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Topology-Based Parameter Identification forDecoupling Material Structure-Process-Property

Relationships

Ian Tolle1,3,Xinqun Huang1,3, Yvonne A. Akpalu2,3, Lealon Martin1,3

11/8/07

1) Department of Chemical and Biological Engineering, Rensselaer Polytechnic Institute

2) Department of Chemistry and Chemical Biology, Rensselaer Polytechnic Institute

3) Rensselaer Nanotechnology Center

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Scattering Overview

Incident beam(wavelength = )

Sample

ScatteredRadiation

DetectorScattering Angle AzimuthalAngle

Dark areas:High Intensity

Light areas:Low Intensity

-Take an Azimuthal slicefrom the center to theedge and plot the intensityas a function of or

ScatteringVector: Complementary

Scattering Techniques: X-Ray (SAXS, WAXS)

Light (SALS) Neutron (SANS)

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Structure Determination

•When X-rays arescattered only slightly,at small angle , thiscorresponds to largeperiod of repetitionby Bragg’s Law.

•X-rays scattered atlarge angles correspondto a small period ofrepetition .

Example: Polymeric structures at multiple lengthscales characterized by several scattering techniquesBragg’s Law

Increasing

Decreasing Length Scale

(Inverse relationship)

Combined data produces a unique material signature

< 1 nm1 - 100 µm

Spherulite Lamellae Unit Cell

L

l

l ~ 10 -20 nm, L >> 10nm

a

b

SALS, USAXS SAXS WAXS

Microscale Nanoscale

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Motivation

q

I(q)

•How do we extract this structuralinformation from scattering data?

•Direct Methods: assume a modeland fit the data.

-Ex: Observed SAXS intensity I(q)which is separated into:

Ib (background intensity)

Id (central diffuse scattering)

Il (lamellar stack)

2/)()()( qqJqJIqI ldb ++=

More structural complexity mayexist in non-lamellar regions

IlId

LamellarStack

Non-LamellarRegions

Il

Murthy, N., et al. Macromolecules 1998, 31, 142

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Motivation

•GOAL: Obtain all possible morphological contributions to theoverall scattering curve across multiple length scales withoutassuming a specific model

•Approach: Model-Free Analysis - Extract different contributions tothe overall scattering data signature directly from experimental data

IlId

LamellarStack

Non-LamellarRegions

Il

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Test System - Branched Copolymers

Ethylene/1-Butene:

Ethylene/1-Hexene:

•Effect of crystallization temperature and short chain branch length

•Samples were prepared by melting at T=160°C in one chamber andthen quickly transferred to a second chamber at the desiredcrystallization temperature (Tc = 83°C, 86°C, 89°C, 92°C, 95°C)

•Time-resolved, simultaneous synchrotron SAXS/WAXS data collected

•Analysis using intensity measured after ~1 hr cooling time

Z. C. Xiao and Y. A. Akpalu. Abstracts Of Papers Of The American Chemical Society, Aug 2005.

70K

70K

Mw

2.0

2.0

Mw

Mn

6.4 %

5.9 %

ComonomerContent(mol %)

0.900

0.900

ρ (g/cm3)

95 °C

95 °C

Tmnominal

CH2CH3

[ ][ ]x y

CH2CH2CH3

[ ][ ]x y

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Ethylene/1-Hexene (SAXS/WAXS)

Normalize WAXS dataw.r.t. amorphous halo

SAXS WAXS

Important:

•Relative WAXS I(q)

•Absolute SAXS I(q)

WAXS Peak corresponds tothe distance, betweencrystallographic planes

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Ethylene/1-Butene (SAXS/WAXS)

As Tc , peakscorresponding to(1 1 0) and (2 0 0)crystallographicplanes decrease

(1 1 0)

(2 0 0)

SAXS WAXS

PE Unit Cell

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Ethylene/1-Hexene (SAXS detail)

Peak position:

Peak intensity:

= number of lamellae

= Long Period

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Ethylene/1-Butene (SAXS detail)

SAXS Intensityincreases andpeak positionshifts to left asTc = 83˚ 92˚

At Tc = 95˚, crystals are farapart and scattering ischaracteristic of diluteparticulate system

-Corresponds to increasing

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Network Materials Analysis

Determine:- Number of parameters which account for the variance

between signatures- Their physical meaning- The network topology

Assumption: Bipartite Network Topology

Composite 1 Composite 2 Composite 3 Composite M

(Signature 1) (Signature 2) (Signature 3) (Signature M)

TopologicalParameter 1

TopologicalParameter 2

TopologicalParameter 3

TopologicalParameter L

TP 1 TP 2 TP 3 TP L…

…

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Network Materials Analysis

Topology Matrix Perturbation MatrixOriginal Data

•We construct thelinear decomposition:

is composed of M row vectors , where

is composed of L row vectors , generated by this decomposition

Elements of represent the connectivity between and

Connection Exists

No Connection

-Based On Network Component Analysis (NCA)Liao and co-workers. PNAS, 100(26), 2003.

