The Combined Role of Thermodynamics and Kinetics in the Growth of Colloidal Binary Superlattice Crystals Talid Sinno, Rayn...
Examples of Engineered Colloidal Self-Assembly Interaction Type Intermolecular External Manoharan  et al., Science  (2003)...
Directed Assembly of  DNA-Functionalized Colloids  S: Spacer L: Linker Inter-particle interactions are driven by:  The DNA...
Timeline of DNA-Mediated  Self-Assembly of Colloidal Particles (1996) Aggregates Mirkin et. al.,  Nature  (1996)  (2005) C...
Resulting Pair-Potential Agrees Well with  Direct Experimental Measurements  Biancanello, P.  et al. ,  Phys. Rev. Lett. ,...
Qualitative Expectations  for the (A:B) Binary Phase Diagram Compositionally Ordered Crystals “ Superlattices” (CP or BCC)...
Optimal Superlattice Crystals (A:B) using  MC Simulated Annealing on a Rigid Lattice BCC (CsCl) Lattice Monte Carlo Optimi...
Perturbation Theory for Free Energy Estimation DNA-mediated Interaction Hard Sphere Free energy from Perturbation theory (...
Free Energy Calculations with PT N AB  = 4,  σ  = 300 nm,  E AB  =  5kT and  E AA  = 0kT The sharp free energy minimum for...
Phase Diagram Based on Ideal Structures (   = 300 nm) CP BCC Repulsion of spacer brushes. E AB r E  ( k B T )
Growth of Ideally Ordered Structures is Not Necessarily Thermodynamically Favorable Correct order:  Site  Surface defect: ...
An Equilibrium Picture for Compositional Ordering  CP BCC Use ordering isotherms as input to perturbation theory free ener...
Phase Diagram Based on Equilibrium Structures (   = 300 nm) 1. Antisite defects are more frequent (lower energy) in CP th...
MC Simulations of Growth  of Binary Superlattice Crystals [1]  Phys. Rev. E. , 54, 6035 (1996)  [2]  J. Chem. Phys. , 102,...
Compositional Ordering is Subject  to Significant Kinetic Limitations BCC CP Overall volume fraction   ~0.15 – 0.3. σ  = ...
Kinetic Model for Compositional Ordering  in Growing Binary Crystals (1) Fluid particles arrive at the crystal surface and...
Kinetic Model for Compositional Ordering  on a CP Crystal Surface Ordering Model for CP Lattices is the available time tha...
Thermodynamic-Kinetic Model for Compositional Ordering The model is able to capture N AB  across a wide range of simulatio...
Compositional Order Parameter  is an Input for Free Energy Calculations Input , Determine Obtain N AB , , Coexistence Betw...
Kinetic Effects Can Significantly  Alter the Equilibrium Growth Picture  =0.1  =0.4 Faster growth conditions lead to inc...
But are they Relevant for Crystal Nucleation?   =0.1  =0.4 Selection of crystal phase by compositional ordering limitati...
Example Solid-Solid Transformation CsCl Seed with Binding Energies Favoring CP BCC CP
Conclusions <ul><li>DNA technology offers a powerful, tunable route for assembling particles into multi-component periodic...
 
Supplementary Slides
What can DNA-Directed Assembly do? Specificity and Tunability Douglas  et al., Nature  (2009)  Rothemund,  Nature  (2006) ...
Observed Crystallization Kinetics from MC (Single Component) where   n  2/3   ~ crystal surface area   Assuming a constant...
Computational Framework for  Studying Colloidal Self-Assembly Monte Carlo Simulations Free Energy Calculations: Perturbati...
Computation of Nucleation Barriers Calculation of Nucleation barrier requires advanced (and  expensive ) simulation techni...
Parameterization of Compositional Ordering Extent  for Use in Perturbation Free Energy Calculations N AB Plane is fitted t...
DNA Hybridization-Mediated Interaction Potential Dolan & Edwards,  P.R.S.  London A,  1974 , 337, 509 h x Mackor,  J. Coll...
DNA-Mediated Colloidal Interaction S S L S: Spacer L: Linker SLS bridge The key components of the  colloidal interaction a...
Identification of solid clusters  J. Chem. Phys., 120  3015
An Effective Measure of Crystalline Order: N AB   For a set A:B ratio, N AB  is defined as the average number of  AB   bon...
Characteristic Travel Length  before Colliding with the Crystal Surface  system volume fraction. DNA-interaction length. a...
Perturbation Theory (PT) for  Fluid and Crystalline Colloids Three terms that must be defined 1) , given in  Phys. Rev. Le...
Developing a Model for DNA-Mediated Interparticle Potential The law of chemical equilibrium relates the equilibrium concen...
Bridge Formation in  Heterogeneously Distributed Environment Mass action can be applied on discrete volume elements: Dutt,...
