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Emulsion Stabilization by Diblock Copolymers: Droplet Curvature Effect Elena N. Govorun*,†,‡ and Igor Erukhimovich†,§ Physics Department, Moscow State University, Moscow 119899, Russia, Topchiev Institute of Petrochemical Synthesis, Russian Academy of Sciences, Leninskii pr. 29, Moscow 117912, Russia, and Nesmeyanov Institute of Organoelement Compounds, Russian Academy of Sciences, Vavilova 28, Moscow 117813, Russia Received April 12, 1999. In Final Form: August 4, 1999
The Cantor-Leibler theory of macroscopic emulsion stabilization by adding diblock copolymer is reexamined with regard for the curvature of the equilibrium droplets. The treatment is based on the assumption that the equilibrium state of the system corresponds to adsorption of most diblock copolymer molecules at the interface which leads to a one-to-one correspondence between the average droplet size, R, and the interfacial area per copolymer molecule, Σ. First we find R in the approximation of zero interfacial curvature and then investigate the equilibrium emulsion characteristics using the interfacial free energy expansion over the interfacial curvature (up to the second-order terms). The curvature effect is shown to lead to small corrections for R as calculated for the flat interface model. As a result, R does not depend strongly (inversely proportional) on the relative difference in copolymer block lengths unlike the results of the previous considerations of the emulsion stabilization. For the cases of a semidilute copolymer layer at the interface and the unpenetrable copolymer layer (“dry brush”) the corrections due to the curvature effect are calculated explicitly. The conditions of a complete diblock copolymer adsorption at the interface are discussed.
1. Introduction It is well-known that the morphology of the demixing homopolymer blends and low-molecular weight mixtures is remarkably influenced by copolymer additives and strongly depends on the copolymer composition and concentration.1-9 Even a minor amount of block copolymer in such mixtures results in a considerable change of their morphology. Indeed, if the copolymer blocks of different types strongly prefer different solvents, then the copolymer tends to adsorb at the interface. The first consequence of this peculiarity of block copolymers is that the phase separation in the demixing low molecular weight system proceeds much more slowly than would occur without block copolymer addition. There is an uncertainty whether this slowing down is only of kinetic origin (such a kinetic slowing down is especially observable under demixing of high molecular weight mixtures in the presence of copolymer) or the size of growing particles of the minor phase is approaching †
Moscow State University. Topchiev Institute of Petrochemical Syntheses. § Nesmeyanov Institute of Organoelement Compounds. ‡
(1) de Gennes, P.-G.; Taupin, C. J. Phys. Chem. 1982, 86, 2294. (2) Fayt, R.; Jerome, R.; Teyssie, Ph. Makromol. Chem 1986, 187, 853. Cigana, P.; Favis, B. D.; Jerome, R. J. Polym. Sci., Part B: Polym. Phys. 1996, 34, 1691. (3) Gue´gan, Ph.; Macosko, C. W.; Ishizone, T.; Hirao, A.; Nakahama, S. Macromolecules 1994, 27, 4993. (4) Macosko, C. W.; Gue´gan, Ph.; Khandpur, A. K.; Nakayama, A.; Marechal, Ph.; Ishizone, T. Macromolecules 1996, 29, 5590. (5) Prokop, R. M.; Hair, M. L.; Neumann, A. W. Macromolecules 1996, 29, 5902. (6) Washington, C.; King, S. M.; Heenan, R. K. J. Chem. Phys. 1996, 100, 7603. (7) Gref, R.; Babak, V.; Bouillot, P.; Lukina, I.; Borodev, M.; Dellacherie, E. Colloids Surf., A: Physicochem. Eng. Aspects 1998, 143, 413. (8) Kawakatsu, T.; Kawasaki, K.; Furusaka, M.; Okabayashi, H.; Kanaya, T. J. Phys.: Condens. Matter 1994, 6, 6385. (9) Cantor, R. Macromolecules 1981, 14, 1186.