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NMA - Procedure

•Require partial knowledge of the connectivity matrix , mostimportantly the location of zeroes, in order to decomposesimultaneously into a unique and , up to a scaling factor

•Minimize the objective function:

is a vector of sequences that defines the topological connectivity

**No prior knowledge of is required**

-Carried out using AMPL platform and SNOPT solver -NEOS Server for Optimization

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NMA - Procedure

•Require partial knowledge of the connectivity matrix , mostimportantly the location of zeroes, in order to decomposesimultaneously into a unique and , up to a scaling factor

•Minimize the objective function:

Network Identifiability Criteria:

must have full-column rank, must have full-row rank

Number of Topological Parameters L ≤ M (# of samples)

Each column of must have at least L-1 zeros

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Principal Component Analysis

% Explained:

91.64176.27351.77390.1670.05510.03970.01590.0150.01120.007

•Compare NMA to the linear decomposition generated by PCA

•Orthogonality assumption for row vectors , no topological basisfor matrix , leads to poor reconstruction of original data

•One useful result - generates the % of the variance explained byeach parameter. 3 Parameters account for 99.7%

Reconstruction:Blue = OriginalRed = Reconstructed

99.7%

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NMA Topology - Optimal Mapping

EB-83

TP 1 TP 2 TP 3

Topological Parameters (TP)

Copolymer Scattering data signatures

UnknownTopology

EB-86 EB-89 EB-92 EB-95 EH-83 EH-86 EH-89 EH-92 EH-95

•Minimize the objective function while iterating through all possible startingtopologies , for 1, 2 and 3 TP’s. The global minimum will produce thebest decomposition and subsequent reconstruction.

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NMA Connectivity Results

•The fit of the reconstruction tothe original data improves withthe addition of parameters

EB-86

1 TP

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NMA Connectivity Results

•The fit of the reconstruction tothe original data improves withthe addition of parameters

EB-86

2 TP

= New zero foundwith the addition ofthe next parameter

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•The fit of the reconstruction tothe original data improves withthe addition of parameters

EB-86

3 TP

= New zero foundwith the addition ofthe next parameter

NMA Connectivity Results

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Optimal Topology Matrix - 1 or 2 TP’s

TP1 TP1

TP2

•Column values of theoptimal matrix are plottedas a function of Tc

•Absolute values not asimportant as observing trends

(Using only 1 Parameter)

(Using 2 Parameters)

Ex: using 2 TP’s we see that TP1 has zero contribution to the 95°samples, while TP2 has a significant contribution.

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Optimal Decomposition - 3 TP’s

TP1

TP2

TP3

•Amorphous, non-lamellar material

•Disordered,smaller crystals(Distribution Function)

•Crystals organizedinto lamellar stack

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Reconstruction, Optimal Topology

•Reconstruction showsexcellent agreementwith experimentaldata for all samples

Blue =

Red =

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Validation: Crystallization Mechanism

2

1

1) Initial crystallization(isolated crystals)

2) Peak max height reachedwhen well-defined lamellarstacks form

3) Peak broadens and shiftsdue to contribution fromsmaller crystals distributedaround thicker crystals inthe lamellar stack

EB-89

Akpalu and Amis, J. Chem. Phys. 113, 2000

3

The decompositiongenerated by NMA separatesthe scattering curve at thelast time step into these twoparameters (TP1 and TP3)

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Contributions

EB-89EB-86

EB-92

Effect of Processing Conditions:

Ex: Contribution to the SAXS curve bystacked lamellar crystals (TP1)decreases with increasing Tc while thatfor smaller crystals (TP2) increases.

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Summary & Conclusions•NMA approach can extract different contributions to the overall scatteringdata signature directly from experimental data without assuming a model,and determine the effect of processing conditions on these contributionsacross multiple length scales.

Effect of Processing Conditions:

Contributions to Structure:

•These scattering contributions can then be compared to existing modelsto determine how best to describe the structure/morphology, possiblyresulting in the development of new models.

Future work:

•Develop generalized models to describe influence of process onstructure/morphology

•Decomposition of time-resolved data into parameters to investigatepolymer crystallization

•Determine how NMA approach can improve the prediction of propertiesusing micromechanical models

Experimental Data:

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Acknowledgements

Funding

NSF, DOE

Thank you! Questions?

•Martin Research Group

•Akpalu Research Group

•NIST, ANL, BNL