DNA-Mediated Interaction Potential Attraction Model Consider two spheres each labeled with  N  statically independent spac...
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The Combined Role of Thermodynamics and Kinetics in the Growth of Colloidal Binary Superlattice Crystals

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  • FIX
  • A is assumed constant flux, while n 2/3 is surface area of the crystal. At the end of this slide, highlight that growth is not the only mechanism important to the interfacial dynamics. Diffusion is also important.
  • This method fails when applied to the nucleation study of DNA-mediated interactions.
  • C l is the linker concentration and σ s is the surface density of the DNA per sphere Types of spacers: Flexible Spacer: Tethered Gaussian Coil Rigid Spacer: Tethered Rigid Rod:
  • Talk about the spacer and linker solution here.
  • Include figure q6q6 distribution function and number of connection graphs
  • C l is the linker concentration and σ s is the surface density of the DNA per sphere Types of spacers: Flexible Spacer: Tethered Gaussian Coil Rigid Spacer: Tethered Rigid Rod:
  • The Combined Role of Thermodynamics and Kinetics in the Growth of Colloidal Binary Superlattice Crystals

    1. 1. The Combined Role of Thermodynamics and Kinetics in the Growth of Colloidal Binary Superlattice Crystals Talid Sinno, Raynaldo T. Scarlett, Marie T. Ung, John C. Crocker Department of Chemical and Biomolecular Engineering University of Pennsylvania Philadelphia, PA 19104, U.S.A NSF-NIRT (CTS-0404259), CBET-0829045
    2. 2. Examples of Engineered Colloidal Self-Assembly Interaction Type Intermolecular External Manoharan et al., Science (2003) Kim et al., Langmuir (2006) Depletion Electrostatic Tierno et al. , PRL (2008) Magnetic Biancanello et al., PRL (2005) DNA-Mediated Capillarity Forces Spheres Surface Modified Spheres Multi-spheres Engineered Particles Hard Sphere RHCP Fluid 0.545 0.494 Zhang et al., Soft Matter (2009) Electric with DNA Leunissen et al., Nature (2005) 1 μ m 20 μ m
    3. 3. Directed Assembly of DNA-Functionalized Colloids S: Spacer L: Linker Inter-particle interactions are driven by: The DNA hybridization energy within each bridge. (Average) number of bridges between particles. 1. Tunable interaction strength 2. Multi-component systems Engineered DNA-sequence Eg. Mirkin, Gang Eg. Crocker Nature (2008) PRL (2005) S S SS bridge S S L SLS bridge
    4. 4. Timeline of DNA-Mediated Self-Assembly of Colloidal Particles (1996) Aggregates Mirkin et. al., Nature (1996) (2005) Crystals Biancanello et al., PRL (2005) (2008) Superlattice Crystals Nykypanchuk et al., Nature (2008) Park et al., Nature (2008) Kim, Scarlett, et al. , Nature Materials (2009) (2009) Solid Solution Crystals (2009) Ordered 2D Nanoparticle Arrays Cheng et al. , Nature Materials (2009) (nm) ( μ m)
    5. 5. Resulting Pair-Potential Agrees Well with Direct Experimental Measurements Biancanello, P. et al. , Phys. Rev. Lett. , (2005) Resulting pair potential is a direct input to simulation (e.g. MD, BD, MC). = Probability profile of the spacer cloud Attractive Energy: h x = Total Gibbs free energy William Rogers, Crocker Lab
    6. 6. Qualitative Expectations for the (A:B) Binary Phase Diagram Compositionally Ordered Crystals “ Superlattices” (CP or BCC) De-mixed Solid-Solution Crystals (CP) 3. Nykypanchuk et al., Nature (2008) 2. Park et al., Nature (2008) 1. Kim, Scarlett, et al. , Nature Materials (2009) Recent experiments E AB < E AA,BB E AB > E AA,BB Fluid Solid Solution Crystals Superlattice Crystals 3 2 1 E AB r E ( k B T )
    7. 7. Optimal Superlattice Crystals (A:B) using MC Simulated Annealing on a Rigid Lattice BCC (CsCl) Lattice Monte Carlo Optimization All optimal CP and BCC superlattices have 4 “A-B” contacts per particle (N AB ). Simple Cubic (SC) superlattices are not stable due to their small number (3) of AB bonds per particle. FCC (CuAu) HCP RHCP CP superlattices have equal free energy (1 st NN interaction model).
    8. 8. Perturbation Theory for Free Energy Estimation DNA-mediated Interaction Hard Sphere Free energy from Perturbation theory (PT) is determined analytically as a perturbation away from a reference state (HS) by u p . Assume linear decomposition of perturbing terms.
