Characterization of Powders and Porous Materials with ... Theory of Adsorption Surface area and porosity Micro-Porosity Characterization of Powders and Porous Materials with Pharmaceutical

OutlineTheory of Adsorption

Surface area and porosityMicro-Porosity

Characterization of Powders and Porous Materialswith

Pharmaceutical Excipinent Case Studies

Jeffrey Kenvin, PhD

Micromeritics Instrument Corporation

July 2008

[email protected] Surface area and porosity



Outline

1 Theory of Adsorption

2 Surface area and porositySurface AreaThicknessPorosityMacro-porosity

3 Micro-Porosity




Adsorption

Definitions

Adsorption → Enrichment in an interfacial layerAdsorbate → Substance in the adsorbed stateAdsorptive → Adsorbable substance in the fluid phaseAdsorbent → Solid material on which adsorption occurs




Definitions cont’d

Preparation

Clean the surface

Remove volatiles1 Water2 CO2

3 Solvents

Controlled environment1 Inert purge or vacuum2 Temperature control

Avoid Phase Changes




Definitions cont’d

Physical Adsorption

Physisorption → Adsorption without chemical bondingpsat or Psat → Saturation pressure (of the cryogen)p◦ or P◦ → Saturation pressure of the adsorptive

Chemical Adsorption

Chemisorption → Adsorption involving chemical bonding




Adsorption

Physical Adsorption

General phenomenon with a relatively low degree of specificity.

Retains identity; desorbs to fluid phase in its original form.

Exothermic adsorption similar to the energy of condensation.

Rapid equilibration; transport limited.

Chemical Adsorption

Dependent on reactivity of adsorbent and adsorptive.

Chemisorbed molecule may react or dissociate.

Energy is similar to energy change for chemical reaction.

Activated process at elevated temperature.




Adsorption

Physical Adsorption

Molecules from the gas phase strike the surface.

At equilibrium the molecule adsorbs, lose the heat ofadsorption (q), and subsequently desorb from surface.

At equilibrium the rate of condensation = the rate ofdesorption

Constant surface coverage at equilibrium.




Adsorption

Physical Adsorption

Surface features change the adsorption potential.

Surface area models neglect the effects of localizedphenomenon.

Curve surfaces or roughness provide enhanced adsorptionpotential.




Multi-Layer Physical Adsorption

As the system pressure isincreased (gas concentrationalso increases) multiple layerssorb to the surface.

The monolayer coverage, adensely packed single adsorbedlayer is used for determiningsurface area.

As pressure is further increasedand adsorption proceeds gascondenses in the pores and thisvolume of condensed adsorptiveis used for characterizingporosity.




Adsorption

Surface area is easily estimated if the number of N2 molecules thatform monolayer is known.

Calculating Surface Area

Surface Area = nm ×Na

w× σA =

Vm

Vg× Na

w× σA

where:

nm = Monolayer quantity, molVm = Monolayer volume, cm3

Vg = Molar volume of gas at STP, 22414 cm3/molNa = Avogadro’s number, 6.023×1023 molecules/molw = sample mass, gσA = Cross-sectional area of the adsorbate, m2




Physisorption - Hardware

Key Features

Stainless steel manifold

1000, 10, 1 torrtransducers

Dedicated vacuum system

Cryogen level control/longdewar life




Physisorption - Special Considerations

Saturation pressure and temperature for N2 or Ar

“Measured” value of p◦

Calculate bath temperature from measured p◦

Saturation pressure and temperature for Kr

“Measured” value of psat of N2

Calculate bath temperature from psatN2

Calculate p◦ from calculated temperature

Saturation pressure and temperature for CO2

“Entered” value of bath temperature

Calculate p◦ from entered temperature




Adsorptive Properties

Gas specific

Non-ideality

Excess N2 at 77 KAll gas accounting calculations

Density conversion factor

Convert from measured volume to a liquid volumet-plot, BJH, pore volume calculations

Cross-sectional area

Size of a nitrogen moleculeBET, Langmuir, BJH, HK

Hard sphere diameter

pressure correction for micro pore analyses




Non-ideality

Accurately account for the Real gas behavior

z(p◦,T p◦

)=

V measured

V ideal=

6.1103

6.3842= 0.957

The ideal gas law only accounts for 95.7% of the nitrogen; there isa 4.3% excess

α =1z − 1

p◦=

10.957 − 1

755.09= 0.0000594 mmHg−1

The non-ideality factor gives us the excess nitrogen per unit ofpressure




Density conversion factor

Convert quantity adsorbed to a pore volume

φ =ρgas

ρliquid=

Vliquid

22.414l/mol=

0.0347

22414= 0.00155

http://webbook.nist.gov/chemistry/fluid/




Adsorption

Isotherms

Quantity adsorbed vs. pressure.

Pressure is usually varied from vacuum to near atmospheric.

Constant temperature.

Quantity adsorbed is normalized for adsorbent mass.

Six isotherm classifications

� Types I, II, and IV - most materials� Type III - uncommon� Type V - rare� Type IV - highly uniform surface




