The Mayo-Lewis Copolymerization Model - rohan.sdsu.edu

Chemistry
Old Models
New Models
The Mayo-Lewis Copolymerization Model
Vadim Ponomarenko
Department of Mathematics and Statistics
San Diego State University
CSRC Colloquium Series
April 27, 2007
http://www-rohan.sdsu.edu/∼vadim/mayolewis.pdf
Chemistry
Old Models
Outline
Chemistry
Old Models
New Models
New Models
Chemistry
Old Models
Outline
Chemistry
Old Models
New Models
New Models
Chemistry
Old Models
New Models
Basics
A polymer is a large molecule, built up from many small
monomers. We study the building process.
Consider the polymer as a long necklace.
Note: Branching (non-linear polymers) is possible, but not
today.
Step Polymerization
All monomers and (partial) polymers are simultaneously
reactive. (attach two small necklaces)
Chain Polymerization (our interest today)
A small amount of reactive (partial) polymer absorbs
monomers one at a time. (add one bead to a necklace)
Chemistry
Old Models
New Models
Basics
A polymer is a large molecule, built up from many small
monomers. We study the building process.
Consider the polymer as a long necklace.
Note: Branching (non-linear polymers) is possible, but not
today.
Step Polymerization
All monomers and (partial) polymers are simultaneously
reactive. (attach two small necklaces)
Chain Polymerization (our interest today)
A small amount of reactive (partial) polymer absorbs
monomers one at a time. (add one bead to a necklace)
Chemistry
Old Models
New Models
Basics
A polymer is a large molecule, built up from many small
monomers. We study the building process.
Consider the polymer as a long necklace.
Note: Branching (non-linear polymers) is possible, but not
today.
Step Polymerization
All monomers and (partial) polymers are simultaneously
reactive. (attach two small necklaces)
Chain Polymerization (our interest today)
A small amount of reactive (partial) polymer absorbs
monomers one at a time. (add one bead to a necklace)
Chemistry
Old Models
New Models
Basics
A polymer is a large molecule, built up from many small
monomers. We study the building process.
Consider the polymer as a long necklace.
Note: Branching (non-linear polymers) is possible, but not
today.
Step Polymerization
All monomers and (partial) polymers are simultaneously
reactive. (attach two small necklaces)
Chain Polymerization (our interest today)
A small amount of reactive (partial) polymer absorbs
monomers one at a time. (add one bead to a necklace)
Chemistry
Old Models
New Models
Chain Polymerization
These are classified into types, based on underlying
reaction:
Ionic (cationic/anionic).
More complex, less common, not today.
Radical
Simple, common. A molecule with a free radical is highly
reactive. It bonds with a monomer, which in turn gains a
free radical.
Chemistry
Old Models
New Models
Chain Polymerization
These are classified into types, based on underlying
reaction:
Ionic (cationic/anionic).
More complex, less common, not today.
Radical
Simple, common. A molecule with a free radical is highly
reactive. It bonds with a monomer, which in turn gains a
free radical.
Chemistry
Old Models
New Models
Chain Polymerization
These are classified into types, based on underlying
reaction:
Ionic (cationic/anionic).
More complex, less common, not today.
Radical
Simple, common. A molecule with a free radical is highly
reactive. It bonds with a monomer, which in turn gains a
free radical.
Chemistry
Old Models
New Models
Radical Chain Polymerization Life Cycle
Step 1: Radicals are born. (initiator radicals)
Processes used: thermal, redox, photochemical, ionizing
radiation, etc.
Very small amount, compared to monomers.
Step 2..1000: Radicals react with monomers.
radical chain + monomer ⇒ longer radical chain
Final Step: Radicals die.
Two radical chains react with each other.
Coupling: they become nonradical and merge
Disproportionation: they become nonradical and don’t
merge
Chemistry
Old Models
New Models
Radical Chain Polymerization Life Cycle
Step 1: Radicals are born. (initiator radicals)
Processes used: thermal, redox, photochemical, ionizing
radiation, etc.
Very small amount, compared to monomers.
Step 2..1000: Radicals react with monomers.
radical chain + monomer ⇒ longer radical chain
Final Step: Radicals die.
Two radical chains react with each other.
Coupling: they become nonradical and merge
Disproportionation: they become nonradical and don’t
merge
Chemistry
Old Models
New Models
Radical Chain Polymerization Life Cycle
Step 1: Radicals are born. (initiator radicals)
Processes used: thermal, redox, photochemical, ionizing
radiation, etc.