thermodynamic equilibrium. In the latter case, a stable emulsion would be formed, the block copolymers being concentrated at the particles surfaces.1-8 The macroscopic emulsion stabilization by diblock copolymers was investigated theoretically, and the final size of the dispersed phase droplets was estimated in a number of papers.9-11 The equilibrium state corresponds to the minimal value of the interfacial free energy determined by an interplay of the interfacial energy of two demixing liquids and the free energy related to the layer of the adsorbed copolymer. Indeed, decrease of the total interfacial area leads to decrease of the interfacial energy but could increase the latter contribution, since the adsorbed copolymer molecules are more stretched, when they occupy less interfacial area. Cantor9 considered a low molecular mixture with diblock copolymer forming a semidilute layer at the interface (the polymer coils in such a layer overlap, but the monomer volume fraction stays small). The equilibrium characteristics of a flat interface were obtained, and the dependence of the free energy of the interface on their curvature was derived. The corresponding results turned out to be selfcontradictory: it was predicted9 that symmetric copolymers could stabilize a flat interface, but the account of a finite curvature would result in radius of the most thermodynamically favorable particles (droplets) inversely proportional to the relative difference in the blocks lengths. Leibler10 considered the macroscopic emulsion in the mixture of homopolymers A and B and diblock copolymer AB. In the case of the dry brush (when one can neglect the penetration of the homopolymer molecules into the copolymer layer formed at the interface), the equilibrium surface density of copolymer chains was calculated for a flat interface. The characteristic droplet radius obtained (10) Leibler, L. Makromol. Chem., Macromol. Symp. 1988, 16, 1. (11) Erukhimovich, I. Ya.; Govorun, E. N.; Litmanovich, A. D. Macromol. Theory Simul. 1998, 7, 233.
10.1021/la990428f CCC: $18.00 © 1999 American Chemical Society Published on Web 10/01/1999
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by Leibler taking into account the curvature dependence of the free energy was also found to be inversely proportional to the relative difference in the blocks lengths. Even though the theoretical results9,10 agree qualitatively as to fundamental asymmetry dependence of the most thermodynamically favorable droplet size, this dependence seems to be in a contradiction with the experimentally observable effective emulsion stabilization by symmetric diblock copolymers.2-4 The polymer mixture of the same type was considered recently by the authors11 for the case of equal lengths of a copolymer block and corresponding homopolymer chains. It was supposed that penetration of homopolymers chains into the diblock copolymer interfacial layer is controlled by the elastic blocks properties. Assuming that all copolymer molecules are adsorbed at the interface, we calculated the equilibrium surface density of the copolymer molecules for a flat interface and estimated the corresponding droplet radius. As consistent with the first part of considerations presented by Cantor9 and Leibler,10 it was shown that the symmetric diblock copolymers do stabilize a flat interface, the calculated numerical values of the equilibrium interfacial area, Σ*, per diblock copolymer chain being in good agreement with experimental results.2-4 The purpose of the present paper is to revise the consideration of the curvature effects9,10 and to show that, in fact, the curvature corrections to the value of Σ* are small. The outline of the paper is as follows. In section 2 we review a general approach to the description of the droplet phase following the papers.1,9-11 In section 3 we apply this approach in zero curvature approximation which results in an improvement of the results obtained by Cantor for the model of dilute brush. In section 4 we reexamine treatment of the curvature influence presented by Cantor and by Leibler. In section 5 we give the results obtained for the dilute brush and the dry brush models taking into account the curvature dependence of the surface free energy. 