Isotherm classifications

I

n ads

II

IV

III

V

P

VI




Isotherm classifications - rearrangement

I

n ads

II

P

IV

III

V VI




Isotherm classifications - similarity

I

n ads

II

P

IV

III

VVI




Surface AreaThicknessPorosityMacro-porosity

Type I - Isotherm

Langmuir Isotherm

Mono-layer adsorption

Micropore filling

Finely divided surface

Limiting amount adsorbedas p/p◦ approaches 1

n ads

P

I





Type I Isotherms - Y & X Zeolite

0

50

100

150

200

250

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Qua

ntity

Ads

orbe

d, c

m3 /g

p/po

Faujasite

Y ZeoliteX Zeolite





Type I Isotherms - Y & X Zeolite

0

50

100

150

200

250

1e-07 1e-06 1e-05 0.0001 0.001 0.01 0.1 1

Qua

ntity

Ads

orbe

d, c

m3 /g

p/po

Faujasite

Y ZeoliteX Zeolite





Type I Isotherms - Fluid Cracking Catalyst

0

10

20

30

40

50

60

70

80

90

100

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Qua

ntity

Ads

orbe

d, c

m3 /g

p/po

Fluid Cracking Catalyst, 0.8nm pores

AdsorptionDesorption





Type I Isotherms - Fluid Cracking Catalyst

0

10

20

30

40

50

60

70

80

90

100

1e-07 1e-06 1e-05 1e-04 0.001 0.01 0.1 1

Qua

ntity

Ads

orbe

d, c

m3 /g

p/po

Fluid Cracking Catalyst, 0.8nm pores






Type I Isotherms - Y Zeolite & FCC

0

50

100

150

200

250

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Qua

ntity

Ads

orbe

d, c

m3 /g

p/po

0.8nm pores

Y ZeoliteFCC





Type I Isotherms - Y Zeolite & FCC

0

50

100

150

200

250

1e-07 1e-06 1e-05 0.0001 0.001 0.01 0.1 1

Qua

ntity

Ads

orbe

d, c

m3 /g

p/po

0.8nm pores

Y ZeoliteFCC





Type I Isotherms - ZSM-5

0

20

40

60

80

100

120

140

160

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Qua

ntity

Ads

orbe

d, c

m3 /g

p/po

ZSM-5, 0.5-0.6nm pores

Adsorption





Type I Isotherms - ZSM-5

0

20

40

60

80

100

120

140

160

1e-08 1e-07 1e-06 1e-05 0.0001 0.001 0.01 0.1 1

Qua

ntity

Ads

orbe

d, c

m3 /g

p/po

ZSM-5, 0.5-0.6nm pores

Adsorption





Type I Isotherms - TS-1

0

20

40

60

80

100

120

140

160

180

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Qua

ntity

Ads

orbe

d, c

m3 /g

p/po

Titano-silicate, 0.5nm pores

Adsorption





Type I Isotherms - TS-1

0

20

40

60

80

100

120

140

160

180

1e-07 1e-06 1e-05 1e-04 0.001 0.01 0.1 1

Qua

ntity

Ads

orbe

d, c

m3 /g

p/po

Titano-silicate, 0.5nm pores

Adsorption





Type I Isotherms - TS-1 & ZSM-5

0

20

40

60

80

100

120

140

160

180

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Qua

ntity

Ads

orbe

d, c

m3 /g

p/po

0.5nm pores

TS-1ZSM-5





Type I Isotherms - TS-1 & ZSM-5

0

20

40

60

80

100

120

140

160

180

1e-08 1e-07 1e-06 1e-05 0.0001 0.001 0.01 0.1 1

Qua

ntity

Ads

orbe

d, c

m3 /g

p/po

0.5nm pores

TS-1ZSM-5





Type I Isotherms - Single Wall Carbon Nanotubes

0

50

100

150

200

250

300

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

Qua

ntity

Ads

orbe

d, c

m3 /g

p/po

Carbon Nanotubes

Adsorption





Type I Isotherms - Single Wall Carbon Nanotubes

0

50

100

150

200

250

300

1e-08 1e-07 1e-06 1e-05 1e-04 0.001 0.01 0.1 1

Qua

ntity

Ads

orbe

d, c

m3 /g

p/po

Carbon Nanotubes





Langmuir

Langmuir Model of Type I Isotherm

dNa

dt= ap(1− θ)− βθ exp

(−E

RT

)= 0, equilibrium

ap(1− θ) = βθ exp

(−E

RT

)b = K exp(E/RT )

where: θ ≡ fraction of surface occupieda ≡ adsorption coefficientβ ≡ desorption coefficientp ≡ equilibrium pressureK ≡ a / β





Langmuir

Langmuir Model

θ

1− θ= bp

θ → 0, Langmuir model → Henry’s law

limθ→0

(θ

1− θ

)= θ

θ = bp





Langmuir

Langmuir Model of Type I Isotherm

n

nm= θ

rearrange the Langmuir model to a more convenient form . . .

n

nm=

bp

1 + bp

where: n ≡ quantity adsorbednm ≡ monolayer capacity





N2 Adsorption on 13x Zeolite

Type I

Crystalline

10 (8) A, pore

Silica-alumina

Ca+ exchangedzeolite

0

20

40

60

80

100

120

140

0 0.2 0.4 0.6 0.8 1

Qua

ntity

Ads

orbe

d, c

m3 /g

STP

Pressure, mmHg

Nitrogen Adsorption, 13x Zeolite

Adsorption






0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0 0.2 0.4 0.6 0.8 1

p/Q

, mm

Hg/

(cm

3 /g S

TP)