Very small amount, compared to monomers.
Step 2..1000: Radicals react with monomers.
radical chain + monomer ⇒ longer radical chain
Final Step: Radicals die.
Two radical chains react with each other.
Coupling: they become nonradical and merge
Disproportionation: they become nonradical and don’t
merge
Chemistry
Old Models
New Models
Radical Chain Polymerization Life Cycle
Step 1: Radicals are born. (initiator radicals)
Processes used: thermal, redox, photochemical, ionizing
radiation, etc.
Very small amount, compared to monomers.
Step 2..1000: Radicals react with monomers.
radical chain + monomer ⇒ longer radical chain
Final Step: Radicals die.
Two radical chains react with each other.
Coupling: they become nonradical and merge
Disproportionation: they become nonradical and don’t
merge
Chemistry
Old Models
New Models
Radical Chain Polymerization Life Cycle
Step 1: Radicals are born. (initiator radicals)
Processes used: thermal, redox, photochemical, ionizing
radiation, etc.
Very small amount, compared to monomers.
Step 2..1000: Radicals react with monomers.
radical chain + monomer ⇒ longer radical chain
Final Step: Radicals die.
Two radical chains react with each other.
Coupling: they become nonradical and merge
Disproportionation: they become nonradical and don’t
merge
Chemistry
Old Models
New Models
Radical Chain Polymerization Life Cycle
Step 1: Radicals are born. (initiator radicals)
Processes used: thermal, redox, photochemical, ionizing
radiation, etc.
Very small amount, compared to monomers.
Step 2..1000: Radicals react with monomers.
radical chain + monomer ⇒ longer radical chain
Final Step: Radicals die.
Two radical chains react with each other.
Coupling: they become nonradical and merge
Disproportionation: they become nonradical and don’t
merge
Chemistry
Old Models
New Models
Radical Chain Polymerization Life Cycle
Step 1: Radicals are born. (initiator radicals)
Processes used: thermal, redox, photochemical, ionizing
radiation, etc.
Very small amount, compared to monomers.
Step 2..1000: Radicals react with monomers.
radical chain + monomer ⇒ longer radical chain
Final Step: Radicals die.
Two radical chains react with each other.
Coupling: they become nonradical and merge
Disproportionation: they become nonradical and don’t
merge
Chemistry
Old Models
New Models
Radical Chain Polymerization Life Cycle
Step 1: Radicals are born. (initiator radicals)
Processes used: thermal, redox, photochemical, ionizing
radiation, etc.
Very small amount, compared to monomers.
Step 2..1000: Radicals react with monomers.
radical chain + monomer ⇒ longer radical chain
Final Step: Radicals die.
Two radical chains react with each other.
Coupling: they become nonradical and merge
Disproportionation: they become nonradical and don’t
merge
Chemistry
Old Models
New Models
Steady-State for Radicals
Radicals are created at a constant rate, and destroyed at a
rate proportional to the square of their concentration.
dx
dt
= a − bx 2 ; has solution x(t) =
As t → ∞, x(t) →
q
q
a
b
a
b.
In practice, ∞ is actually quite small.
√
tanh((t + c) ab).
Chemistry
Old Models
New Models
Steady-State for Radicals
Radicals are created at a constant rate, and destroyed at a
rate proportional to the square of their concentration.
dx
dt
= a − bx 2 ; has solution x(t) =
As t → ∞, x(t) →
q
q
a
b
a
b.
In practice, ∞ is actually quite small.
√
tanh((t + c) ab).
Chemistry
Old Models
New Models
Steady-State for Radicals
Radicals are created at a constant rate, and destroyed at a
rate proportional to the square of their concentration.
dx
dt
= a − bx 2 ; has solution x(t) =
As t → ∞, x(t) →
q
q
a
b
a
b.
In practice, ∞ is actually quite small.
√
tanh((t + c) ab).
Chemistry
Old Models
New Models
Polymerization Summary
Molecule Concentrations
Molecule
Monomers
Radicals
Polymers
Initial
high
none
none
In Progress
decreasing
tiny
increasing
Final
low
none
high
We want to understand the rate at which this happens. ‘kinetics’
Only monomer concentrations are efficiently measured,
through various spectroscopies (IR, UV, NMR).
