POLYMER MODELING IN INDUSTRY
Jozef Bicerano, Ph.D. Bicerano & Associates, LLC
For further details on most of the work summarized in these slides, see J. Bicerano, S. Balijepalli, A. Doufas, V. Ginzburg, J. Moore, M. Somasi, S. Somasi, J. Storer and T. Verbrugge, “ ” J. Macromol. Sci.-Polymer Reviews, 44, 53-85, 2004.
August 13, 2004 1 INTEGRATED MODELING APPROACH
. New product and/or process development in industry requires attention to many (and often contradictory) considerations: Formulation design. Raw material costs. Processing costs. Product performance targets. Market trends. Governmental regulations.
. A multidisciplinary and integrated modeling approach is desirable since the relevant materials science encompasses many phenomena.
August 13, 2004 2 Multidisciplinary Nature of Industrial Modeling
CPU Numerical Methods Problem Definition Software Code Customer Requirements n o i t
i Numerical User Interface & l n
i Modeling Tools a f o e G D
e m h e l T b o
r Model Formation P
First Principles Deterministic Phenomenology Stochastic Empiricism
August 13, 2004 3 “ ”
Chemical structures of system components Modeling at molecular scale (polymers, molecular fluids, surfactants, fillers, etc.) (QSPR, atomistic simulations)
Modeling at mesoscale (self-consistent mean field theory Binary interaction parameters (Flory-Huggins or its for thermodynamic equilibrium; mesoscale dynamic analogues) between all distinct pairs of system components simulations for evolution as a function of time)
Phase diagrams, interfacial tensions, phase inversion points, Modeling at the macroscale (computational model-generated three-dimensional snapshots of morphology as a fluid dynamics, solid mechanics) function of time and of thermodynamic equilibrium morphology
Understanding and predictions of bulk sample behavior (morphological stability, flow characteristics, physical and mechanical properties)
August 13, 2004 4 BRIEF REVIEWS OF SELECTED PROJECTS
. Mechanical properties of thermoplastics (Jonathan Moore). . Polymer/clay nanocomposites (Valeriy Ginzburg). . Polyol templating (Sudhakar Balijepalli). . Flow-induced crystallization and polymer process modeling (Antonios Doufas and Madan Somasi). . High-throughput polymer design (Sweta Somasi and Jozef Bicerano). . Controlled release of drugs from hydrophilic polymers (Sweta Somasi, Irina Graf, Steve Ceplecha, Daniel Simmons and Jozef Bicerano). . Branched/network polymer structures (Tom Verbrugge). . Water vapor transport in a polymer matrix composite (Joey Storer, Jozef Bicerano and Dave Moll).
August 13, 2004 5 MODELING MECHANICAL PROPERTIES OF THERMOPLASTIC POLYMERS Microscopic Properties Experimental Data Intra-Chain Energies Inter-Chain Energies Mechanical t Crystallinity via Finite Element Property Models Analysis Tie Chains Long Chain Branching t
The basic goal is to relate material properties to their origins in microscopic phenomena and to their influence on the performance of products in end-use applications.
August 13, 2004 6 Physical Picture for Glassy Thermoplastic Polymers
. A mosaic of nanoscale clusters: Distribution of differing cluster viscoelastic characteristics. Fluctuations of the Eyring-type activation energy for yielding induce dynamic inhomogeneities.
. “ ”
August 13, 2004 7 Distribution of Yield Characteristics Related to d/d
Bisphenol-A polycarbonate MFR ~ 10
233 < T (K) < 338
1.6x10-3 < strain rate (sec-1) < 1.6x10-1
August 13, 2004 8 Strain Rate and Temperature Dependences
Dashed lines are calculated. Solid lines are experimental data.
August 13, 2004 9 Thermoplastic Mechanical Property Modeling Summary
. The model requires three stress-strain curves as input for calibration. . It captures the stress-strain curves up to yield accurately at different rates and temperatures for bisphenol-A polycarbonate. . Limited testing suggests that it works well for other glassy polymers, such as ABS, PC/ABS, and HIPS. . It also provides insights into the fundamental issue of glass-former "fragility" in the glassy state and a practical means to assess dynamic inhomogeneities in polymeric glasses. . For further details, see J. Moore, M. Mazor, S. Mudrich, J. Bicerano, A. Patashinski and M. Ratner, ’ , Society of Plastics Engineers, 1961-1965 (2002).