Pressure, mmHg

Langmuir Transformation, 13x Zeolite

13X

Linearized Langmuir Model → 620m2/g , b = 563.5/mmHg

p

n=

1

bnm+

p

nm






0

20

40

60

80

100

120

140

0 0.2 0.4 0.6 0.8 1

Qua

ntity

Ads

orbe

d, c

m3 /g

STP

Pressure, mmHg

13x Zeolite

13XLangmuir model





Type II - Isotherm

BET Isotherm

Non-porous surface

Uniform surface

Multilayer adsorption

Infinite adsorption asp/p◦ approaches 1

n ads

P

II





Type II Isotherms - Silica 1000A

0

5

10

15

20

25

30

35

40

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Qua

ntity

Ads

orbe

d, c

m3 /g

p/po

Silica, 100nm pores






Type II Isotherms - Silica 1000A

0

5

10

15

20

25

30

35

40

1e-07 1e-06 1e-05 0.0001 0.001 0.01 0.1 1

Qua

ntity

Ads

orbe

d, c

m3 /g

p/po

Silica, 100nm pores





Type II Isotherms - SRB D5 Carbon Black

0

10

20

30

40

50

60

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Qua

ntity

Ads

orbe

d, c

m3 /g

p/po

SRB D5 Carbon Black

Adsorption





Type II Isotherms - SRB D5 Carbon Black

0

10

20

30

40

50

60

1e-07 1e-06 1e-05 0.0001 0.001 0.01 0.1 1

Qua

ntity

Ads

orbe

d, c

m3 /g

p/po

SRB D5 Carbon Black





Brunauer, Emmett, and Teller

Surface Area

Stephen Brunauer

Paul Emmett

Edward Teller

N2 adsorption

Fixed Nitrogen Laboratory,1938

Second most cited reference over a 50 year period





BET Assumptions

Surface Area

Multi-layer adsorption

Non-porous, Uniform surface

Heat of adsorption for the first layer is higher than successivelayers.

Heat of adsorption for second and successive layers equals theheat of liquefaction

Lateral interactions of adsorbed molecules are ignored





BET Model

First Layer

a1pθ0 = b1θ1 exp

(−E1

RT

)where: a1pθ0 ≡ rate of condensation

b1θ1 exp(−E1RT

)≡ rate of evaporation

θ0 ≡ fraction of bare surfaceθ1 ≡ fraction of covered surface

Rate of condensation = rate of desorption

pθ0 =b1

a1θ1 exp

(−E1

RT

)[email protected] Surface area and porosity




BET Model

All Layers

pθ0 =b1

a1θ1 exp

(−E1

RT

)

pθ1 =b2

a2θ2 exp

(−E2

RT

)pθ2 =

b3

a3θ3 exp

(−E3

RT

)...

pθi−1 =bi

aiθi exp

(−Ei

RT

)

Ei>1 = El

pθ0 =b1

a1θ1 exp

(−E1

RT

)

pθ1 =b2

a2θ2 exp

(−El

RT

)pθ2 =

b3

a3θ3 exp

(−El

RT

)...

pθi−1 =bi

aiθi exp

(−El

RT

)

bi/ai = g

pθ0 =b1

a1θ1 exp

(−E1

RT

)

pθ1 = gθ2 exp

(−El

RT

)pθ2 = gθ3 exp

(−El

RT

)...

pθi−1 = gθi exp

(−El

RT





BET Model

Sum of Surface Fractions

θ0 + θ1 + θ2 + θ3 + . . .+ θi + · · · = 1

Total Quantity Adsorbed

n = nm (1θ1 + 2θ2 + 3θ3 + . . .+ iθi + · · · )

Multilayer has Infinite Thickness

p

p◦= 1, i → inf






Type II Isotherm

n

nm=

Cx

(1− x)(1− x + Cx)

where: n ≡ quantity adsorbednm ≡ monolayer capacity

C ≡ adsorption coefficient, ≈ exp E1−ElRT

x ≡ relative pressure at equilibrium, p/p◦






Linearized BET

p

n(p◦ − p)=

1

nmC+

C − 1

nmC× p

p◦

1 Plot pn(p◦−p) vs. p

p◦ , 0.05 ≤ pp◦ ≤ 0.30

2 nm = 1slope+intercept

3 C = 1 + slopeintercept

4 C > 0





N2 Adsorption on Macro-Porous Silica

Type II

Amorphous

1000A, pore

Desorption

Lack ofHysteresis

0

5

10

15

20

25

30

35

40

0 0.2 0.4 0.6 0.8 1

Qua

ntity

Ads

orbe

d, c

m3 /g

STP

Relative Pressure, p/po

Nitrogen Isotherm, Lichrosphere 1000






N2 - BET Transformation - 25.7 m2/g

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0 0.05 0.1 0.15 0.2 0.25 0.3

1/Q

(po /p

-1)


Linear BET, Lichrosphere 1000

Lic 1000





N2 Adsorption on Carbon Black

Type II

Amorphous

Nano-particle

0

10

20

30

40

50

60

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Qua

ntity

Ads

orbe

d, c

m3 /g

p/po

SRB D5 Carbon Black

Adsorption





SRB D5 Carbon Black - 21.2 m2/g

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0 0.05 0.1 0.15 0.2 0.25 0.3

1/(q

ads(

po /p -

1))

p/po

SRB D5





Special considerations

Low Surface Area Materials

Krypton - p◦ is 1/300 of N2 p◦

Eliminate “void space errors”

Approximately same quantity adsorbed for Kr or N2

Same error for “void space”

Error is proportional to peVfs

Typical N2 experiment 35 - 220 mmHg

Typical Kr experiment 0.01 - 0.5 mmHg

Additonal Hardware

Turbo-pump

10 torr transducer





Alternate approach to Kr

Balanced Tube Design

Balance tubeeliminates “voidspace” errors

Rapid analysis





Special Case - Single Point Surface Area

Linearized BET Transformation

p

n(p◦ − p)=

1

nmC+

C − 1

nmC× p

p◦

Assume C →∞ (C > 100)

limC→∞

(1

nmC

)= 0

limC→∞

(C − 1

nmC

)=

1

nm

Single point estimate of nm

nm = n

(1− p

p◦





Physisorption - Dynamic Adsorption Hardware

Key Features

Rapid Analysis

Wide range of materials

High-sensitivity

Low cost





Physisorption - Dynamic Adsorption

Analysis Tips

1 Desorption peak isused for Vm

2 Simple calibration

N2 injections

3 Not limited to N2

adsorption

Kr ,Ar ,CO2, . . .