Chemistry
Old Models
New Models
Polymerization Summary
Molecule Concentrations
Molecule
Monomers
Radicals
Polymers
Initial
high
none
none
In Progress
decreasing
tiny
increasing
Final
low
none
high
We want to understand the rate at which this happens. ‘kinetics’
Only monomer concentrations are efficiently measured,
through various spectroscopies (IR, UV, NMR).
Chemistry
Old Models
New Models
Polymerization Summary
Molecule Concentrations
Molecule
Monomers
Radicals
Polymers
Initial
high
none
none
In Progress
decreasing
tiny
increasing
Final
low
none
high
We want to understand the rate at which this happens. ‘kinetics’
Only monomer concentrations are efficiently measured,
through various spectroscopies (IR, UV, NMR).
Chemistry
Old Models
Outline
Chemistry
Old Models
New Models
New Models
Chemistry
Old Models
Single Monomer Polymerization
With one type of polymer, the relevant DE is:
dx
dt
= −k rx
x is the monomer concentration
r is the (constant) radical concentration
k is the goal (rate constant)
Too simple for chemists to screw up.
Also, of limited interest.
New Models
Chemistry
Old Models
Single Monomer Polymerization
With one type of polymer, the relevant DE is:
dx
dt
= −k rx
x is the monomer concentration
r is the (constant) radical concentration
k is the goal (rate constant)
Too simple for chemists to screw up.
Also, of limited interest.
New Models
Chemistry
Old Models
Single Monomer Polymerization
With one type of polymer, the relevant DE is:
dx
dt
= −k rx
x is the monomer concentration
r is the (constant) radical concentration
k is the goal (rate constant)
Too simple for chemists to screw up.
Also, of limited interest.
New Models
Chemistry
Old Models
New Models
Two-Monolymer Polymerization Terminal Model
With two type of monomers, we divide radicals into rx , ry
based on last monomer added.
dx
dt
= −kxx rx x − kyx ry x
dy
dt
= −kxy rx y − kyy ry y
x, y are the monomer concentrations
kxx , kyx , kxy , kyy are the goal (rate constants)
rx + ry is the (constant) radical concentration
Chemistry
Old Models
New Models
Two-Monolymer Polymerization Terminal Model
With two type of monomers, we divide radicals into rx , ry
based on last monomer added.
dx
dt
= −kxx rx x − kyx ry x
dy
dt
= −kxy rx y − kyy ry y
x, y are the monomer concentrations
kxx , kyx , kxy , kyy are the goal (rate constants)
rx + ry is the (constant) radical concentration
Chemistry
Old Models
New Models
The Value of Kinetics
dx
dt
= −kxx rx x − kyx ry x
dy
dt
= −kxy rx y − kyy ry y
x, y are the monomer concentrations
kxx , kyx , kxy , kyy are the goal (rate constants)
rx + ry is the (constant) radical concentration
kyx , kxy ∼
= 0 Block Copolymerization; homogeneous chains
kxx , kyy ∼
= 0 Alternating Copolymerization
kyx ∼
= kxy ∼
= kxx ∼
= kyy Random Copolymerization
Chemistry
Old Models
New Models
The Value of Kinetics
dx
dt
= −kxx rx x − kyx ry x
dy
dt
= −kxy rx y − kyy ry y
x, y are the monomer concentrations
kxx , kyx , kxy , kyy are the goal (rate constants)
rx + ry is the (constant) radical concentration
kyx , kxy ∼
= 0 Block Copolymerization; homogeneous chains
kxx , kyy ∼
= 0 Alternating Copolymerization
kyx ∼
= kxy ∼
= kxx ∼
= kyy Random Copolymerization
Chemistry
Old Models
New Models
The Value of Kinetics
dx
dt
= −kxx rx x − kyx ry x
dy
dt
= −kxy rx y − kyy ry y
x, y are the monomer concentrations
kxx , kyx , kxy , kyy are the goal (rate constants)
rx + ry is the (constant) radical concentration
kyx , kxy ∼
= 0 Block Copolymerization; homogeneous chains
kxx , kyy ∼
= 0 Alternating Copolymerization
kyx ∼
= kxy ∼
= kxx ∼
= kyy Random Copolymerization
Chemistry
Old Models
New Models
The Value of Kinetics
dx
dt