August 13, 2004 10 MULTISCALE MODELING OF POLYMER/CLAY NANOCOMPOSITES
SCFT DFT PALMYRA Mechanical Properties ’ Phase Effective behavior, clay-clay morphology interactions Clay (chemical structure), Clay aspect ratio, organic modifier (chemical organoclay loading structure, amount, MW), polymer (structure, MW) WAXS and SAXS TEM
August 13, 2004 11 Modeling Clay Intercalation or Exfoliation: Mean-Field Theory of Polymers in the Galleries
Dispersing Unmodified (Virgin) Clay in Polymer: • Calculate polymer-induced clay-clay interactions: ] Immiscible Entropy -- polymer pushed out a e r
Enthalpy -- polymer pulled in a
t • Enthalpy needs to dominate i n
• Best way -- use polar head group u / y
(such as MaPP for PP/clay hybrids) g
r Intercalated e n E
For Success With Organically Modified Clays: [ • Organic modifier chains should be sufficiently long. Exfoliated • Organic modifier and polymer should be miscible. • Polymer should be attracted to clay surface. Distance, nm Vaia and Giannelis, Macromolecules, 30, 7990-7999 (1997). August 13, 2004 12 Modeling Clay Dispersion Morphology
Clay-Clay Potentials (SCMFT) Dispersion Phase Behavior (DFT)
Immiscible Two-Phase Increase in End-Functionalized (Part. Exf., Part. Stacks) Polymer Adsorption Enthalpy (units: kT) Onto Clay Surface Isotropic Aligned Aligned Exf. Exf. Int. Exfoliated Iso. Gel Clay Stacks
Organoclay Loading Ginzburg and Balazs, Adv. Mater., 12, 1805-1809 (2000).
. Clay surface coverage and organoclay loading determine the equilibrium morphology, but in practice the morphology is often non-equilibrium. . Morphology can be deduced experimentally from WAXS, SAXS and TEM.
August 13, 2004 13 Verifying Dispersion Morphology Using WAXS
System 1: <10% single layers,XRD Intensity the rest in stacks of 20-30 System 2: ~50% single layers,XRD Intensity the rest in stacks of 6-8 80000platelets each with interlayer separation ~ 3.7 nm 30000platelets each with interlayer separation ~ 3.9 nm
70000 25000 60000
20000 s s 50000 t t i i n n Exper u E xper u
. 40000 . 15000 b C a lc b Calc r r a a
, ,
I 30000 I 10000 20000
5000 10000
0 0 0 2 4 6 8 0 2 4 6 8 2 T h e ta, d e g 2 Theta, deg
Calculations based on a model by Vaia and Liu, Polymeric Materials: Science & Engineering, 85, 14-15 (2001). . XRD fitting model can determine degree of exfoliation in clay dispersions. . Results in qualitative agreement with transmission electron micrographs.
August 13, 2004 14 Predicting Mechanical Properties of Nanocomposites Using Finite Element Models
2.0 No Stacks, Aligned, In-plane • For randomly oriented disks, 1.9 50%Stacks, Aligned, In-plane no significant dependence of 1.8 100%Stacks, Aligned, In-plane
) No Stacks, Random
x ’ i
r 1.7 50% Stacks, Random t a 100% Stacks, Random m ( 1.6 • For aligned disks, in-plane E
/
) modulus improves with
e 1.5 t i s
o increased exfoliation.
p 1.4 m o c
( 1.3 E See also: D. A. Brune and J. Bicerano, 1.2 Polymer, 43, 369-387 (2002), where 1.1 closed-form equations were used to 1.0 model more idealized morphologies. 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 Clay Loading, Weight %
. ’ . Similar methods are applicable to predicting other properties. August 13, 2004 15 Nanocomposite Modeling Summary
. Implemented multiscale modeling framework relating properties of nanocomposites to their formulations: Polymer/Clay Miscibility -- Mean-Field Theory of Polymers. Phase Behavior of Dispersions -- DFT of Colloids. Morphology Verification -- XRD Analysis and Modeling. Property Prediction -- Finite Element Analysis. . Modeling framework was applied successfully to: Design compatibilizer formulations to improve clay dispersion in aqueous and other solutions, as well as in polymer melts. Estimate mechanical properties of specific nanocomposites.