-1.5

-1

-0.5

0

0.5

1

1.5

0 1 2 3 4 5 6 7 8 9

TC

D S

igna

l, V

time, minutes

Dynamic N2 Adsorption/Desorption, SiO2-Al2O3

N2 Adsorption

N2 Desorption





Physisorption - Special Case - Magnesium Stearate

0

5

10

15

20

25

30

35

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Qua

ntity

Ads

orbe

d, c

m3 /g


Static N2 Adsorption, Magnesium Stearate





Physisorption - BET Surface Area Magnesium Stearate

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22

0 0.05 0.1 0.15 0.2 0.25 0.3

p/(Q

*(po -p

))


Static N2 Adsorption, Magnesium Stearate

7.14 m2/g

9.3 m2/g





Excipients

Pharmaceutical excipients

Substances other than the pharmacologically active drug orprodrug which are included in the manufacturing process or arecontained in a finished pharmaceutical product dosage form.

Commonly used excipients

Calcium stearate

Gelatin

Lactose

Magnesium stearate

Microcrystalline cellulose

Silicon dioxide

Sodium stearate

Stearic acid

Sucrose

Talc

Titanium dioxide





Investigate the impact of preparation temperature

Both excipients and APIs may be sensitive to temperature.

The preparation (removal of moisture, solvents, and ambientgases) is often performed at or near room temperature.

1 Purge with inert gas (nitrogen) or evacuate2 Control temperature at 40◦C3 Duration 4 - 24 hours





Oxides

Commonly used oxides

1 Aluminum oxideAlumina - Al2O3

2 Silicon dioxideSilica - SiO2

3 Titanium dioxideTitania - TiO2

Not chemically active

Stable

Do not dissolve

Range of particle sizes

Range of surface area andporosity





Effect of preparation temperature on the SSA of Al2O3

0.3

0.32

0.34

0.36

0.38

0.4

0.42

0.44

20 40 60 80 100 120

Sur

face

Are

a, m

2 /g

Preparation Temperature, °C

α-alumina

Surface area increases 33% as preparation temperature is increasedfrom 40 to 100◦C as CO2 is removed.





Effect of preparation temperature on the SSA of SiO2

367

368

369

370

371

372

373

374

375

376

20 40 60 80 100 120

Sur

face

Are

a, m

2 /g


Silicon Dioxide

Surface area increases 3% as preparation temperature is increasedfrom 40 to 60◦C as H2O is removed.





Effect of preparation temperature on the SSA of TiO2

8.95

9

9.05

9.1

9.15

9.2

9.25

9.3

50 100 150 200 250 300

Sur

face

Are

a, m

2 /g


Titanium Dioxide

A slight increase in SSA is observed for the titania.





Glidants and Lubricants

for making tablets

1 Stearic Acid

2 Calcium Stearate

3 Magnesium Stearate

4 Sodium Stearate

5 Talc





Effect of temperature on the SSA of Stearic Acid

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

20 40 60 80 100 120

Sur

face

Are

a, m

2 /g


Stearic Acid

The loss is surface area indicates the melting of the stearic acidsmall particles → large particles.





Effect of temperature on the SSA of Ca2+ Stearate

5.6

5.8

6

6.2

6.4

6.6

6.8

7

20 40 60 80 100 120

Sur

face

Are

a, m

2 /g


Calcium Stearate

SSA drops as temperature increases.





Effect of temperature on the SSA of Mg2+ Stearate

2.5

2.6

2.7

2.8

2.9

3

3.1

3.2

20 40 60 80 100 120

Sur

face

Are

a, m

2 /g


Magnesium Stearate

SSA decreases 20% as temperature increases.





Effect of temperature on the SSA of Na+ Stearate

7.35

7.4

7.45

7.5

7.55

7.6

7.65

7.7

20 40 60 80 100 120

Sur

face

Are

a, m

2 /g


Sodium Stearate

SSA reaches at optimum at 60◦C and then decreases withincreasing temperature.





Effect of temperature on the SSA of Stearates

0

1

2

3

4

5

6

7

8

20 40 60 80 100 120

Sur

face

Are

a, m

2 /g


Na+, Ca2+, Mg2+ Stearate, and Stearic Acid

Na+ StearateCa2+ StearateMg2+ StearateStearic Acid





Effect of temperature on the SSA of Talc

7.35

7.4

7.45

7.5

7.55

7.6

7.65

7.7

20 40 60 80 100 120

Sur

face

Are

a, m

2 /g


Talc





Other Excipients

Binders and fillers

1 Gelatin - solution binder

2 Lactose - binder

3 Microcrystalline Cellulose - binder

4 Sucrose - filler





Effect of temperature on the SSA of Gelatin

0.205

0.21

0.215

0.22

0.225

0.23

0.235

0.24

0.245

0.25

20 40 60 80 100 120

Sur

face

Are

a, m

2 /g


Gelatin





Effect of temperature on the SSA of Lactose

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

20 40 60 80 100 120

Sur

face

Are

a, m

2 /g


Lactose Monohydrate

Significant increase in SSA as preparation temperature is increased.





Effect of temperature on the SSA of MicrocrystallineCellulose

1.48

1.5

1.52

1.54

1.56

1.58

1.6

1.62

20 40 60 80 100 120

Sur

face

Are

a, m

2 /g


Microcrystalline Cellulose





Effect of temperature on the SSA of Sucrose

0.16

0.17

0.18

0.19

0.2

0.21

0.22

0.23

20 40 60 80 100 120

Sur

face

Are

a, m

2 /g


Sucrose





Type IV - Isotherm

Mesoporous Materials

Multi-layer adsorption

Reduced saturationpressure in pores

Hysteresis

� Shape� Tortuosity

n ads

P

IV





N2 Adsorption on Amorphous Silica-Alumina

Type IV

Amorphous

100A, pore

Desorption

Hysteresis

0

50

100

150

200

250

300

350

400

450

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Qua

ntity

ads

orbe

d, c

m3 /g

p/po

Amorphous Silica Alumina, 11nm pores







0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

1/(q

ads(

po /p -

1))

