-Consistent Field Theories for Inhomogeneous (Co)polymer blends
Friederike Schmid
Universität Mainz, Germany

2. Copolymer Systems +
Polybutadiene
Polystyrene

3. Styrene-butadiene copolymers
Copolymer Systems
Styrene-butadiene copolymers
(Spitzner, Magerle)

4. Self-Organizing Copolymer Systems
Förster, Plantenberg Angew. Chemie 41, 688 (2002)

5. Example Diblock Copolymer Melts
Ordered Phases
(up to 2005)
2005
BCC
Gyroid
Theoretical prediction
of another phase: Fddd
Tyler and Morse
2007
Experimental verification
Takenaka et al
Fddd
Hexagonal
Lamellar
10 nm

6. Purpose of this tutorial
Show you how Tyler and Morse did this.
Give a "simple", incomplete introduction into the self-consistent field theory for inhomogeneous polymers
Content
General Introduction in Polymer Modeling
Self-Consistent Field Theory

7. General Introduction in Polymer Modeling

8. Length scales
(Example: Polystyrene)

9. Coarse Graining Ab initio (atoms + electrons) Molecular models (atoms)
Coarse grained models
(molecule units, molecules)
Mesoscopic models
(continuous models)
Quantitative results on specific materials.
Small system sizes
Generic insight into the properties of material classes.
(Semiquantitative)

10. A Paradigm of Polymer Theory
The Gaussian Chain
Starting point:
Some arbitrary non-interacting ('ideal') chain

11. A Paradigm of Polymer Theory
The Gaussian Chain
Starting point:
Some arbitrary non-interacting ('ideal') chain
Coarse-Grain

12. A Paradigm of Polymer Theory
The Gaussian Chain
Starting point:
Some arbitrary non-interacting ('ideal') chain
Coarse-Grain
Coarse-Grain further
 Random Walk!

13. The Gaussian Chain: Discrete Version
Gaussian Limit Theorem:
Length d of coarse-grained bonds that connect every nth monomer is distributed according to
The same distribution is reproduced by the
effective Hamiltonian
 Discrete Gaussian Chain model
P(d) ~ exp − 𝑑 2 2 𝑛 𝜎 2

14. The Gaussian Chain: Continuous Version
Discrete Gaussian chain:
'Continuous limit': Polymer described as space curve R(s)
 Continuous Gaussian Chain model

15. Other Popular Chain Models
Freely Jointed Chain (discrete model)
Wormlike Chain (continuous model, stiff chain)
Lattice Models
etc.
with

16. Interacting Chains: Weak Interactions
Weak Interactions: 'Virial Expansion'
 Additional Contribution to the Hamiltonian
with the 'monomer density':
'Ideal chain': v=w=…=0  'Random walk' statistics, Rg ~ N1/2
Single 'real chain' in good solvent: v > 0
 Chains swell, 'Self-avoiding walk' statistics Rg ~ N0.588

17. Dense Melts Ideality assumption (Flory):
In dense melts, polymers behave as if they were ideal!
Results from the cancellation of two effects:
Intrachain interactions → Chain swelling
Interchain interactions → Chain compression
Caveats:
Ideality assumption not correct for chains of finite length.
(Wittmer et al 2004, 2007; Grzywacz et al 2007, Virnau 2016)
But: Remains valid for infinitely long chains. (?)
Shall be applied from now on.

18. AB-Blends: "Standard Model"
Most popular model in theories of (co)polymer blends
Continuous Gaussian chains:
Two types of monomers (j = A or B):
Incompressible: Constraint everywhere.
Flory-Huggins Interactions:
with

19. Discussion of "Standard Model" I
Incompressibility
Realization of incompressibility in a discretized version of the model.
Continuum limit?
My personal view: Continuum limit is unphysical!
Discrete coarse-graining length is part of the model.
Alternative: Tackle continuum limit, full field theory
 Ultraviolet divergencies, renormalization necessary, …
(David Morse)

20. Discussion of "Standard Model" II
Topological interactions
are missing in the "Standard Model"
(Interactions are soft, chains may cross each other.)
Affects
Dynamic properties (entanglements, reptation) 
Static properties of ring polymers
Static properties in ultrathin films (two dimensions)
Does not affect:
Static properties of linear chains in three dimensions 

21. Discussion of "Standard Model" III-a
Flory-Huggins Parameter 
Origin: Flory-Huggins Theory of binary homopolymer blends
Predicts demixing phase diagram
Example (NB =2NA )

22. Discussion of "Standard Model" III-b
Flory-Huggins Parameter 
Effective parameter, connects with the 'real world'
Methods for experimental determination (using samples with different degrees of polymerization)
Comparison with real demixing phase diagrams
Differential scanning calorimetry – enthalpies of mixing
Small angle neutron scattering (SANS) – fitting scattering function to mean-field prediction (RPA, to be discussed later).

23. Discussion of "Standard Model" III-c
Flory-Huggins Parameter 
Fully exhaustive theory of  is still lacking !
(despite many efforts over decades)
Contributions:
Monomer incompatibility
Equation of state (Volume per monomer depends on composition)
Chain correlations, violations of 'ideality', chain end effects
Composition correlations (nonrandom mixing and nonrandom packing)

24. Discussion of "Standard Model" III-d
Flory-Huggins Parameter 
… as a result, depends on:
Monomer type 
Temperature and pressure 
Local composition 
Chain length 
 The very concept of a -parameter should perhaps be questioned (e.g., Tambasco et al)
But: We'll use it anyway 