    9. 9. Free Energy Calculations with PT N AB = 4, σ = 300 nm, E AB = 5kT and E AA = 0kT The sharp free energy minimum for crystals allows decoupling from fluid – and direct comparison between CP and BCC solid phases. Fluid BCC RHCP a b Simulation: BCC PT: BCC
    10. 10. Phase Diagram Based on Ideal Structures (  = 300 nm) CP BCC Repulsion of spacer brushes. E AB r E ( k B T )
    11. 11. Growth of Ideally Ordered Structures is Not Necessarily Thermodynamically Favorable Correct order: Site Surface defect: Antisite CP: BCC: Probability of an (equilibrium) particle existing on a “wrong” site ( antisite ) Select surface particles, switch their identity, measure energy change . Antisites are more expensive on the BCC (CsCl) surface
    12. 12. An Equilibrium Picture for Compositional Ordering CP BCC Use ordering isotherms as input to perturbation theory free energy calcs.
    13. 13. Phase Diagram Based on Equilibrium Structures (  = 300 nm) 1. Antisite defects are more frequent (lower energy) in CP than in BCC super-lattices. 2. It may be quite difficult to grow perfectly ordered CP superlattice crystals.
    14. 14. MC Simulations of Growth of Binary Superlattice Crystals [1] Phys. Rev. E. , 54, 6035 (1996) [2] J. Chem. Phys. , 102, 3015 (2004) <ul><li>NVT-MC </li></ul><ul><li>Randomly assign colloids x,y,z coordinates </li></ul><ul><li>Eliminate overlaps within the system, [1] </li></ul><ul><li>Create a seed lattice (BCC or FCC) . </li></ul><ul><li>Insert within equilibrated fluid, </li></ul><ul><li>modified [1] </li></ul><ul><li>Fix seed coordinates </li></ul><ul><li>Periodically identify particles as </li></ul><ul><li>solid or fluid, [2] </li></ul><ul><li>NVT-MC </li></ul><ul><li>Release seed coordinates </li></ul><ul><li>Periodically identify particles as </li></ul><ul><li>solid or fluid and identify NAB, [2] </li></ul><ul><li>NVT-MC </li></ul>Build and Equilibrate the Fluid Seed the Fluid with Desired Lattice Equilibrate the Static Seed Evolve the Seed and Fluid System
    15. 15. Compositional Ordering is Subject to Significant Kinetic Limitations BCC CP Overall volume fraction  ~0.15 – 0.3. σ = 100 nm σ = 300 nm σ = 980 nm 1. All BCC runs exhibited equilibrium ordering. 2. CP runs show lowering ordering than expected from equilibrium isotherm. 3. Effect is consistent across a large range of particle sizes.
    16. 16. Kinetic Model for Compositional Ordering in Growing Binary Crystals (1) Fluid particles arrive at the crystal surface and randomly occupy a site. (2) Particles hop between sites (S) & antisites (AS). a. Bond breakage ( bb ) b. Surface diffusion ( sd ) “ Bond” Energies
    17. 17. Kinetic Model for Compositional Ordering on a CP Crystal Surface Ordering Model for CP Lattices is the available time that a particle has to sample the surface Initial Condition Long Time Limit
    18. 18. Thermodynamic-Kinetic Model for Compositional Ordering The model is able to capture N AB across a wide range of simulation parameters. σ = 100 nm σ = 300 nm σ = 980 nm open symbols: CP filled symbols: BCC ii . . , , , i iii ii i . . , , , iii . . , , ,
    19. 19. Compositional Order Parameter is an Input for Free Energy Calculations Input , Determine Obtain N AB , , Coexistence Between BCC and CP: Perturbation Theory BCC CP Free energy comparisons are made on the basis of the realizable ordering in the different crystal phases.  = 0.3
    20. 20. Kinetic Effects Can Significantly Alter the Equilibrium Growth Picture  =0.1  =0.4 Faster growth conditions lead to increasingly compositionally defective CP crystals Under the right conditions, simply changing the growth conditions can shift the “equilibrium” phase.
