solubility and crystal radius r nc liquid solid a a + b liquid a + b a r
TRANSCRIPT
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)