2. The Droplet Phase Stabilized Due to the Presence of Diblock Copolymers We consider a mixture of two immiscible liquids R and β and diblock copolymer AB. The mixture is an emulsion consisting of minor R-phase particles in the β-phase, the particles being stabilized by diblock copolymer chains concentrated mainly at the R/β interface. We restrict ourselves to the strong segregation limit assuming that all diblock copolymer junction points are tethered to a thin surface. Let A and B blocks be placed in the R and β phases correspondingly. The conditions of the complete diblock copolymer adsorption at the interface will be discussed further (see also ref 11). Let V, S, and mc be the volume of the mixture, the total interfacial area, and the number of diblock copolymer chains in the mixture, respectively. The free energy of the mixture may be written as the sum of the bulk and interfacial contributions
F ) Fbulk + SfS
(1)
where fS is the interfacial energy per unit area which depends on the value of the interfacial area per one diblock copolymer chain Σ ) S/mc. The free energy minimum corresponds to the condition
σ)
|
∂fS ∂F ∂(SfS(S)) ) )0 ) fS(Σ) + Σ ∂S ∂S ∂Σ S)mcΣ
(2)
where σ denotes the surface tension. The existence of the solution of eq 2 Σ ) Σ* implies the presence of the stable dispersed phase. Let γ be the surface tension between the two lowmolecular liquids. The interfacial free energy can be written as follows
FS ) fS(Σ)S ) γS + FAbrush + FBbrush
(3)
here FAbrush and FBbrush are the free energies of the A and B brushes. We assume that the surface tension does not depend on the copolymer coverage at the interface. This condition holds if the fraction of the covered interface is low enough. Then the condition (3) of the free energy minimum can be rewritten in the form1,9,10
γ ) π ) -∂ f/∂Σ
(4)
here π is the lateral brush pressure, f ) (FAbrush + FBbrush)/ mc is the brush free energy per copolymer chain. Let φ be the volume fraction of the minor R component. If the dispersed phase consists of the spherical droplets of radius R and concentration n, then1,11
R ) 3φ/ncΣ*
(5a)
n ) (ncΣ*)3/36πφ2
(5b)
where nc ) mc/V is the average concentration of copolymer chains. Recall that the expressions (5) are obtained under the condition of the complete adsorption of the diblock copolymers at the interface. Now to obtain the droplets radius we should find the equilibrium interfacial area per a diblock copolymer chain, Σ. 3. The Zero Curvature Approximation (Flat Interface) To obtain the equilibrium value of the interfacial area per a copolymer chain, we should derived the expression for the interfacial free energy. In the zero approximation, we neglect the curvature dependence of the brush free energy. The statistical segment lengths a and monomer volumes v are assumed to be the same for both A and B copolymer blocks, the block lengths are equal to NA and NB correspondingly, and the total copolymer chain length Nc ) NA + NB. The interaction between polymer units and solvent molecules is described by the Flory-Huggins parameters χA ) χAR and χB ) χBβ. We consider the case of good solvents with χi < 1/2. The copolymer layer at the interface is described by the well-known model proposed by Alexander12 and de Gennes.13 Let φic be the volume fraction of the i monomers in the homogeneous copolymer layer (the brush) of the width Hi. For the A side
φAc(z) )
{
φAc, 0 < z < HA 0, z > HA
(6)
Here z is the distance from the interface in the direction perpendicular to the interface (z ) 0). The B side is described in the same way for z < 0. Then φic ) vNi/ΣΗi. As follows from the incompressibility condition, the solvent volume fraction in the brush φis ) 1 - φic. (12) Alexander, S. J. Phys. (Paris) 1977, 38, 983. (13) de Gennes, P.-G. Macromolecules 1980, 13, 1069.
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The free energies of the A and B brushes consist of the elastic energy of the copolymer blocks, their energy of interaction with the solvent (R and β respectively), and the free energy of mixing of the copolymer blocks with the solvent
Fibrush ) mc fibrush ) mc(fiel + fiint + fimix)
(7)
The free energy of the copolymer blocks mixing with solvent per a copolymer block for the uniform brush11,14
fimix ) kT
SH s φ ln φis ) kTNi mcv i
φis ln φis
(8)
φic
k is the Boltzmann constant. The elastic energy per a copolymer block is taken in the form11,12