    21. 21. But are they Relevant for Crystal Nucleation?  =0.1  =0.4 Selection of crystal phase by compositional ordering limitations appears to extend to (at least very shortly after) the nucleation stage. Additional considerations arise from solid-solid transformations… BCC CP
    22. 22. Example Solid-Solid Transformation CsCl Seed with Binding Energies Favoring CP BCC CP
    23. 23. Conclusions <ul><li>DNA technology offers a powerful, tunable route for assembling particles into multi-component periodic structures. </li></ul><ul><li>But - the equilibrium phase diagram is a poor predictor of what is possible – kinetics play an important role in setting not just the quality but also the very phase of the resultant crystal. </li></ul><ul><li>The simulations required to probe the coupling between kinetics and thermodynamics are expensive – simple mechanistic models can greatly reduce this cost. </li></ul>
    24. 25. Supplementary Slides
    25. 26. What can DNA-Directed Assembly do? Specificity and Tunability Douglas et al., Nature (2009) Rothemund, Nature (2006) He et al. , Nature (2008) Park et al., Nano Lett. (2005) DNA-origami Nanoscaffold Polyhedra Engineered Nano-shapes 3D Structures 2D Structures
    26. 27. Observed Crystallization Kinetics from MC (Single Component) where n 2/3 ~ crystal surface area Assuming a constant flux of particles from the fluid to the crystal surface, the rate of crystal growth equals Growth kinetics can be described by a single constant. For a sphere
    27. 28. Computational Framework for Studying Colloidal Self-Assembly Monte Carlo Simulations Free Energy Calculations: Perturbation Theory (PT) Mechanistic Models Fluid BCC RHCP Bulk Fluid Crystal Growth Nucleation Thermodynamics and kinetics of nucleation and growth N=1 N=2 N=12
    28. 29. Computation of Nucleation Barriers Calculation of Nucleation barrier requires advanced (and expensive ) simulation techniques (e.g. parallel tempering and umbrella sampling) Critical Cluster
    29. 30. Parameterization of Compositional Ordering Extent for Use in Perturbation Free Energy Calculations N AB Plane is fitted to simulation data with  0.3. The compositional order is parameterized to determine the effect of kinetic limitations on the binary phase diagram. Simulated CP ordering (spheres)
    30. 31. DNA Hybridization-Mediated Interaction Potential Dolan & Edwards, P.R.S. London A, 1974 , 337, 509 h x Mackor, J. Coll. Sci ., 1951 , 6, 492 Gibbs free energy from the Nearest Neighbor Model plus the loss in rotational entropy: REPULSIVE TERM The entropic repulsion due to compression of the grafted DNA is ATTRACTIVE TERM = Probability profile of the ligand cloud as a function of x
    31. 32. DNA-Mediated Colloidal Interaction S S L S: Spacer L: Linker SLS bridge The key components of the colloidal interaction are: (2) Interaction strength determined by number of SLS bridges formed between colloids, <n> . <ul><li>The spacer and linker are bound by </li></ul><ul><li>the DNA-hybridization energy. </li></ul>A: Adenine T: Thymine C: Cytosine G: Guanine T A G C T A C G T A Spacer Linker
    32. 33. Identification of solid clusters J. Chem. Phys., 120 3015
    33. 34. An Effective Measure of Crystalline Order: N AB For a set A:B ratio, N AB is defined as the average number of AB bonds between 1 st nearest neighbors (NN) and without double accounting particles. A:B = 50:50 CP crystals, number of 1 st NN: 12 BCC crystals, number of 1 st NN: 8 Random disorder, N AB = 3 Prefect order, N AB = 4 Random disorder, N AB = 2 Prefect order, N AB = 4
    34. 35. Characteristic Travel Length before Colliding with the Crystal Surface system volume fraction. DNA-interaction length. adjustable parameter.
    35. 36. Perturbation Theory (PT) for Fluid and Crystalline Colloids Three terms that must be defined 1) , given in Phys. Rev. Lett. , 94 , 5, 058302 2) , the hard sphere free energy is given by J. Chem. Phys. , 51 , 635 (1969) for fluids and for crystals by Mol. Phys. , 97 , 1273 (1999). 3) for fluid given by Phys. Rev. A , 5 , 939 (1972) and for crystals Phys. Rev. E , 54 , 1261 (1996). Applied to a Lennard-Jones (LJ) fluid. VW : Phys. Rev. A , 5 , 939 (1972) T * =0.75 T * =1.15 MC PT VW PT <ul><li>Verify that our implementation of PT with a LJ fluid. </li></ul><ul><li>Switch from LJ fluid to binary crystals. </li></ul>PT calculations: Since all colloids sit at lattice sites, remains unchanged. Assume linear decomposition of perturbing terms.
    36. 37. Developing a Model for DNA-Mediated Interparticle Potential The law of chemical equilibrium relates the equilibrium concentrations as Consider three reactive species uniformly distributed in volume V . s l s sls
    37. 38. Bridge Formation in Heterogeneously Distributed Environment Mass action can be applied on discrete volume elements: Dutt, A. K.; Datta, A. J. Phys. Chem. A, 1998 x h
    38. 39. DNA-Mediated Interaction Potential Attraction Model Consider two spheres each labeled with N statically independent spacers at a separation h. Let p = probability a GIVEN spacer forms a bridge Probability no bridges form: Probability one or more bridges form: Partition Function of bound states relative to free states: At a given separation, E a equals the average number of bridges at equilibrium times k B T . Biancanello, P. et al. , Phys. Rev. Lett. , 94 , 5, 058302 1 2 N . . . h ~ 30 nm

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