p/po

BET Surface Area = 215.5 m2/g





N2 Adsorption on MCM-41

Type IV

MesoporousSilica

40A,cylindricalpore

Desorption

Lack ofHysteresis 0

100

200

300

400

500

600

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Qua

ntity

ads

orbe

d, c

m3 /g

p/po

Silica, 4 nm pores







0

0.0002

0.0004

0.0006

0.0008

0.001

0.0012

0.0014

0.0016

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

1/(q

ads(

po /p -

1))

p/po

BET Surface Area = 926.8





Standard Isotherms

Thickness

Monolayer region is sensitive to isotherm shape

Multilayer region is not sensitive to isotherm shape

Multilayer region is less dependent on the adsorbent structure

Multilayer Thickness

t = d ′n

nm

where: d ′ ≡ thickness of the monolayer, d ′ = 3.54A for N2





t-Plot → “Rules of Thumb”

Plot Va vs. t

Slope of a linear region corresponds to area

Intercept from a linear region is a pore volume

Based on BET surface area

n ads

thickness, Å thickness, Å





t-Plot → “Micro Porous Sample”

Plot Va vs. t

Slope corresponds to external (matrix) area

Intercept is the micro pore volume

t-curve is critical

n ads

thickness, Å

Flat Surface

External Area

µ Pore Vol

thickness, Å





t-Plot → “Meso Porous Sample”

Plot Va vs. t

Low ”t” slope is pore area

Intercept is meso pore volume

High ”t” slope is external area

n ads

thickness, Å

Flat Surface

External Area

µ Pore Vol

thickness, Å

Flat Surface

External Area

Pore Area

Meso Pore Vol





Statistical t-Curve

Halsey

t = 3.54×

(−5

ln pp◦

) 13

Harkins and Jura

t =

(13.99

0.034− log10pp◦

) 12





Statistical t-Curve

Jaroniec et. al.

t =

(60.65

0.03071− log10pp◦

)0.3968

Broekhoff de Boer

F (te) = 2.303R

(−16.11

t2e

+ 0.1682e−0.1137te

), F (t) = R ln

(p

p◦

)





Statistical t-Curve

Exteranl Surface Area of Carbon Black

Special application of t-plot to determine the external area ofcarbon.

Replaces the traditional CTAB

cetyltrimethyl ammonium bromide

Carbon STSA

t = 0.88

(p

p◦

)2

+ 6.45

(p

p◦

)+ 2.98





The t-Method

t-Plot

Statistical curves

0

5

10

15

20

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Thi

ckne

ss, a

ngst

rom

s

p/po

HalseyHarkins and Jura

Jaroniec et. al.Broekhoff de Boer





The t-Method

t-Plot

Silica surface with 1000 A pores

DFT used to determine monolayer capacity

0

5

10

15

20

25

30

35

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Thi

ckne

ss, a

ngst

rom

s

p/po





The t-Method

t-Plot


BET used to determine monolayer capacity

0

5

10

15

20

25

30

35

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Thi

ckne

ss, a

ngst

rom

s

p/po

DFTBET





The t-Method

t-Plot


Silane treatment to remove OH−1

0

5

10

15

20

25

30

35

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Thi

ckne

ss, a

ngst

rom

s

p/po

DFTODMS





The t-Method → Microporous sample - 13X

t-Plot → external area = 201.3 m2/g

Faujasite - 13X

Silica surface with 1000 A pores - DFT used for Vm

0

20

40

60

80

100

120

140

160

0 0.5 1 1.5 2 2.5

Qua

ntity

Ads

orbe

d, c

m3 /g

Thickness, angstroms

Micropore filling

External area





The t-Method → Microporous sample - Y Zeolite


Faujasite - Y Zeolite

Silica reference curve

0

50

100

150

200

250

0 2 4 6 8 10 12 14 16

q ads

, cm

3 /g

thickness, Å





The t-Method → Microporous sample - FCC


FCC - Y Zeolite & binder


0

10

20

30

40

50

60

70

80

0 2 4 6 8 10 12 14 16

q ads

, cm

3 /g

thickness, Å





The t-Method → Microporous sample - ZSM-5


ZSM-5


0

20

40

60

80

100

120

140

160

0 2 4 6 8 10 12

q ads

, cm

3 /g

thickness, Å





The t-Method → Microporous sample - TS-1


Titano-silicate


0

20

40

60

80

100

120

140

160

180

200

0 2 4 6 8 10 12 14 16

q ads

, cm

3 /g

thickness, Å





The t-Method → Mesoporous sample - MCM 41

t-Plot → pore area = 699.4 m2/g

40 A mesoporous silica, cylindrical pores


0

100

200

300

400

500

600

700

0 2 4 6 8 10 12 14

Qua

ntity

Ads

orbe

d, c

m3 /g


Pore area





The t-Method → Mesoporous sample - MCM 41


40 A mesoporous silica, cylindrical pores


0

100

200

300

400

500

600

0 2 4 6 8 10 12 14

Qua

ntity

Ads

orbe

d, c

m3 /g


External area





The t-Method → Mesoporous sample - Silicaalumina

t-Plot → external area = 143 m2/g

110 A mesoporous silica


0

50

100

150

200

250

300

350

400

0 2 4 6 8 10 12 14

Qua

ntity

Ads

orbe

d, c

m3 /g






The t-Method

0

100

200

300

400

500

600

0 2 4 6 8 10 12 14

Qua

ntity

Ads

orbe

d, c

m3 /g


40 angstrom30 angstrom

110 angstrom8 angstrom





The t-Method → SRB D5 - Carbon Black


D5

STSA for Carbon Black

0

2

4

6

8

10

12

0 1 2 3 4 5 6 7 8 9

q ads

, cm

3 /g

thickness, Å

SRB D5





Excipients - t-plot

Pharmaceutical excipients

Use the t-plot method to determine if select excipients exhibitmicro porosity or meso porosity