August 13, 2004 16 POLYOL TEMPLATING
. Predict phase diagram, including transitions.
. Many applications in nanotechnology.
August 13, 2004 17 Assumptions in Lattice Mean-Field Model (developed by Per Linse et al., Lund University)
. Space divided into equal-sized lattice sites in two dimensions. . One species per lattice site. . Flexible polymers. . Mean-field approximation, with nearest neighbor interactions accounted for via Flory-Huggins interaction () parameters. . Aqueous solutions of PEO and PPO homopolymers display a lower critical solution temperature. . “ ” “ ”
August 13, 2004 18 Main Features of Phase Diagram for an AB-EO Triblock / Water System
. Lyotropic Phase Transitions: 360 L Caused by concentration
340 changes (for example, H1 L I 2 ] H as is increased). 2 polymer K
[ 320
T L . 300 Thermotropic Phase Transitions: I Caused by temperature changes 1 H 280 1 (for example, H1 L I2 as 0 0.2 0.4 0.6 0.8 1 T is increased). polymer
Ordered Phases: Disordered Phase: L
Micellar cubic: I1 Hexagonal: H1 Lamellar: L Reverse hexagonal: H2 Reverse micellar cubic: I2 August 13, 2004 19 Thermotropic Behavior From Increasing Effective Hydrophobicity With Temperature
Low temperature . Water-continuous phases with hydrophobic hydrophilic high curvature dominate. . Lamellar phase is narrow. . “ ” Intermediate temperature
hydrophobic hydrophilic . Lamellar phase is stable.
High temperature
hydrophobic hydrophilic . “ ”
August 13, 2004 20 Block Length Effects on Thermotropic Transitions
(CH ) -PO -EO 2 14 12 8 (CH2)14-PO12 -EO17 360 360 L 340 340
] I 2 K ] H Increasing the [ L 320 2 K [ T
320 T L EO block length 300 I H 2 2 300 I 280 H L 1 H extends the 1 280 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 temperature polymer polymer stability of ordered (CH2)14-PO12 -EO34 (CH2)14-PO12 -EO68 phases and favors 360 L 360 H 1 water-continuous 340 340 L ] H 1 ) phases. K [ K
320 ( L 320 T T 300 300 I 1 280 280 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 polymer polymer August 13, 2004 21 Polyol Templating Summary
. Lattice mean-field theory was used to predict the lyotropic and thermotropic phase transitions of polyol surfactants with water, as functions of chemical composition, block length and concentration.
. This work is discussed in greater detail in several publications: N. P. Shusharina, S. Balijepalli, H. J. M. Grünbauer and P. Alexandridis, ACS Polymer Preprints, 43(2), 354-355 (2002). N. P. Shusharina, P. Alexandridis, P. Linse, S. Balijepalli and H. J. M. Grünbauer, Eur. Phys. Jour. E, 10, 45-54 (2003). N. P. Shusharina, S. Balijepalli, H. J. M. Grünbauer and P. Alexandridis, Langmuir, 19, 4483-4492 (2003). P. Alexandridis, N. P. Shusharina, K.-T. Yong, K.-K. Chain, S. Balijepalli and H. J. M. Grünbauer, ACS Polymer Preprints, 44(2), 218-219 (2003).
August 13, 2004 22 MICROSTRUCTURAL / CONSTITUTIVE FLOW-INDUCED CRYSTALLIZATION MODEL Amorphous phase Semi-crystalline phase
No No x Statistical segments Mesoscale picture u T, v No (1-x) c =
FIC Kinetic theory
Modified Giesekus Model Rigid Rod Model Model
Flexible molecules, represented here as non-linear Rigid rods (with anisotropic drag) that orient and elastic dumbbells with finite extensibility (Peterlin grow at the expense of flexible portions of chains. approximation to get closed form equation).