fiel ) 3kT(hi2 + hi-2 - 2) hi2 ) Hi2/(2Nia2)
(9)
coefficients
hi2 )
SH χ φ c(1 - φic) ) χiNi(1 - φic) mcv i i
(10)
Now the total brush contribution to the free energy per copolymer chain takes the form
Fibrush ) Ni(((φic)-1 - 1) ln(1 - φic) + χi(1 - φic)) + mckT 3(hi2 + hi-2 - 2) (11) For φic , 1 and hi . 1 our expression (11) takes the form presented by Cantor9
Fi(C) 1 ) 3hi2 - Ni(1 - χi) + Niφic(1 - 2χi) (12) mckT 2 Thus, our expression for the brush free energy is a generalization of that of Cantor to arbitrary values of the brush volume fraction, φic. The elastic terms in eqs 11 and 12 become indistinguishable only in the limit Hi . Ni1/2a. The elastic term (9) in the case of a dilute brush (Σ f ∞) has the minimum at hi )1 that corresponds to the mean square end-to-end distance for a Gaussian chain anchored to a flat surface by one end in a half-space Hi ) (2Ni)1/2a. The minimum conditions for the brush free energy (11) given a fixed value of Σ are
u(χi,φic) ∂Fibrush ∼ 6(hi2 - hi-2) + Ni )0 ∂Hi φc
(ΣHi)2 v2Ni
) (2(Σ ˜ φic)2/Ni)-1 (14)
2(vNi)2 (aΣ)2
u(χA,φAc) u(χB,φBc) ∂FS ∼Σ ˜ γ˜ + NA + N ) 0 (15) B ∂Σ φ c φ c A
B
the reduced value of the interfacial tension between the liquids, γ˜ ) γv/akT, being introduced. Now, to determine the equilibrium brush characteristics, it is necessary to solve the simultaneous equations (13) and (15) numerically. For the symmetric case (NA ) NB ) N, χA ) χB ) χ, φAc ) φBc ) φc) eqs 13 and 15 can be rewritten as
N u(χ,φc) γ˜ φc
Σ ˜ ) -2
6
(
(16)
)
8Nu2(χ,φc) u(χ,φc) γ˜ +N ) 0 (17) 2 γ˜ φc 8Nu (χ,φc)
It can be seen from eqs 16 and 17 that the equilibrium values of the brush copolymer volume fraction and the interfacial area per one diblock copolymer chain are determined in this case by all three parameters γ˜ , N, and χ. In comparison with the mixture of homopolymers stabilized by diblock copolymer chains,11 the penetration of solvent molecules into the copolymer brush in a lowmolecular mixture increases the energy of mixing. As a result, we expect the lesser equilibrium values of the copolymer volume fraction in the brush, φic, and greater interfacial area per copolymer chain, Σ, than those in a homopolymer mixture. Earlier Cantor9 derived the equilibrium interfacial characteristics assuming that the brush free energy has the form (11). For the fixed Σ the minimization over Hi gives
(
H i ) Nia φic )
(
(
)
1 - 2χi 6Σ ˜
1/3
)
6 (1 - 2χi)Σ ˜2
1/3
1/3 2/3 fi(C) 3 6 (1 -2χi) - (1 - χi) ) Ni kT 4 Σ ˜ 2/3
(18)
)
(19)
pi ) Ni/Nc. The minimization of the interfacial free energy over Σ gives
i
i ) A, B
2Nia
) 2
and the reduced specific interfacial area, Σ ˜ ) Σa/v, being introduced. The minimum condition under fixed values of HA and HB is
where hi is the stretching coefficient for a tethered block. The interaction energy per copolymer block has the form15
fiint )
Hi2
(13)
where u(χ,x) ) ln(1 - x) + x(1 + χx), the stretching (14) Scheutjens, J. M. H. M.; Fleer, G. J. J. Chem. Phys. 1979, 83, 1619. (15) Flory, P. J. Principles of polymer chemistry; Cornell University Press: Ithaca, NY, 1953.
Σ ˜ 0*(C) )
() ( ) 3 4
1/5
Nc γ˜
3/5
G13/5
(20)
here G1 ) pA(1 - 2χA)2/3 + pB(1 - 2χB)2/3. For the symmetric case, the dependence of the equilibrium interfacial area per a copolymer chain on χ and N is shown in Figure 1 for both considered models.
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copolymer asymmetry is also negligible for χA ) χB. If the interaction energy contributions of A and B blocks are different (|χA - χB|Nc . 1), then the equilibrium value Σ* becomes strongly dependent on the block length ratio. Let us now discuss the influence of the interfacial tension, γ, on the droplet size in the blend. If the interaction between R and β liquids becomes less favorable, then the total interfacial area should decrease and, in turn, the droplets size should increase. For a fixed copolymer architecture and χi , 1, both considered models yield the droplet size proportional to γ3/5 (from the eqs 5a and 20). 4a. The Curvature Effect
Figure 1. Equilibrium interfacial area per copolymer chain Σ* vs the copolymer blocks length N ) NA ) NB for the different values of the Flory-Huggins parameter χ ) χA ) χB; γv/(akT) ) 1. The solid and dashed curves corresponds to the present expression for the free energy and that of Cantor.