= −kxx rx x − kyx ry x
dy
dt
= −kxy rx y − kyy ry y
x, y are the monomer concentrations
kxx , kyx , kxy , kyy are the goal (rate constants)
rx + ry is the (constant) radical concentration
kyx , kxy ∼
= 0 Block Copolymerization; homogeneous chains
kxx , kyy ∼
= 0 Alternating Copolymerization
kyx ∼
= kxy ∼
= kxx ∼
= kyy Random Copolymerization
Chemistry
Old Models
New Models
Completing the Model
dx
dt
= −kxx rx x − kyx ry x
dy
dt
= −kxy rx y − kyy ry y
x, y are the monomer concentrations
kxx , kyx , kxy , kyy are the goal (rate constants)
rx + ry is the (constant) radical concentration
Additional DE:
drx
dt
dr
= − dty = kyx ry x − kxy rx y = 0 (Mayo, Lewis) BAD IDEA
Chemistry
Old Models
New Models
Completing the Model
dx
dt
= −kxx rx x − kyx ry x
dy
dt
= −kxy rx y − kyy ry y
x, y are the monomer concentrations
kxx , kyx , kxy , kyy are the goal (rate constants)
rx + ry is the (constant) radical concentration
Additional DE:
drx
dt
dr
= − dty = kyx ry x − kxy rx y = 0 (Mayo, Lewis) BAD IDEA
Chemistry
Old Models
New Models
Completing the Model
dx
dt
= −kxx rx x − kyx ry x
dy
dt
= −kxy rx y − kyy ry y
x, y are the monomer concentrations
kxx , kyx , kxy , kyy are the goal (rate constants)
rx + ry is the (constant) radical concentration
Additional DE:
drx
dt
dr
= − dty = kyx ry x − kxy rx y = 0 (Mayo, Lewis) BAD IDEA
Chemistry
Old Models
New Models
Completing the Model
dx
dt
= −kxx rx x − kyx ry x
dy
dt
= −kxy rx y − kyy ry y
x, y are the monomer concentrations
kxx , kyx , kxy , kyy are the goal (rate constants)
rx + ry is the (constant) radical concentration
Additional DE:
drx
dt
dr
= − dty = kyx ry x − kxy rx y = 0 (Mayo, Lewis) BAD IDEA
Chemistry
Old Models
New Models
Mayo Lewis Copolymerization Model
Divide dx/dt by dy /dt, and use the assumption drx /dt = 0
dx
dy
kx =
kxx
kxy , ky
=
=
x(kx x+y )
y (x+ky y )
kyy
kyx ,
‘copolymerization equation’
new rate constants
kx measures the degree to which a radical ending in x
preferentially bonds with x again. (still meaningful)
Chemistry
Old Models
New Models
Mayo Lewis Copolymerization Model
Divide dx/dt by dy /dt, and use the assumption drx /dt = 0
dx
dy
kx =
kxx
kxy , ky
=
=
x(kx x+y )
y (x+ky y )
kyy
kyx ,
‘copolymerization equation’
new rate constants
kx measures the degree to which a radical ending in x
preferentially bonds with x again. (still meaningful)
Chemistry
Old Models
New Models
Mayo Lewis Copolymerization Model
Divide dx/dt by dy /dt, and use the assumption drx /dt = 0
dx
dy
kx =
kxx
kxy , ky
=
=
x(kx x+y )
y (x+ky y )
kyy
kyx ,
‘copolymerization equation’
new rate constants
kx measures the degree to which a radical ending in x
preferentially bonds with x again. (still meaningful)
Chemistry
Old Models
New Models
Using The Copolymerization Equation
dx
dy
=
x(kx x+y )
y (x+ky y )
Basic use: Take multiple measurements of x, y throughout.
dx
at each paired measurement. With two such,
Estimate dy
can solve 2 × 2 linear system to find kx , ky .
y
x
Advanced use: Change variables (e.g. to x+y
, x+y
) and/or
manipulate algebraically. Find kx , ky through regression.
Many variations exist.
Scat with racing stripes: Expand model to consider last two
monomers added.
Chemistry
Old Models
New Models
Using The Copolymerization Equation
dx
dy
=
x(kx x+y )
y (x+ky y )
Basic use: Take multiple measurements of x, y throughout.
dx
at each paired measurement. With two such,
Estimate dy
can solve 2 × 2 linear system to find kx , ky .
y
x
Advanced use: Change variables (e.g. to x+y
, x+y
) and/or
manipulate algebraically. Find kx , ky through regression.
Many variations exist.
Scat with racing stripes: Expand model to consider last two
monomers added.