August 13, 2004 23 Modeling Fiber Spinning and Film Blowing Processes
Mass throughput W Flattened Tube Spinneret Nip rolls
z = 0 v = v z z r z o 1 D = Do T = To P Must solve the coupled
RL mass, momentum and Quench air energy conservation and constitutive/crystallization v , T c a z equations numerically. L Frost line L 1 R(z)
2 H(z) Freeze point z Cooling Air r
z2 z = L
vz = vL D = DL Fiber Spinning Take-up roll Film Blowing Air
August 13, 2004 24 Fiber Spinning Model Validation
PET 5947 m/min NYLON 6,6 100 100 5100 m/min 6000 0.04 T x 0.1 oC 80 e r ] u
n 5000 t i a e r m 0.03 10 / c e e 60 n c p m 4000 v
[ z e n
m g e y e t n
n g i i T
c r 3000 0.02 n , i
40 f r o r l e f e r e t i e D r e V i B 2000 velocity 1 l m B a a i 0.01 20 i
x birefringence 1000 D A n x 103 0 0.00 0 0 20 40 60 80 100 120 140 160 0.1 Distance from Spinneret [cm] 0 40 80 120 160 Distance from Spinneret [cm]
The model predictions are in good agreement with high-speed spinning data for the velocity, diameter (necking), temperature, and flow birefringence of Nylon and PET.
August 13, 2004 25 Linear Stability Analysis for Fiber Spinning
To predict the onset and evolution of instabilities (draw resonance), the system of equations is linearized around the steady state and the time evolution of prescribed disturbances is monitored. This evolution is dictated by the eigenvalues of the system.
4 20 Unstable (Draw Ratio=100) DrawRatio=50 0 10 s s
R -4 e r 0 t S
-8 Stable (Draw Ratio=50) -10 Draw Ratio=100
-12 -20 0 100 200 300 400 0 0.5 1 1.5 2 Time
Eigenspectrum for two Stress disturbance evolution at end of different take-up speeds spinline at most dominant eigenvalue for the two take-up speeds August 13, 2004 26 Blown Film Model Validation
Prediction of bubble velocity and temperature profiles
1.0 160
140 0.8 120 ] ]
s (1) / C o m [ [
0.6 100 e y r t i u c t o a l r 80 e e
V data, run 56 p data, run 50 (2) l 0.4 m a
i data, run 56
model, run 56 e x
T 60
A data, run 57 data, run 57 model, run 57 model, run 50 0.2 data, run 50 40 model, run 56 (3) model, run 50 model, run 57 20 0.0 0 1 2 3 4 5 0.0 0.2 0.4 0.6 0.8 1.0 Distance from Die [m] Distance from Die [m]
The model can predict film velocity and temperature profiles accurately and capture the plateau of the film temperature due to crystallization.
August 13, 2004 27 Blown Film Model Validation (continued)
8e+5
7e+5 3 y t i n i l l
6e+5 a CD macroscopic t
s CD molecular y r ]
c relative crystallinity
a 5e+5 d P n [ 2
a s s 4e+5 n e o i r t s s n
1 e
1 run 56, model 3e+5 t x T run 57, model e run 50, model e 1 v i
2e+5 t a l e
1e+5 R
0 0 0 1 2 3 4 5 0 1 2 3 4 Distance from Die [m] Distance from Die [m] . . Predicted axial stress profiles T11 Locking-in of the macroscopic and along bubble tangential direction. molecular relative extensions in hoop “ ”
August 13, 2004 28 Flow-Induced Crystallization Modeling Summary
. Model is mesoscopically-based, applicable to a variety of kinematics (uniaxial or biaxial extension, shear), and able to simulate polymer processes of great industrial and academic interest quantitatively.
Predicts necking and the associated softening (decrease of the extensional viscosity at high draw ratios) behavior in fiber spinning for the first time.
Predicts stresses in and microstructures of fibers and films at the freeze point, which are closely related to the mechanical and physical properties.
Predicts effects of fabrication conditions and molecular architecture on draw resonance during non-isothermal fiber spinning.
Predicts rheometric data (such as Meissner, Rheotens, shear) in absence of crystallization.
. See the many publications of Doufas et al. for further details.
August 13, 2004 29 HIGH-THROUGHPUT POLYMER DESIGN
“ ” Researcher guesses some Properties are predicted to candidate repeat units that select the best candidates for may perhaps give polymers synthesis, which are limited satisfying performance to being just a subset of the criteria for target application. ’
“ ” High-throughput (combinatorial) methods are used to generate a vast “ ”
. The approach attempted at Dow was based on J. Bicerano, Prediction of Polymer Properties, third edition, Marcel Dekker, New York (2002), as implemented in the SYNTHIA software module marketed by Accelrys, Inc.