In the preceding section we have considered the flat interface neglecting the curvature influence on the interfacial properties. Here we will reexamine the equilibrium characteristics of the droplet system taking into account that the interface is slightly curved (the droplets radius R . Hi). Now the copolymer A blocks are concentrated in the inside spherical layer of the thickness HA and B blocks are in the outside layer of the thickness HB. For the sake of simplicity we assume, following the original Cantor consideration,9 that for each layer the copolymer volume fraction φic is constant. The volumes of the spherical layers of one droplet are Vi ) 4πR2Hi (1 + δiHi/R + 1/3(Hi/R)2), here δA ) -1 and δB ) 1. The number of copolymer blocks in the each layer is 4πR2/Σ. Then the copolymer volume fractions in the brush are
φic )
(
)
Ni v Hi Hi2 + 1 + δi ΣHi R 3R2
-1
(21)
Now, one should minimize the free energy FS defined by expression 3 as to the parameters HA, HB, and Σ, keeping in mind the relations (21). The minimization of the brush free energy (11) over the brush thicknesses Hi gives (up to the second-order terms in the expansion over R-1)
6 Figure 2. Dependence of the equilibrium interfacial area per diblock copolymer chain Σ* on the block lengths NA and NB for χA ) χB ) 0 (solid lines) and χA ) 0, χB ) 0.4 (dashed lines); γv/(akT) ) 1. The pairs of lines correspond to the equal values of Σ*v/a.
It can be seen that for χ ) χA ) χB, the curves corresponding to both aforesaid models are almost identical only for χ , 1. With the increase of χ the copolymer blocks become more incompatible with the corresponding solvent and the copolymer brush becomes more dense. For the Cantor model, the equilibrium copolymer volume fraction φc scales as (1 - 2χ)-3/5 (from (18) and (20)). Hence, the condition φc , 1 used by Cantor becomes invalid when χ increases, and one should thoroughly account for the energy of mixing of copolymer blocks with solvent if the interaction parameter χ is not very small. The equilibrium interfacial area per diblock copolymer chain for the semidilute brush model of Cantor (20) is proportional to (Nc)3/5(1 - 2χ)2/3 and does not depend on the copolymers asymmetry characterized by pi. The dependence of the equilibrium interfacial area per copolymer chain on the copolymer block lengths for the present model is shown on the square diagram (Figure 2), each line corresponding to the certain constant value of Σ*. It can be seen that the dependence of Σ* on the
(
Hi2
2Nia2
-
)
2Nia2 Hi2
+ Ni
(
u(χi,φic) φic
)
Hi Hi2 1 + δi )0 R 3R2
i ) A, B
(22)
As far as the minimization of the free energy FS over Σ is concerned, one should take into account that the droplet radius R cannot be treated as an independent parameter as was done by Cantor.9 In fact, the quantities R and Σ are interrelated by formula (5a) which could be rewritten, for the sake of convenience, in the form
R ) 3φNcv/φcopΣ It follows
γ˜ Σ ˜+
∑ Ni
i)A,B
(
u(χi,φic) φic
1 + δi
Hi R
(23)
-
Hi2
)
3R2
) 0 (24)
here φcop ) ncNcv is the average volume fraction of the diblock copolymer in the mixture (the conventional natural restriction φcop , φ being assumed). Now, to determine the equilibrium value of the interfacial area per copolymer chain and the corresponding droplet radius, we solve the simultaneous eqs 22-24 numerically. The resulting values of the interfacial area per diblock copolymer chain turn out to be only slightly less than that those for the flat interface.