Chemistry
Old Models
New Models
Using The Copolymerization Equation
dx
dy
=
x(kx x+y )
y (x+ky y )
Basic use: Take multiple measurements of x, y throughout.
dx
at each paired measurement. With two such,
Estimate dy
can solve 2 × 2 linear system to find kx , ky .
y
x
Advanced use: Change variables (e.g. to x+y
, x+y
) and/or
manipulate algebraically. Find kx , ky through regression.
Many variations exist.
Scat with racing stripes: Expand model to consider last two
monomers added.
Chemistry
Old Models
New Models
Critiquing The Copolymerization Equation
dx
dy
Critical Assumption:
=
drx
dt
x(kx x+y )
y (x+ky y )
= kyx ry x − kxy rx y = 0.
Note that kyx , ry , kxy , rx are all constants. Hence
some constant α independent of t.
x
y
= α, for
The model collapses; x and y must react exactly in their
initial proportions. Hence all rate constants are equal (x, y
indistinguishable) ‘random model’.
Chemistry
Old Models
New Models
Critiquing The Copolymerization Equation
dx
dy
Critical Assumption:
=
drx
dt
x(kx x+y )
y (x+ky y )
= kyx ry x − kxy rx y = 0.
Note that kyx , ry , kxy , rx are all constants. Hence
some constant α independent of t.
x
y
= α, for
The model collapses; x and y must react exactly in their
initial proportions. Hence all rate constants are equal (x, y
indistinguishable) ‘random model’.
Chemistry
Old Models
New Models
Critiquing The Copolymerization Equation
dx
dy
Critical Assumption:
=
drx
dt
x(kx x+y )
y (x+ky y )
= kyx ry x − kxy rx y = 0.
Note that kyx , ry , kxy , rx are all constants. Hence
some constant α independent of t.
x
y
= α, for
The model collapses; x and y must react exactly in their
initial proportions. Hence all rate constants are equal (x, y
indistinguishable) ‘random model’.
Chemistry
Old Models
New Models
Critiquing The Copolymerization Equation, II
dx
dy
Critical Assumption:
=
drx
dt
x(kx x+y )
y (x+ky y )
= kyx ry x − kxy rx y = 0.
Can we derive the equation from the starting DEs without
the critical assumption?
This has been done (e.g. probabilistic methods).
It doesn’t help. The copolymerization equation is
equivalent to the critical assumption. It cannot be saved.
Chemistry
Old Models
New Models
Critiquing The Copolymerization Equation, II
dx
dy
Critical Assumption:
=
drx
dt
x(kx x+y )
y (x+ky y )
= kyx ry x − kxy rx y = 0.
Can we derive the equation from the starting DEs without
the critical assumption?
This has been done (e.g. probabilistic methods).
It doesn’t help. The copolymerization equation is
equivalent to the critical assumption. It cannot be saved.
Chemistry
Old Models
New Models
Critiquing The Copolymerization Equation, II
dx
dy
Critical Assumption:
=
drx
dt
x(kx x+y )
y (x+ky y )
= kyx ry x − kxy rx y = 0.
Can we derive the equation from the starting DEs without
the critical assumption?
This has been done (e.g. probabilistic methods).
It doesn’t help. The copolymerization equation is
equivalent to the critical assumption. It cannot be saved.
Chemistry
Old Models
New Models
Critiquing The Copolymerization Equation, II
dx
dy
Critical Assumption:
=
drx
dt
x(kx x+y )
y (x+ky y )
= kyx ry x − kxy rx y = 0.
Can we derive the equation from the starting DEs without
the critical assumption?
This has been done (e.g. probabilistic methods).
It doesn’t help. The copolymerization equation is
equivalent to the critical assumption. It cannot be saved.