Commonly used excipients

Calcium stearate

Gelatin

Lactose

Magnesium stearate

Microcrystalline cellulose

Silicon dioxide

Sodium stearate

Stearic acid

Sucrose

Talc

Titanium dioxide





Effect of preparation temperature on the porosity ofCalcium Stearate

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0 2 4 6 8 10

Qua

ntity

, cm

3 /g

thickness, Å

Calcium Stearate





Effect of preparation temperature on the porosity ofMagnesium Stearate

0

0.5

1

1.5

2

2.5

0 2 4 6 8 10

Qua

ntity

, cm

3 /g

thickness, Å

Magnesium Stearate





Effect of preparation temperature on the porosity ofMicrocrystalline cellulose

0

0.2

0.4

0.6

0.8

1

1.2

0 2 4 6 8 10

Qua

ntity

, cm

3 /g

thickness, Å

Microcrystalline Cellulose





Effect of preparation temperature on the porosity of SiO2

0

50

100

150

200

250

0 2 4 6 8 10

Qua

ntity

, cm

3 /g

thickness, Å

Silicon Dioxide





Effect of preparation temperature on the porosity of Talc

0

1

2

3

4

5

6

0 2 4 6 8 10

Qua

ntity

, cm

3 /g

thickness, Å

Talc





Effect of preparation temperature on the porosity of TiO2

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0 2 4 6 8 10

Qua

ntity

, cm

3 /g

thickness, Å

Titanium Dioxide





Capillary Condensation

Mesoporous

Adsorbed layer

Condensed phase

Liquid Nitrogen

t t

r r





BJH Calculations

tt

rr

Pore Width

Hydraulic radius

Kelvin equation

Adsorbed Layer

Thickness equation or curve





BJH Calculations

Combine Kelvin Equation with the Thickness of the Adsorbed Layer

rp = rk + t

wp = 2× (rk + t)

where: wp ≡ pore width (diameter)rp ≡ pore radiusrk ≡ hydraulic radiust ≡ thickness of the adsorbed layer





BJH Calculations

Kelvin Equation for Cylindrical Pores

Calculate the hydraulic radius for capillary condensation inMeso-pores.

Cylindrical geometry is the standard for BJH calculations.

Pore size > 20 A, (reduced precision below 75 A)

RT lnp

p◦= −2γv l

rk

where: γ ≡ surface tensionv l ≡ liquid molar volumerk ≡ hydraulic radius





BJH Calculations

Kelvin Equation for Slit-shaped Pores

Calculate the hydraulic radius for capillary condensation inMeso-pores.

Hydraulic radius is 2D, width of an infinite slit.

RT lnp

p◦= −γv

l

rk

where: γ ≡ surface tensionv l ≡ liquid molar volumerk ≡ hydraulic radius





BJH Example Data

AmorphousSilicaAlumina

Surface area -214 m2/g

0

50

100

150

200

250

300

350

400

0 0.2 0.4 0.6 0.8 1

Qua

ntity

Ads

orbe

d, c

m3 /g

ST

P


Nitrogen Isotherm, Amorphous Silica-Alumina






BJH Example Data


BET Surfacearea - 214m2/g

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

10 100 1000

Cum

ulat

ive

Por

e V

olum

e, c

m3 /g

D, angstroms





BJH Example Data



0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

10 100 1000

Cum

ulat

ive

Por

e V

olum

e, c

m3 /g

dV/d

D

D, angstroms





BJH Example Data



0

50

100

150

200

250

300

10 100 1000

Cum

ulat

ive

Por

e A

rea,

m2 /g

D, angstroms





BJH Example Data



0

50

100

150

200

250

300

10 100 1000

Cum

ulat

ive

Por

e A

rea,

m2 /g

dSA

/dD

D, angstroms





BJH Pore Calculations Adsorption Data



0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

10 100 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

pore

vol

ume,

cm

3 /g

dV/d

(log(

D))

, (cm

3 /g)/

Å

width, Å





BJH Pore Calculations Desorption Data



0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

10 100 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

pore

vol

ume,

cm

3 /g

dV/d

(log(

D))

, (cm

3 /g)/

Å

width, Å





BJH Example Data

MesoporousSilica

Surface area -926.8 m2/g

0

100

200

300

400

500

600

0 0.2 0.4 0.6 0.8 1

Qua

ntity

Ads

orbe

d, c

m3 /g

ST

P


Nitrogen Adsorption, MCM-41

Adsorption





BJH Example Data


BET Surfacearea - 926.8m2/g

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

10 100 1000

Cum

ulat

ive

Por

e V

olum

e, c

m3 /g

D, angstroms





BJH Example Data



0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

10 100 1000

Cum

ulat

ive

Por

e V

olum

e, c

m3 /g

dV/d

D

D, angstroms





BJH Example Data



0

100

200

300

400

500

600

700

800

900

1000

10 100 1000

Cum

ulat

ive

Por

e A

rea,

m2 /g

D, angstroms





BJH Example Data



0

100

200

300

400

500

600

700

800

900

1000

10 100 1000

Cum

ulat

ive

Por

e A

rea,

m2 /g

dSA

/dD

D, angstroms





Mercury Intrusion Porosimetry





Sample cell - Penetrometer





Low Pressure Analysis





High Pressure Analysis





Sample cell - Penetrometer





Washburn equation

Intrusion Force

P · A = P · πd2

4

Resistance Forcefriction = πdγ cos(θ)

Force Balance

P · πd2

4= πdγ cos(θ)

Washburn Equation

d =4γ cos(θ)

P





“Ink-bottle” Pores

Trap Hg in the sample - extrusion rarely follows intrusion





“Ink-bottle” Pores





Effect of equillibrium time





Pore size distributions





Alumina

Total Intrusion Volume 1.2166 mL/gTotal Pore Area 305.880 m2/gMedian Pore Diameter (Volume) 0.0171 µmMedian Pore Diameter (Area) 0.0084 µmAverage Pore Diameter (4V/A) 0.0159 µmBulk Density at 0.56 psia 0.6636 g/mLApparent (skeletal) Density 3.4454 g/mLPorosity 80.7390 %Stem Volume Used 60 %





Summary Data

Total Intrusion Volume - capacitance to volume measurement

Total Pore Area - we have used the Washburn equation tocalculate a size for each pressure. This diameter is then usedwith the incremental volume to determine the area of acylinder.

