7446
Langmuir 2001, 17, 7446-7447
Additions and Corrections A Process Model for Latex Film Formation: Limiting Regimes for Individual Driving Forces Alexander F. Routh and William B. Russel Langmuir 1999, 15, 7762-7773. With the definition of the stress relaxation modulus, eq 8, and the viscous limit defined in eq 26, we err from the definition of a Newtonian fluid by a factor of 2. To correct this, we insert the factor 2 into subsequent equations. This changes the classical limits: Viscous particle sintering because of surface tension, eq 27, for comparison with eq 1 becomes
R ) γt/2R0η
〈σ〉xxp )
t′)
(
)
F πGR02
(30)
2γ n + γeFδ(z) ( Fδ(x2 + y2)ez R
(16)
ν
∑ * ) ∫θ)0∫β)04π* sin β dθ dβ contacts
(45)
This error carries through the derivation. Although many of the numerical factors in the equations change, the physical interpretation remains the same. Here, we list the changes to the appropriate equations:
ν 4V
π R0(1 - cos2 β)F(R)[sin2 β(exex + ∫β)0
[(
)∫
7 212 3ν 156 18 88 t
3
dth
+G hσ jt +
t
0
+ O(G5) (48)
(
)
( )
7γ j d λh d d d +G h ) +G h λh 5 dth η j dth dth dth
(51)
0
[
2
O(3)
]]
+ O(G5) (47)
28σtR0 3νφmγwa
This absorbs the effect of the error for capillary stresses but not for surface tension. Equation 53 becomes
(
)
( )
7 d λh d d G h λh d + +G h ) 5 dth η jγ j dth dth γ j dth
(53)
The limit for deformation solely by wet sintering, when G h . 1, is λh/γ j e 7/5.
[(
[
)
λh 29th 29th2 η j-1 + ht 1 + + (G h ht - 1 + e-Gh ht ) × η j 36 792 G h 7(η j - 1) 11th 295th2 13th + 1+ + 1+ [2 - 2G h ht + 2 18 792 22 36G h 119(η j - 1) (6G h ht - 6 - 3G h 2ht 2 + G h 2ht 2 - 2e-Gh ht ] 264G h3
σ j t(th) )
(
)]
(
)
]
G h 3ht 3 + 6e-Gh ht ) + O(th4) (56)
The limit for capillary deformation when . 1 is λh e 42.9. d dth
+G h )
- * * )
dt′ -
d 4 14γ 5 35 dt′ + + 1∫0tG(t - t′)dt′ 5R0 4 216
119 528
]]
∫ G(t -
∫
d G(t - t′) 2 dt′ + dt′
)∫ G(t - t′)dt′d
9 7 154 22
(
t
0
595 4752
3νφmγ (1 + O()) ) 20R0 3νφm t d G(t - t′) 2 dt′ (1 + O()) (49) 56 0 dt′
σt +
eyey) + 2 cos2 βezez] sinβ dβ (46) 〈σ〉zzp )
[
σ jt )
The major error resides in eq 45 which should read
〈σ〉p )
dt′ -
14γ 3 52 d 4 dt′ + + O(3) 1dt′ 3R0 4 72
This should be compared with eq 3. Equation 16 should, from dimensional arguments, read
π
3
(29)
R ) γ/R0G
2π
t
0
d G(t - t′) 2 dt′ + dt′
The scaling for the top stress in Table 1 should read
1/2
Force free elastic particles under surface tension deform according to
n‚σ )
t
0
)∫ G(t - t′)dt′d
(
dσ jt
R )
)∫
7 5 154 22
(27)
Equation 29 for elastic particles compressed by a force F, for comparison with eq 2, becomes
[(
3ν 5 352 1280 18 264
5G h λh d 7γ j dth
(60)
7γ j 7γ j ln ) ht 5G h λh * 5λh
(61)
14γpw j γ j 14 η 14γ j ) ≈ 5R0G′∞ 5 η j-1G h λh 5G h λh
λh 7 70 35 0.504G h λh + ln < + γ j 5 9G γ j h 9G h
(
)
(62) (63)
Therefore, the limit for wet sintering when G h . 1 and η j . 1 reduces to λh/γ j e 7/5.
10.1021/la0110038 CCC: $20.00 © 2001 American Chemical Society Published on Web 10/11/2001
Additions and Corrections
[(
Langmuir, Vol. 17, No. 23, 2001 7447
)(
)
11th 295th2 1 + ht + (e-Gh ht - 1) + 18 792 G h 7 13th 2 2 1+ ht + 2(1 - G h ht - e-Gh ht ) 36 22 G h
σ j t(th) ) λh 1 +
(
(
)(
))]
(
6 G h 2ht 2 119 3 ht + 3 G h ht - 1 + e-Gh ht 264 2 G h 3
2
)
h h p ) 1 - 2th + 4
+ O(th ) (64) 3
λh < 20G h /[2.70 - 2.22G h - 0.40G h + 0.47G h 2.70e-0.36Gh (1 + 0.18G h - 0.47G h 2)] (65)
[
∂ 7γ j ) ∂th 5λh
]( (
(84)
) )
7γ j φmht 5λh 2φ0 - φm exp -1 7γ j φm 5λh(φm - φ0) (86)
In Table 2, the times for complete film formation need to be multiplied by a factor of 6/7, except in the capillary deformation regime. A corrected Figure 12 is shown. All other figures do not change by a discernible amount.
The limit for capillary deformation when G h . 1 and η j. 1 is λh e 42.9.
7γpwH
aTg(T) ) ht comp )
5E˙ R0η0(Tg) 0.26η0R0E˙ Hγpw
aTg(T) ) ht comp ) t* +
42.9γwaH E˙ R0η0(Tg)
5η0R0E˙ (1 - φm - ht *) 7Hγpa
0.26η0R0E˙ ht comp ) Hγpa
(76)
(77)
(78)
(79) Figure 12. Generalization of time for complete deformation of time for complete deformation in film.
(81) LA0110038
