SOLUBILITY AND CRYSTAL RADIUS R nc Liquid Solid a a + b Liquid a + b a r.

SOLUBILITY AND CRYSTAL RADIUS Rnc

Liquid

Solid

a

a + bLiquid a + b

a

r

Csinf = 10 g/cm3 (37°C, water)

Carbon Nitrogen Oxigen Sulphur

Mw = 308.5

NIMESULIDE(non steroidal antiinflammatory drug)

(crystal cell side = 0.87 nm)

0

5

10

15

20

25

0 5 10 15 20 25

t(min)

C(

g/c

m3)

Csinf

polymer

nanocrystrals

amorphous

NIMESULIDE RELEASE FROM CROSSLINKED PVP(water 37°C)

Liquid a + b

a

r

rTRv

nfis

snc

ssl

eC

C1

Kelvin equation9

sl = solid-liquid surface tension

vs = solid solute molar volume

R = universal gas constant

T = temperature

Csnc = nanocrystal solubility

Csinf = macrocrystal solubility

It holds for an ideal solution

lv

Vapour

sv sl

Solid substrate

Liquid drop

EQUAZIONE DI YOUNG

sllvsv cos sllvsv Per sostanza pura = 0 ===>

Melting temperature and enthalpy dependence on crystal radius

solid

liquid vapor

0 lvslsvvls UdUdUdUdUdUdUdss

k

i

si

si

sss VdPndμSdTUd 1

llk

i

li

li

lll VdPndμSdTUd 1

vvk

i

vi

vi

vvv VdPndμSdTUd 1

sv2

svsv1

svsvsvk

i

svi

svi

svsvsv cdCcdCAdγndμSdTUd 211

sl2

slsl1

slslslk

i

sli

sli

slslsl cdCcdCAdγndμSdTUd 211

lv2

lvlv1

lvlvlvk

i

lvi

lvi

lvlvlv cdCcdCAdγndμSdTUd 211

sl = solid-liquid interfacial tension

sv = solid-vapour interfacial tension

lv = liquid-vapour interfacial tension

Asv = solid-vapour interfacial area

Asl = solid-liquid interfacial area

Alv = liquid-vapour interfacial area

lvc1 liquid-vapour surface first curvature

lvc2 liquid-vapour surface second curvature

slc2 solid-liquid surface second curvature

slc1 Solid-liquid surface first curvature

svc2 solid-vapour surface second curvature

svc1 solid-vapour surface first curvature

lvsvslC ,,,21

constants

For a sphere:

lvlv rc 11

lvsvsllvsvsl cc ,,,,21

svsv rc 11

slsl rc 11

rsl, rsv, rlv curvature radii

TTTTTTT lvslsvvls

Closed system

ivslsvv

ilii μμμμμμμ l

iiis

thermal equilibrium

chemical equilibrium

Remembering that:

vllsvs PPPPPP Ps

Pl Pv

1)

svsllv γγγ 2) Young eq. for a pure substance

svlvlvsvslslvlvlss dγdγddd AAAAPPVPPVU

s

ssl

s

svslslls

V

A

V

AAPP

d

dγ

d

dγ

v

vlv

v

svlvlvlv

V

A

V

AAPP

d

dγ

d

dγ

mechanical equilibrium

svslsl

s

ssl

s

svslslls

RRV

A

V

AAPP

22γ

d

dγ

d

dγ

svlvlv

v

vlv

v

svlvlvlv

RRV

A

V

AAPP

22γ

d

dγ

d

dγ

2sv

2sl

svsl π2π2 RRAAAs

SLV

Rsl

Rsv3sv

3sl π

3

2π

3

2RRV s

slsldπ4d RRAsl svsvsv dπ4d RRA

sl2sl

sl dπ2d RRV sv2sv

sv dπ2d RRV

01

ssk

ii

si

s PdVdμnTdS01

llk

ii

li

l PdVdμnTdS

01

vvk

ii

vi

v PdVdμnTdS

Considering the Gibbs-Duhem equation

k = 1 ===> only one component (pure substance)

s s sd d d 0l l ls s T v P v P

ld d d 0l v l v vs s T v P v P

1 2

3

2 1

1 3

dd d d γ

d

ss l sl

s

AP P

V

dd d d γ

d

vv l lv

v

AP P

V

From the mechanical equilibrium conditions, it follows:

dd d d γ

d

s l s sl sl

s l s l s

s s v AP T

v v v v V

dd d d γ

d

l v v vl lv

l v l v v

s s v AP T

v v v v V

d dd d γ d γ

d d

s l l v v v s slv sl

s l l v l v v s l s

s s s s v A v AT

v v v v v v V v v V

then:

Assuming vl and vs << vv l v s l

v s l

s s s s

v v v

f

d dd d γ d γ

d d

v ss l lv s sl

v s

A AS T v v v

V V

f

2 2 2 2d d γ d γs l lv s sl

lv sv sl svS T v v v

R R R R

Rnc

Rlv ≈

TWO LIMITING CONDITIONS

Rsv does not exist

Rlv ≈ Rsl =Rnc

RncRnc

RncRnc

Rlv

ncnc Rd

RdT

T

hTs

2γ

ρ

12γ

ρ

1

ρ

1dd sl

s

lv

lsmr

mrmr

ncR

dTT

h 1

ρ

γγ

ρ

1

ρ

12d

s

sllv

lsmr

mr

ncR

nc

T

T RdT

T

h1

0s

sllv

ls

mr 1

ρ

γγ

ρ

1

ρ

12d

mr

m

ncRdT

T

hTs

2γ

ρ

1dd sl

smr

mrmrmr

ncR

dTT

h 1

ρ

γ2d

s

sl

mr

mr

ncR

nc

T

T RdT

T

h1

0s

slmr 1

ρ

γ2d

mr

m

ncs

slmr

ρ

γ2d

mr

mR

TT

hT

T

ls

lvs

slnc

mr

ρ

1

ρ

1γ

ρ

γ2d

mr

mR

TT

hT

T

Rnc

Rlv ≈

Rlv ≈ Rsl =Rnc

RncRnc

RncRnc

Rlv

Xncr ≈ 1Many nanocrystals

Xncr ≈ 0Very few nanocrystals

s

slncr

lslv

s

slncrnc

m

ρ

γ1

ρ

1

ρ

1γ

ρ

γ2d

mr

m

XXR

TT

hT

T

r

General equation

hmr and Tmr dependence on Rnc and Xcnr requires an iterative solution of these equations assuming a starting value of Xcnr