25. Extensions of the 'Standard Model'
Other chain models, for example
Wormlike chains (for semiflexible polymers)
Ideal lattice chains (→ 'Scheutjens-Fleer theory')
Self-avoiding chains (→'Szleifer theory')
Realistic chain models
Other interactions, for example
Long-range interactions (e.g., electrostatics)
Anisotropic interactions (e.g., liquid crystalline order)
Compressible models, using equation-of-state information

26. Direct Simulation of the 'Standard Model'
Most straightforward way of studying the 'Standard Model':
Simply put it on the computer
Requires:
Discretized formulation
For convenience: Compressibility
(Laradji, Guo, Zuckermann 1994, Besold et al 1999, Marcus Müller et al, Kostas Daoulas, Juan de Pablo et al,…)

27. Example Switching transition in a polymer brush (Shuanhu Qi)
(L. Klushin, A. Skvortsov, A. Polotsky, S. Qi, FS, PRL 2014)
(Shuanhu Qi)

28. The Self-Consistent Field Theory (SCF Theory): Basics

29. Starting Point: "Standard Model" (Recall)
For simplicity: Single component copolymer melt
Melt of AB diblock copolymers  with A-monomer fraction f
Hamiltonian:
- Gaussian chains:
- Interaction energy: with
- Incompressibility constraint:
Monomer densities:

30. Partition Function (Canonical, one component melt)
Path integral over all possible chain conformations R(s)
Flory-Huggins interactions
Incompressibility

31. SCF Equations in a Nutshell I
Basic idea (intuitive picture):
Chains are treated as independent paths in an average (self-consistent) field created by the other chains.
Self-consistent fields:
Incompressibility
Flory-Huggins interactions
(factors 1/N for convenience)

32. SCF Equations in a Nutshell II
Independent copolymer chains in the field WA/B(r)
Define segment dependent field:
 Single chain partition function
Monomer density distribution:

33. Derivation I: Density Functional Theory
Basis: (Hohenberg-Kohn theorem)
Free energy of an inhomogeneous system can be described
by a functional F[j(r)] that only depends on local densities.
→ Task: "Guess" F[j(r)]
Ansatz: F[j(r)] = Fid[j (r) ] + HI[A,B] + H[A,B]
Fid : Reference system of ideal, noninteracting chains
HI = 0 dr AB : AB interactions
H= /2 dr (A+B-0)2 : (in)compressibility (→)

34. Derivation I: Density Functional Theory
Reference system: Noninteracting chains
Starting point: Systems with external fields WA,WB
Free energy:
→ Densities:
Legendre transform Wj → j
 Density functional of reference system:

35. Derivation I: Density Functional Theory
Full density functional (everything taken together)
Minimize this functional with respect to (r)
(with Lagrange parameter  to account for incompressibility)
 Gives SCF equations
and expression for the free energy.

36. Derivation II: Field-Theoretic Approach
Start from partition function
Insert delta-functions

37. Derivation II: Field-Theoretic Approach
Gives fluctuating field theory
with
SCF approximation: Saddle point integral
→ Minimize F with respect to Wj, j ,
 Gives same SCF equations than density functional. and, same free energy F.

38. Derivation II: Field-Theoretic Approach
Analysis of saddle point approximation:
Rewrite
with : Ginzburg parameter
 Saddle point approximation is good, if C is large.

39. Two Approaches to the SCF Theory
Density Functional
Field theoretic approach
Good starting point for a more detailed incorporation of microscopic structure (packing, local monomer correlations, etc.) → Adjust reference system
Established starting point for dynamical studies (dynamic density functional theories)
Range of possible underlying models is restricted (incorporating hard-core inter- actions is difficult) .
Good starting point for a sys- tematic study of long-range ("ultrared" ) fluctuation effects.
Otherwise, both approaches result in equivalent theory!

40. Practical Implementation I
Tasks along the way
Evaluate single-chain partition function Q of chains in the external fields WA/B
Evaluate average densities A/B(r)
Evaluate fields Wj :
Trick: Feynman-Kac formula
Define propagator G(r0,r1,s): Restricted partition function for chains of length s with end monomers at ri
G(r0,r1, s) satisfies diffusion equation → can be evaluated.
Using G(r0,r1), calculate Q and densities A/B(r). ? ? √

41. Practical Implementation II
More specifically
Define partial partition functions: q(r,s) = dr0 G(r0,r,s) and q+(r,s) = dr1 G(r,r1,N-s)
Diffusion equations
with q(r,0) = q+(r,0) = 1
 Full partition function: Q = dr q(r,N) = dr q+(r,N)
Densities:

42. Practical Implementation III
Structure of a typical (simple) SCF program
Starting from given density profiles and guess for 
Calculate fields
Solve diffusion equations for q(r,s), q+ (r,s)
Calculate single chain partition function Q
Calculate 'new' density profiles
Check whether SCF equations are fulfilled within the desired accuracy.
If No: New guess for densities and ; go back to 1)
If Yes: Stop – You made it 

43. Application example: Diblock copolymers
S (BCC)
G (Gyroid)
Experimental phase diagram
L (Lamellar)
C (Hexagonal)
(Bates et al)

44. Application example: Diblock copolymers
Experimental phase diagram
SCF phase diagram
(Bates et al)
(Matsen et al, 1996)
SCF obtains correct sequence of phases at similar N 
(but: gets order of transition wrong)

45. Application example: Diblock copolymers
Monte Carlo phase diagram
SCF phase diagram
(Matsen et al, 2006)
(Matsen et al, 1996)
SCF obtains correct sequence of phases at similar N 
(but: gets order of transition wrong)

46. Summary and Conclusions
'Standard model' of polymer theory
Gaussian chains and Flory-Huggins type interactions
Self-consistent field theory
Successful to the point of very often being quantitative.
Good tool to predict/analyze nanostructured phases in copolymer melts and blends.
But: Beware of fluctuation effects.