Chemistry
Old Models
Outline
Chemistry
Old Models
New Models
New Models
Chemistry
Old Models
New Models
Decoupled Model
Assume vastly more rx reactions.
kxx rx x kyx ry x, kxy rx y kyy ry y
dx
dt
∼
= −kxx rx x
−kxx rx x ≥
−kxy rx y ≥
1
kx yx 1+
≥
dx
dy
∼
= kx yx
dx
dt
dy
dt
dx
dy
dy
dt
∼
= −kxy rx y
≥ −kxx rx x(1 + )
≥ −kxy rx y (1 + )
≥ kx yx (1 + )
Analytic solution:
x(t)
x(0)
=
y (t)
y (0)
kx
Chemistry
Old Models
New Models
Decoupled Model
Assume vastly more rx reactions.
kxx rx x kyx ry x, kxy rx y kyy ry y
dx
dt
∼
= −kxx rx x
−kxx rx x ≥
−kxy rx y ≥
1
kx yx 1+
≥
dx
dy
∼
= kx yx
dx
dt
dy
dt
dx
dy
dy
dt
∼
= −kxy rx y
≥ −kxx rx x(1 + )
≥ −kxy rx y (1 + )
≥ kx yx (1 + )
Analytic solution:
x(t)
x(0)
=
y (t)
y (0)
kx
Chemistry
Old Models
New Models
Decoupled Model
Assume vastly more rx reactions.
kxx rx x kyx ry x, kxy rx y kyy ry y
dx
dt
∼
= −kxx rx x
−kxx rx x ≥
−kxy rx y ≥
1
kx yx 1+
≥
dx
dy
∼
= kx yx
dx
dt
dy
dt
dx
dy
dy
dt
∼
= −kxy rx y
≥ −kxx rx x(1 + )
≥ −kxy rx y (1 + )
≥ kx yx (1 + )
Analytic solution:
x(t)
x(0)
=
y (t)
y (0)
kx
Chemistry
Old Models
New Models
Decoupled Model
Assume vastly more rx reactions.
kxx rx x kyx ry x, kxy rx y kyy ry y
dx
dt
∼
= −kxx rx x
−kxx rx x ≥
−kxy rx y ≥
1
kx yx 1+
≥
dx
dy
∼
= kx yx
dx
dt
dy
dt
dx
dy
dy
dt
∼
= −kxy rx y
≥ −kxx rx x(1 + )
≥ −kxy rx y (1 + )
≥ kx yx (1 + )
Analytic solution:
x(t)
x(0)
=
y (t)
y (0)
kx
Chemistry
Old Models
New Models
Decoupled Model
Assume vastly more rx reactions.
kxx rx x kyx ry x, kxy rx y kyy ry y
dx
dt
∼
= −kxx rx x
−kxx rx x ≥
−kxy rx y ≥
1
kx yx 1+
≥
dx
dy
∼
= kx yx
dx
dt
dy
dt
dx
dy
dy
dt
∼
= −kxy rx y
≥ −kxx rx x(1 + )
≥ −kxy rx y (1 + )
≥ kx yx (1 + )
Analytic solution:
x(t)
x(0)
=
y (t)
y (0)
kx
Chemistry
Old Models
New Models
Decoupled Model
Assume vastly more rx reactions.
kxx rx x kyx ry x, kxy rx y kyy ry y
dx
dt
∼
= −kxx rx x
−kxx rx x ≥
−kxy rx y ≥
1
kx yx 1+
≥
dx
dy
∼
= kx yx
dx
dt
dy
dt
dx
dy
dy
dt
∼
= −kxy rx y
≥ −kxx rx x(1 + )
≥ −kxy rx y (1 + )
≥ kx yx (1 + )
Analytic solution:
x(t)
x(0)
=
y (t)
y (0)
kx
Chemistry
Old Models
New Models
Weaknesses of Decoupled Model
dx
dy
∼
= kx yx
We assumed kxx rx x kyx ry x,
kxy rx y kyy ry y .
We may force rx ry by x(0) y (0), hopefully.
However, what if kxx kyx , or kxy kyy ?
The error is difficult to determine, and yet has a large
effect on the outcome interval.
Need to have x(0) y (0).
Chemistry
Old Models
New Models
Weaknesses of Decoupled Model
dx
dy
∼
= kx yx
We assumed kxx rx x kyx ry x,
kxy rx y kyy ry y .
We may force rx ry by x(0) y (0), hopefully.
However, what if kxx kyx , or kxy kyy ?
The error is difficult to determine, and yet has a large
effect on the outcome interval.
Need to have x(0) y (0).
Chemistry
Old Models
New Models
Weaknesses of Decoupled Model
dx
dy
∼
= kx yx
We assumed kxx rx x kyx ry x,
kxy rx y kyy ry y .
We may force rx ry by x(0) y (0), hopefully.
However, what if kxx kyx , or kxy kyy ?
The error is difficult to determine, and yet has a large
effect on the outcome interval.
Need to have x(0) y (0).