Median Pore Diameter1 by Volume - the diameter is calculated at the pressure

corresponding to 50% of the total intrussion volume.2 by Area - the diameter is calculated at the pressure

corresponding to 50% of the total intrussion Area.

Average pore diameter - 4V/A





Density and porosity

Bulk density - actually an envelope density determined by thequantity of Hg in the penetrometer empty (calibrated) versusthe quantity at low pressure with the sample.

Apparent density - similar to true density - density of thematerial determined at high pressure and subject tocompressibility effects.

Porosity - using both the bulk and apparent density todetermine percentage of void space = 100*(1 - ρB/ρA)






0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0.001 0.01 0.1 1 10 100 1000

Intr

usio

n, m

l/g

Pore width, µm

DV

Volume






0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0.001 0.01 0.1 1 10 100 10000.0

50.0

100.0

150.0

200.0

250.0

300.0

350.0In

trus

ion,

ml/g

Por

e ar

ea, m

2 /g

Pore width, µm

DV DA

VolumeArea






0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0.001 0.01 0.1 1 10 100 10000.0

50.0

100.0

150.0

200.0

250.0

300.0

350.0In

trus

ion,

ml/g

Por

e ar

ea, m

2 /g

Pore width, µm

DV DA

4V/A

VolumeArea






0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0.001 0.01 0.1 1 10 100 10000.0

0.5

1.0

1.5

2.0

2.5In

trus

ion,

ml/g

Log

diffe

rent

ial i

ntru

sion

Pore width, µm

Cumulative IntrusionLog Differential Intrusion





Porosity of tablets

Tableting process influences pore size and ultimately dissolution

Use the mercury intrusion to evaluate the porosity of tablets

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

1 10 100 1000 10000 100000

Intr

usio

n V

olum

e, c

m3 /g

Pressure, psia

Excedrin Tablet

IntrusionExtrusion





Intrusion volume of Excedrin tablet

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.001 0.01 0.1 1 10 100

Intr

usio

n V

olum

e, c

m3 /g

Pressure, psia

Excedrin Tablet

Intrusion





Intrusion and pore size of an Excedrin tablet

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.001 0.01 0.1 1 10 100 0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

Intr

usio

n V

olum

e, c

m3 /g

Log

Diff

eren

tial I

ntru

sion

Size, µm

Excedrin Tablet





Effect of breaking the Excedrin tablet

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.001 0.01 0.1 1 10 100 0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

Intr

usio

n V

olum

e, c

m3 /g

Log

Diff

eren

tial I

ntru

sion

Size, µm

Excedrin

TabletBroken





Acetaminophen tablet

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.001 0.01 0.1 1 10 100 0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

Intr

usio

n V

olum

e, c

m3 /g

Log

Diff

eren

tial I

ntru

sion

Size, µm

Acetaminophen Tablet

TabletBroken





Acetaminophen caplet

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.001 0.01 0.1 1 10 100 0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