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4b. The Curvature Effect: The Semidilute Brush (Cantor’s) Model
5. Equilibrium Interfacial Characteristics for the Dry Brush Model
In the previous section we used numerical calculations to obtain the corrections to the interfacial area per copolymer chain originating from the curvature dependence of the free energy. For the semidilute brush model of Cantor we can obtain these corrections explicitly. If the copolymer volume fractions in the both brushes are small, then the expression (12) for the brush free energy can be used. The minimization of the total interfacial free energy over Hi and over the interfacial area gives two equations
Let us also calculate the equilibrium interfacial characteristics for the dry brush model (φic ) 1) studied by Leibler.10 The model is applicable in some cases for the homopolymers A and B mixtures stabilized by diblock copolymers AB, but its consideration reveals the connection between the choice of the brush model and the role of the block asymmetry. The brush free energy per a copolymer chain has the form10
(
2
2
)
f(L) ) fAbrush + fBbrush )
Hi Hi - Niφic(1 - 2χi) 1 + δi )0 2 R 3R2 Ni a 6Hi
i ) A, B γ˜ Σ ˜+
1
(
Hi
∑ Niφi (1 - 2χi) 1 + δi R c
2i)A,B
Hi2
)
2
3R
(25) )0
(26)
(
)
(27)
The solution of eq 25 up to the second-order terms over R-1 is
(
)(
1 - 2χi Hi ) Nia 6Σ ˜
1/3
(
)
2Ni2a2 1 - 2χi 16Σ ˜ 9R2
)
2/3
(28)
From eqs 27, 28, and 23 the equilibrium interfacial area per a copolymer chain is
∆)-
˜ 0*(C)(1 + ∆) Σ ˜ *(C) ) Σ
(29a)
( ) ( )( )
(29b)
1 4 135 3
2/5
φcop φ
2
Nc γ˜
4/5
G1-1/5G3
G3 ) pA3(1 - 2χA)4/3 + pB3(1 - 2χB)4/3. The correction to the interfacial area per a diblock copolymer chain calculated for a flat interface within Cantor’s model (20) is second order in R-1 ∼ φcop/φ. Even though this correction grows with increase of Nc, the condition |∆| , 1 is satisfied for any realistic values of the system parameters: Nc , 1620γ˜ (φ/φcop)5/2(1 - 2χ)-3/2 (χ ) χA ) χB). For the semidilute brush model of Cantor in the case χA ) χB ) χ, we obtain the expression for the droplets radius from eqs 23 and 29
R/a )
(
φγ˜ 3/5 18Nc φcop 1 - 2χ
)( 2/5
1+
4 3
()
2/5
( ) ( ))
pA3 + pB3 Nc 135 γ˜
4/5
φcop φ
2
(30) The corrections to the droplets radius obtained via numerical calculations for our model and semidilute brush model (eq 30) turn out to be rather small for the reasonable values of the ratio x ) φcop/φ. Actually, they are much less than the difference in the results of both considered models for a flat interface.
)
(31)
Deriving Hi from eq 21 and using the condition φic ) 1, one can rewrite expression 31 as (including the terms proportional to R-2)
f(L) )
(
(NA2 - NB2)a 3kT N + 1 + 2 2 Σ NΣ ˜R ˜2
)
2 3 3 13 (pA + pB )(Na) (32) 3 Σ ˜ 2R2
where φic is the copolymer volume fraction in the ith brush given by eq 21 and the droplet radius R is given by (24). It follows from eqs 25 and 26 2 HB2 3 HA + Σ ˜) γ˜ N a2 N a2 A B
(
2 HB2 3kT HA + 2 N a2 N a2 A B
Now, using the condition γ ) -∂f (L)/∂Σ, we can easily obtain from eqs 23 and 32
Σ ˜ *(L) )
( )( 3Nc γ˜
1/3
1+
( )(
1 φcop 81 φ
2
2φcop(NA - NB) + φNc
13(pA3 + pB3) -
))
2 4 (NA - NB) 9 N2 c
(33)
The term of the zeroth order in x ) φcop/φ corresponds to the flat interface. Note that for the Leibler’s brush model the first nonvanishing curvature correction is first order in x and proportional to the relative difference in the block lengths, whereas such a correction (29) for Cantor’s model is only second order in x and does not depend on |NA - NB| so strongly. This distinction is due to the large difference of the semidilute and dry brush models. 6. Discussion of the Validity of the Adopted Assumptions Now, let us list again the assumptions we used in the present work and the conditions ensuring their validity. 1. Expression 32 does not include the term describing the interaction between the copolymer block units and those of the corresponding homopolymer (or liquid) at the brush-liquid interface. For the flat interface such a term may be incorporated directly via a proper redefinition of the effective surface tension, γ˜ , appearing in expression 33 for the equilibrium reduced area per chain, Σ ˜ . This would lead to a slight modification of the curvature correction. 2. The approximation of a slightly curved interface (Hi , R) was used in all considered approaches.9-11 As it can be estimated from eqs 18, 20, and 23 for χi ) 0, in the limit of a semidilute brush this condition is equivalent to
Nc , γ˜
( ) 3φ piφcop
5/2
(34)
It means that our approximation is valid when the copolymer chains are not very long and the copolymer
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on the degree of asymmetry of block copolymer chains, ) 0.5 - NA/Ncop. This is contrary to the previous claim9 that the average droplet radius is inversely proportional to . The reason for such a discrepancy is the fact that the stabilized emulsion has a developed interface where all copolymer chains are adsorbed. Therefore, any change of the equilibrium interfacial area per copolymer chain is followed by a corresponding change in the droplet size satisfying the relation (5a) or (23). It is the explicit account of this relationship which distinguishes our treatment from the previous ones. Acknowledgment. The authors thank INTAS and RFBR (Grant INTAS-RFBR No. 95-0082) for financial support. E.G. acknowledges also an additional support by RFBR (No. 97-03-32695a). Figure 3. Free energy of the diblock copolymer chain in the major bulk phase fbulk (dashed line) and that of the adsorbed copolymer chain fads (solid line) vs the copolymer block length N in the symmetric case (N ) NA ) NB, χA ) χB ) 0, χAβ ) 0.75, φ ) 0.3, φcop ) 0.03).