mrmpl

lv

s

svncmmr ρ

γ

ρ

γ3TTC

Rhh

[M. Zhang, et al., Physical Review B 62 (2000) 10548]

rh Th

Xncr = Xncr1A

Yes Solution: Xncr, hmr(Rnc), Tmr(Rnc)No

newncr

oldncrncr λλ1 XXX

s

slncr

lslv

s

slncrnc

m

ρ

γ1

ρ

1

ρ

1γ

ρ

γ2d

mr

m

XXR

TT

hT

T

r

Trmmr hhhh

Numerical solution of:

Trddmmix

dmrcgcalcncr ωω,

ω,

hhTh

ThX

1Ancr

calcncr XX

?

d(drug mass fraction)

hmix

(mixture melt. enthalpy)

0 1

hmd

(drug melt. enthalpy)

dhr+hT)

dmmix ω,rTh dmmix ω, Th

Nanocrystals size distribution

mr

mrr h

HddV

volume occupied by crystals ranging in [Rnc – (Rnc+dRnc)]

mr

ncmr

mr

mr

mrnc

mrncr 11

hdR

dT

dT

Hd

hdR

Hd

dR

dV

mrncmr

mrncmr

1

mr

mr 11

hdR

dT

v

Q

hdR

dT

dT

dt

dt

Hd

vQ

nc

ncr

ncnc

dd

d

d

d

ncmax

ncmax

RR

V

R

V

RfR

R

r

Solubility dependence on crystal radius Rnc

sd

sd

lddd

ld γ fffXf

thermodynamic equilibrium

Liquid(a+b)

a

dl

d

sd

d γ

1

f

fX drug solubility

ldf

fugacity of pure drug in the state of under-cooled liquid at the system temperature (T) and pressure (P)

sd

ld41 ln ffRTG

1Solid drug

nanocrystalsT, P

2Solid drug

nanocrystalsTmr, P

3Liquid drug Tmr, P

4Under-cooled liquid drug

T, P

isobaric heating

Isobaric-isotermic melting

isobaric cooling

414141 STHG

TTcTcHT

T

mrsp

sp21

mr

d

TTcTT

cS

T

T

mrsp

sp

21 lndmr

mr32 hH

mrmr32 ThS

mrlp

lp43

mr

d TTcTcHT

T

mrlp

lp

43 lndmr

TTcTT

cS

T

T

433221433221414141 SSSTHHHSTHG

T

T

R

c

T

T

RT

h

T

TX

Rc

mrp

mr

mr

mrdd 11exp

γ

1p

d is calculated knowing macro-crystal solubility in the desired solvent

Case study: nimesulide + crosslinked polyvinylpyrrolidone co-ground

Ratio 1:3 Co-grinding time: 1, 2 and 4 hours

3

3.2

3.4

3.6

3.8

4

4.2

4.4

4.6

4.8

5

100 110 120 130 140 150

T (°C)

Hea

t fl

ow (

mW

)

0

5

10

15

20

25

30

35

Hea

t fl

ow (

mW

)

T mr = 129.6 °C; h cg = 8700 J/Kg

T mr = 126.6 °Ch cg = 8500 J/Kg

T mr = 148.7 °Ch m = 109000 J/Kg

nimesulide1 hour

0.5 hours

DSC analysis

3

3.1

3.2

3.3

3.4

3.5

3.6

3.7

3.8

3.9

4

100 110 120 130 140 150

T (°C)

Hea

t fl

ow (

mW

)

0

5

10

15

20

25

30

35

Hea

t fl

ow (

mW

)

Tmr = 118.0 °Ch cg = 5100 J/Kg

T mr = 112.0 °Ch cg = 2500 J/Kg

T mr = 148.7 °Ch m = 109000 J/Kg

nimesulide2 hours

4 hours

DSC analysis

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

1 2 3 4 5R nc(nm)

f(R

nc)(

1/n

m)

X ncr[0.5h] = 0.40 ± 0.01

X ncr[1h] = 0.40 ± 0.01

X ncr[2h] = 0.33 ± 0.01

X ncr[4h] = 0.14 ± 0.03

Nanocrystals differential size distribution

hmr and Tmr dependence on Rnc and Xncr

75000

80000

85000

90000

95000

100000

105000

110000

0.1 1 10 100

Rnc

(nm)

h

mr(

KJ/

g)

95

105

115

125

135

145

155

165

175

Tm

r(°C

)

1

0.4

0

h mr[X ncr =1]

h mr[X ncr =0.4]

h mr[X ncr =0] mr[X ncr =0.4]

(crystal cell side = 0.87 nm)

Nanocrystals solubility dependence on Rnc and Xncr

1

10

0.1 1 10 100 1000R nc(nm)

CsN

/CsN

∞

Xncr = 0

Xncr = 0.4

Xncr = 1

O-F equation

(crystal cell side = 0.87 nm)