Chemistry
Old Models
New Models
Weaknesses of Decoupled Model
dx
dy
∼
= kx yx
We assumed kxx rx x kyx ry x,
kxy rx y kyy ry y .
We may force rx ry by x(0) y (0), hopefully.
However, what if kxx kyx , or kxy kyy ?
The error is difficult to determine, and yet has a large
effect on the outcome interval.
Need to have x(0) y (0).
Chemistry
Old Models
New Models
Partially Decoupled Model
Assume extreme relative concentrations of x, y and rx , ry :
r
x y , rx ry . More precisely, yx , ryx ∈ [0, ].
dx
dt
dy
dt
drx
dt
r
A∼
= −kxx
r
B∼
= −kxy
C∼
=0
= rx x(−kxx − kyx ryx ) = Arx x
= rx y (−kxy − kyy ryx ) = Brx y
r
= rx x(−kxy yx + kyx ryx ) = Crx x
−kxx ≥ A ≥ (−kxx − kyx )
−kxy ≥ B ≥ (−kxy − kyy )
kyx ≥ C ≥ −kxy A/B ∼
= kx
Chemistry
Old Models
New Models
Partially Decoupled Model
Assume extreme relative concentrations of x, y and rx , ry :
r
x y , rx ry . More precisely, yx , ryx ∈ [0, ].
dx
dt
dy
dt
drx
dt
r
A∼
= −kxx
r
B∼
= −kxy
C∼
=0
= rx x(−kxx − kyx ryx ) = Arx x
= rx y (−kxy − kyy ryx ) = Brx y
r
= rx x(−kxy yx + kyx ryx ) = Crx x
−kxx ≥ A ≥ (−kxx − kyx )
−kxy ≥ B ≥ (−kxy − kyy )
kyx ≥ C ≥ −kxy A/B ∼
= kx
Chemistry
Old Models
New Models
Partially Decoupled Model
Assume extreme relative concentrations of x, y and rx , ry :
r
x y , rx ry . More precisely, yx , ryx ∈ [0, ].
dx
dt
dy
dt
drx
dt
r
A∼
= −kxx
r
B∼
= −kxy
C∼
=0
= rx x(−kxx − kyx ryx ) = Arx x
= rx y (−kxy − kyy ryx ) = Brx y
r
= rx x(−kxy yx + kyx ryx ) = Crx x
−kxx ≥ A ≥ (−kxx − kyx )
−kxy ≥ B ≥ (−kxy − kyy )
kyx ≥ C ≥ −kxy A/B ∼
= kx
Chemistry
Old Models
New Models
Partially Decoupled Model
Assume extreme relative concentrations of x, y and rx , ry :
r
x y , rx ry . More precisely, yx , ryx ∈ [0, ].
dx
dt
dy
dt
drx
dt
r
A∼
= −kxx
r
B∼
= −kxy
C∼
=0
= rx x(−kxx − kyx ryx ) = Arx x
= rx y (−kxy − kyy ryx ) = Brx y
r
= rx x(−kxy yx + kyx ryx ) = Crx x
−kxx ≥ A ≥ (−kxx − kyx )
−kxy ≥ B ≥ (−kxy − kyy )
kyx ≥ C ≥ −kxy A/B ∼
= kx
Chemistry
Old Models
New Models
Partially Decoupled Model, II
The system
dx
dt
= Arx x,
dy
dt
= Brx y ,
drx
dt
= Crx x,
has an analytic solution.
x(t) = x(0) 1−αAe1−αA
x(t)(1−αA)Ct
B/A
y (t) = y (0) 1−αAe1−αA
x(t)(1−αA)Ct
rx (t) = αx(0)C −αA+e1−αA
−x(t)(1−αA)Ct
x(t)
x(0)
=
y (t)
y (0)
A/B
x(t)
x(0)
=
y (t)
y (0)
kx
Chemistry
Old Models
New Models
Partially Decoupled Model, II
The system
dx
dt
= Arx x,
dy
dt
= Brx y ,
drx
dt
= Crx x,
has an analytic solution.
x(t) = x(0) 1−αAe1−αA
x(t)(1−αA)Ct
B/A
y (t) = y (0) 1−αAe1−αA
x(t)(1−αA)Ct
rx (t) = αx(0)C −αA+e1−αA
−x(t)(1−αA)Ct
x(t)
x(0)
=
y (t)
y (0)
A/B
x(t)
x(0)
=
y (t)
y (0)
kx
Chemistry
Old Models
New Models
Partially Decoupled Model, II
The system
dx
dt
= Arx x,
dy
dt
= Brx y ,
drx
dt
= Crx x,
has an analytic solution.