August 13, 2004 30 CERIUS2/SYNTHIA Approach
Dow software enumerating all possible repeat unit structures that can be built “ ”
Enter all repeat units and property predictions by using SYNTHIA once into a database
August 13, 2004 31 High-Throughput Polymer Design Summary
SYNTHIA was endowed with combinatorial modeling capabilities!
Enumeration Program Builds repeat units for CERIUS2 Backbone Program Changes torsion angles to 180o or 0o and stores all such conformations CERIUS2 Minimizes energies of conformations Compare Program Compares conformations, finds one with maximum extension, saves it SYNTHIA Predicts properties at T= 200K to 500K for stored conformations, saves output in database-compatible XML format Database Stores conformations and predictions in a readily searchable format (work not completed on this component of system)
August 13, 2004 32 High-Throughput Polymer Design: Work in Progress
. Bicerano & Associates, LLC, is working with DTW Associates, Inc., to develop a much more effective new modeling approach to high-throughput polymer design.
. This approach will be implemented as a new module (HTPD) which will be made commercially available in ’ ™
August 13, 2004 33 CONTROLLED RELEASE OF DRUGS FROM HYDROPHILIC POLYMERS
Swelling of tablet Dissolution and formation of gel of polymer
Simultaneous Processes: • Diffusion of water into tablet. Drug HPMC • Diffusion of drug/filler out of tablet. Filler • Dissolution of polymer matrix. • Swelling of tablet as water enters. • Formation of gel. Diffusion of Water water, drug and filler
August 13, 2004 34 Literature Model (Siepmann and Peppas)
Main Advantages:
• Physically reasonable, yet mathematically simple.
• Well tested with different drugs, polymer grades, and release media.
• Parameters which are not drug dependent are published and can be used directly, so that only two parameters must be determined by fitting to data.
Water w ˆ D c ) t w w Diffusivity
d cw Drug ˆ tDdcw ) Dc D0c exp(c (1 )) cs c p K a Polymer t p s Solve these equations and at each time step compute volume and remesh.
August 13, 2004 35 User Interface
Interface to Graphical either input representation values for model of experimental parameters or results and model choose them to predictions be fitted by using a state-of- Interface to the-art particle choose input swarm optimizer parameters (tablet size, loading, etc.)
Standard deviation – August 13, 2004 36 Results With Siepmann-Peppas Model
5.E-06 100 Phenylpropanolamine ) 4.E-06 s /
2 Ephedrine 80 4.E-06 m c
( 3.E-06
%
y , t i d 3.E-06 Terbutaline
60 v e i s s Aminophylline a 2.E-06 u e f l f i e
r 2.E-06
40 D
g g u
u 1.E-06 r r d
D Reproterol 20 5.E-07 Salbutamol 0.E+00 0 3 4 5 6 7 0 2 4 6 8 time, h Cavity Sirface Area (nm2)
Fitted controlled release data for six different drugs of same family (water-soluble bronchodilators) from a literature reference, from a hydroxypropylmethylcellulose matrix (50% drug loading by weight), by using the Siepmann-Peppas model.
The drug diffusivity shows dependence on “ ” “ ”
August 13, 2004 37 MODELING BRANCHED/NETWORK POLYMERS
. This work is aimed towards understanding the following relations: reactant structures network structure network reaction conditions properties
. One can then predict industrially important parameters (such as the time to gel, critical gel ratio, polymer molecular weight and viscosity buildup).
. One can also gain insights on aspects of branched and/or network architectures where analytical tools fail (such as the compositions and sequence distributions of the elastic chain segments between crosslinks, dangling ends, loops).
. Most importantly, success in this effort can shorten the product development time by providing models to guide the experimental work.
August 13, 2004 38 Available Theories
Different theories originate from different assumptions about non-ideal behavior.