Intr

usio

n V

olum

e, c

m3 /g

Log

Diff

eren

tial I

ntru

sion

Size, µm

Acetaminophen Caplet

CapletBroken





Tablets and Caplets

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.001 0.01 0.1 1 10 100

Intr

usio

n V

olum

e, c

m3 /g

Size, µm

Excedrin, Acetaminophen Tablets and Caplets

ExcedrinAcetaminophen TabAcetaminophen Cap





Tablets and Caplets

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.001 0.01 0.1 1 10 100

Log

Diff

eren

tial I

ntru

sion

, cm

3 /g

Size, µm

Excedrin, Acetaminophen Tablets and Caplets

ExcedrinAcetaminophen TabAcetaminophen Cap




Structures

Common Structures

1 ZSM-5

� Nan[AlnSi96−nO192], n > 27� MFI - Structure Code

2 13x

� (Na2,Ca,Mg)29[Al58Si134O184]� FAU - Structure Code

3 H-Y

� H53.3[Al53.3Si138.7O357.3]� FAU - Structure Code




Structures

Common Structures

1 ZSM-5

� Nan[AlnSi96−nO192], n > 27� MFI - Structure Code

2 13x

� (Na2,Ca,Mg)29[Al58Si134O184]� FAU - Structure Code

3 H-Y

� H53.3[Al53.3Si138.7O357.3]� FAU - Structure Code




Adsortives

Nitrogen

Argon




N2 Adsorption on Faujasite, 13X

FAU Structures

13x

H-Y

0

20

40

60

80

100

120

140

160

1e-008 1e-007 1e-006 1e-005 0.0001 0.001 0.01

Qua

ntity

Ads

orbe

d, c

m3 /g

p/po

CationNa+




N2 Adsorption on Faujasite, H-Y

FAU Structures

13x

H-Y

0

50

100

150

200

250

1e-007 1e-006 1e-005 0.0001 0.001 0.01 0.1 1

Qua

ntity

Ads

orbe

d, c

m3 /g

p/po

CationH+




N2 Adsorption on Faujasite

FAU Structures

13x

H-Y

0

50

100

150

200

250

1e-008 1e-007 1e-006 1e-005 0.0001 0.001 0.01 0.1 1

Qua

ntity

Ads

orbe

d, c

m3 /g

p/po

CationH+

Na+




Ar Adsorption on ZSM-5

0

20

40

60

80

100

120

140

160

180

200

1e-007 1e-006 1e-005 0.0001 0.001 0.01 0.1 1

SiO2:Al2O330:155:180:1

280:1

MFI Structure

SiO2:Al2O3 ratios -30, 55, 80, and 280

Argon isothermscollected at 77 K

Significanttransition




Ar Adsorption on ZSM-5

0

50

100

150

200

250

1e-007 1e-006 1e-005 0.0001 0.001 0.01 0.1 1

SiO2:Al2O330:155:180:1

280:1

MFI Structure

SiO2:Al2O3 ratios -30, 55, 80, and 280

Argon isothermscollected at 77 K

Significanttransition




Microporosity - Dubinin

Common Models

Dubinin

Horwath and Kawazoe

Density Functional Theory

Generalized Form

A = RT ln

(po

p

)W = W0 exp

[−(

A

βE0

)n]




Microporosity - HK Slit Pores

Common Models

Dubinin

Horwath and Kawazoe


Slit-shape Pore Geometry

RT ln

(p

po

)= K

[(NaAa + NAAA)

σ4 (l − d)

]×[

σ4

3 (l − d/2)3− σ10

9 (l − d/2)9− σ4

4 (d/2)4+

σ10

9 (d/2)9

]




Microporosity - HK Cylindrical Pores

Common Models

Dubinin

Horwath and Kawazoe


Cylindrical-shape Pore Geometry

RT ln

(p

po

)=

3

4πK

[(NaAa + NAAA)

d4

]×

inf∑k=0

[1

2k + 1

(1− d

rp

)2k

−

(21

32αk

(d

rp

)10

− βk

(d

rp

)4)]




Microporosity - DFT

Common Models

Dubinin

Horwath and Kawazoe


Integral Equation of Adsorption

Q(p) =

∫dH q(p, h) f (H)




ZSM-5 Pore Size Distribution

H-K Model

Saito-Foley Model

Cylindrical PoreGeometry

Nitrogen, 77 K

0

20

40

60

80

100

120

140

160

180

200

1e-008 1e-007 1e-006 1e-005 0.0001 0.001 0.01 0.1 1

Qua

ntity

Ads

orbe

d, c

m3 /g

p/po

SiO2:Al2O330:1





H-K Model

Saito-Foley Model


Nitrogen, 77 K

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

4 5 6 7 8 9 10 11 12 13

dV/d

W, c

m3 /g

-A

W, A

SiO2:Al2O330:1





H-K Model

Saito-Foley Model


Nitrogen, 77 K

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

4 5 6 7 8 9 10 11 12 13

dV/d

W, c

m3 /g

-A

W, A

SiO2:Al2O330:155:180:1

280:1





H-K Model

Saito-Foley Model


Argon, 87 K

0

20

40

60

80

100

120

140

160

1e-007 1e-006 1e-005 0.0001 0.001 0.01 0.1 1

Qua

ntity

Ads

orbe

d, c

m3 /g

p/po

SiO2:Al2O3280:1





H-K Model

Saito-Foley Model


Argon, 87 K

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

4 5 6 7 8 9 10 11 12 13 14 15

dV/d

W, c

m3 /g

-A

W, A

SiO2:Al2O3280:1





H-K Model

Saito-Foley Model


Argon, 87 K

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

4 6 8 10 12 14 16

dV/d

W, c

m3 /g

-A

W, A

SiO2:Al2O3280:155:180:1




DFT Pore Size Distribution

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

1 10 100 1000

dV/d

W, c

m3 /g

-A

W, A

SiO2:Al2O3ZSM-5 30:1

DFT Model

Tarazona Model


Nitrogen, 77 K

ZSM-5





0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

1 10 100 1000

dV/d

W, c

m3 /g

-A

W, A

SiO2:Al2O3ZSM-5 30:1

H-Y 5:1

DFT Model

Tarazona Model


Nitrogen, 77 K

ZSM-5

H-Y





0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

10 100 0

2

4

6

8

10

12

14

16

18

20

pore

vol

ume,

cm

3 /g

dV/d

(log(

D))

, (cm

3 /g)/

Åwidth, Å

DFT Model

Cylindrical PoresOxide Model


Nitrogen, 77 K

MCM-41




X Zeolite

0

0.05

0.1

0.15

0.2

0.25

1 10 100 0

1

2

3

4

5

6

7

pore

vol

ume,

cm

3 /g

dV/d

(log(

D))

, (cm

3 /g)/

Å

width, Å




Y Zeolite

0

0.05

0.1

0.15

0.2

0.25

0.3

1 10 100 0

0.5

1

1.5

2

2.5

pore

vol

ume,

cm

3 /g

dV/d

(log(

D))

, (cm

3 /g)/

Å

width, Å




Summary

Chemistry effects

Composition influnces the adsorption potential - for example thealkaline zeolites adsorbed nitrogen at very low pressure.

Structure

Adsorption potential also follows pore size - for example a 5A poreadsorbs nitrogen at lower pressures than an 8A pore.

Surface Area

Surface area increases as pore size decreases




Summary cont’d

Preparation

The preparation temperature strongly influences surface area. Asprep temperature was increased for oxides the SSA increased; whilethe opposite was observed for stearates - increasing preptemperature reduced the surface area.

Porosity

Mesoporosity is a function of pressure and not as stronglydependent upon chemistry like the micro porous materials.

Porosity of Excipients

Common excipients do not exhibit significant levels of porosity aswe observed with silica.




Summary cont’d

Macro porosity

Mercury porosimetry allows us to understand the larger pores in amaterial

Tablets

The pore size of consumer tablets is very consistent and breaking atablet does not provide additional access to the API. This shouldprovide a consistent release of the API.


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