x(t) = x(0) 1−αAe1−αA
x(t)(1−αA)Ct
B/A
y (t) = y (0) 1−αAe1−αA
x(t)(1−αA)Ct
rx (t) = αx(0)C −αA+e1−αA
−x(t)(1−αA)Ct
x(t)
x(0)
=
y (t)
y (0)
A/B
x(t)
x(0)
=
y (t)
y (0)
kx
Chemistry
Old Models
Partially Decoupled Model, III
More precisely,
x(t)
x(0)
=
y (t)
y (0)
A/B
−kxx ≥ A ≥ (−kxx − kyx )
−kxy ≥ B ≥ (−kxy − kyy )
Error Range:
A
B
− k? ≤ kx ≤
kx =
kxx
kxy , ky
=
A
B (1
+ ky k? )
kyy
kyx , k?
=
kyx
kxy
New Models
Chemistry
Old Models
Partially Decoupled Model, III
More precisely,
x(t)
x(0)
=
y (t)
y (0)
A/B
−kxx ≥ A ≥ (−kxx − kyx )
−kxy ≥ B ≥ (−kxy − kyy )
Error Range:
A
B
− k? ≤ kx ≤
kx =
kxx
kxy , ky
=
A
B (1
+ ky k? )
kyy
kyx , k?
=
kyx
kxy
New Models
Chemistry
Old Models
Weaknesses of Partially Decoupled Model
x(t)
x(0)
A
B
− k? ≤ kx ≤
∼
=
A
B (1
y (t)
y (0)
kx
+ ky k? )
Error range depends on ky , k? . Can estimate ky through
dual experiment, but k? is a mystery.
Sensitive to measurement error for x(0), y (0).
Need to have x y , rx ry .
New Models
Chemistry
Old Models
Weaknesses of Partially Decoupled Model
x(t)
x(0)
A
B
− k? ≤ kx ≤
∼
=
A
B (1
y (t)
y (0)
kx
+ ky k? )
Error range depends on ky , k? . Can estimate ky through
dual experiment, but k? is a mystery.
Sensitive to measurement error for x(0), y (0).
Need to have x y , rx ry .
New Models
Chemistry
Old Models
Weaknesses of Partially Decoupled Model
x(t)
x(0)
A
B
− k? ≤ kx ≤
∼
=
A
B (1
y (t)
y (0)
kx
+ ky k? )
Error range depends on ky , k? . Can estimate ky through
dual experiment, but k? is a mystery.
Sensitive to measurement error for x(0), y (0).
Need to have x y , rx ry .
New Models
Chemistry
Old Models
Weaknesses of Partially Decoupled Model
x(t)
x(0)
A
B
− k? ≤ kx ≤
∼
=
A
B (1
y (t)
y (0)
kx
+ ky k? )
Error range depends on ky , k? . Can estimate ky through
dual experiment, but k? is a mystery.
Sensitive to measurement error for x(0), y (0).
Need to have x y , rx ry .
New Models
Chemistry
Old Models
Future Work
• Solutions without restricting relative concentrations.
(analytic? approximate? numerical?)
• Extend previous models to 3 types of monomers.
• Detailed analysis of Mayo-Lewis model.
(can we save the old data?)
• Propagate mathematical knowledge through chemistry
community.
New Models
Chemistry
Old Models
Future Work
• Solutions without restricting relative concentrations.
(analytic? approximate? numerical?)
• Extend previous models to 3 types of monomers.
• Detailed analysis of Mayo-Lewis model.
(can we save the old data?)
• Propagate mathematical knowledge through chemistry
community.
New Models
Chemistry
Old Models
Future Work
• Solutions without restricting relative concentrations.
(analytic? approximate? numerical?)
• Extend previous models to 3 types of monomers.
• Detailed analysis of Mayo-Lewis model.
(can we save the old data?)
• Propagate mathematical knowledge through chemistry
community.
New Models
Chemistry
Old Models
Future Work
• Solutions without restricting relative concentrations.
(analytic? approximate? numerical?)
• Extend previous models to 3 types of monomers.
• Detailed analysis of Mayo-Lewis model.
(can we save the old data?)
• Propagate mathematical knowledge through chemistry
community.
New Models