Statistical Technique Mean field behavior Miller-Macosko Average Properties ARS & Rate Theory – – – – – –
Accelrys Software Mean field behavior Pre-gel Cyclization
– – – st and 2nd kinetics – – – DryAdd, DMC Monte Carlo Technique 3D-Percolation Detailed information Long Correlation Effects
August 13, 2004 39 Thermoplastic Polyurethane Pultrusion: Process
. Pultrusion is a continuous process where the developing composit “ ” . “ ” . Subsequent rapid cooling results in polymerization reactions becoming favored again.
5 sec 10 sec
0.001 0.001 s 230C-220C s s 0.0009
s 0.0009 220C-230C e e 0.0008 0.0008 n
n 240C-250C k 0.0007 k
0.0007 c i c 230C-240C
i 0.0006
0.0006 h h t
0.0005 t 0.0005 t t
r 0.0004 250C-260C 0.0004 r a
a 0.0003
0.0003 p
p
20 mm f 0.0002 0.0002 l f l
a 0.0001 0.0001 a 240C-250C 260C-270C
H 0 0 H 1 2 3 4 5 6 7 8 9 10 11 1 2 3 4 5 6 7 8 9 10 11 Half part width Half part width
August 13, 2004 40 Thermoplastic Polyurethane Pultrusion: Reactions
-OH + -NCO -Urethane-
-NCO + -NH2 -Urea-
-NCO + -NCO -NCN- + CO2
-NCO + H2O -NH2 + CO2
-Urethane- -NH2 + CH2=CH- + CO2
August 13, 2004 41 Thermoplastic Polyurethane Pultrusion: Mw Prediction
80000 800 0.14
70000 700 0.12 60000 600 Time = 25 sec 0.1 50000 500 Time = 85 sec 0.08 40000 400 0.06 30000 300
0.04 20000 200
10000 100 0.02
0 0 0 0 50 100 150 200 250 300 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 4.2 4.4 4.6 4.8 5.0 5.2 5.4
Time [s] Log M [g/mol]
. Applied temperature profiles (left) all exhibit the same temperature rise and fall rates, but differ in length of the plateau that the temperature resides at prior to cooling and that determines the extruder residence time. . The assumed reaction mechanism was implemented with the dynamic Monte Carlo method. . The starting Mw was ~75 kg/mole, and a shorter residence time was found to favor a higher final Mw. . Snapshots of weight distributions of formed polymers at time 25 sec and 85 sec are shown at the right. . See T. A. M. Verbrugge and C. F. J. den Doelder (to be published) for further details.
August 13, 2004 42 MODELING WATER VAPOR TRANSPORT
. Determine the filler [Linde Type A (LTA) zeolite] loadings needed to “ ” Schematic Illustration of Initial Morphological Hypothesis Micrograph
August 13, 2004 43 Accessible Volume Calculation
. Compute accessible volume of LTA. Assume probe radius r=1.4 Å or 0.75 Å (representing roughly the longest and shortest dimensions of a water molecule). Use the CERIUS2 crystal library and internal volume calculator. Multiply by volume % of LTA loaded into polymer.
Occupiable volume (r=1.4 Å) Accessible Volume (r=1.4 Å)
August 13, 2004 44 Water Vapor Transport Modeling Summary
. The observed water vapor transmission rates were compared with the calculated water-accessible LTA zeolite volume fraction in the blends.
Percolation Model for LTA MVTR in D2045A at 5% PEG probe 1.4 probe 0.75 Linear (probe 1.4) 90 0.25 Linear (probe 0.75) 80 Expon. (MVTR) l i
m 0.2 70 7 e l @ b
60 i y s a 0.15 s d e / 50 c 2 c ^ A
m 40 /
0.1 %
m . l g
30 o R V T
V 20 0.05 M 10 0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 Vol % LTA
. The sudden onset of water vapor transmission was seen to coincide simply with the percolation threshold for water accessibility with a probe radius of “ ” August 13, 2004 45 SUMMARY AND CONCLUSIONS
. Polymer modeling has reached the state of maturity where it can serve as a powerful component of an industrial R&D program on polymer product and process development.
. Such work, at its most effective, attempts to address all of the essential formulation, process and product issues within a multidisciplinary and multiscale modeling paradigm.
. A few polymer modeling projects from Dow Chemical were reviewed to illustrate some of the many types of industrially relevant problems that can be solved.
August 13, 2004 46