concentrations are low enough (more generally, when the chains are much smaller than the droplet radius R). Otherwise, it is necessary to account for the copolymer layer free energy more thoroughly including the dependence of the copolymer volume fraction in the brush on the distance from the interface. 3. At last, the crucial assumption we used is that of complete diblock copolymer molecule adsorption at the interface. To validate this assumption, one should compare the equilibrium free energy of the adsorbed diblock copolymer molecule, fads, and the free energy of copolymer molecule in bulk phase, fbulk, which gives the following adsorption criteria
∆f ) fbulk - fads . kT
(35)
where expressions for fads and fbulk are given in the Appendix with due regard for the fact that the A and B blocks form globules in the β and R bulk phases correspondingly. In Figure 3 the dependences of the free energies fbulk and fads on the copolymer blocks length for the symmetric case are shown. It is seen that the condition (35) really holds for sufficiently long copolymer blocks. In other words, the formation of a unimolecular micelle in the bulk phase (a copolymer chain with one of its blocks forming a globule) is strongly disadvantageous in comparison with adsorption of this copolymer chain at interface. 7. Conclusion Summarizing, in this paper we have reexamined the earlier works1,9-11 concerning the emulsion stabilization by diblock copolymers. The present treatment covers both a semidilute copolymer layer at the interface9 and unpenetrable copolymer layer (“dry brush”)10 as well as the intermediate regime characterized by finite volume fractions of the copolymer within the interfacial layer. It enabled us to refine the results of Cantor9 for a flat interface (in particular, the values of the equilibrium interfacial area per copolymer molecule appeared to be somewhat larger than those found by Cantor). But the main result of our paper is that we found the curvature corrections to the stable emulsion (droplet phase) characteristics to be small and slightly dependent
Appendix To establish the conditions of the complete diblock copolymer adsorption at the interface, one should compare the free energy of an adsorbed diblock copolymer molecule and the entire molecule in the bulk phase. If a diblock copolymer chain is placed, for example, in the β-bulk phase, then two chemically different blocks would have different conformations: the A block forms a globule and the B block is a coil. Let us consider the coil conformation energy as the zero level. The polymer-solvent interaction energy for the diblock copolymer chain in a coil conformation in the β phase is fcoil/kT ) NAχAβ + NBχBβ (χAβ > 1/2, χBβ < 1/2). The transition of the block from a coil to a globule conformation leads to the free energy decrease. In the volume approximation near the transition temperature, the globule free energy of the copolymer A block in the bulk β phase fbulk is16 fbulk ) fcoil - kTNAB2/(4C) with B and C being the second and third virial coefficients appearing in the expansions of the energetic contributions into monomer chemical potential µ* and pressure p* for the “broken links” system: µ*/kT ) 2φB + 3φ 2C, p*v/kT ) φ 2B + 2φ 3C. For the lattice model we use in the present paper, B ) χAβ 1/2, C ) 1/6. Then the free energy of a diblock copolymer molecule in the β bulk phase is
fbulk/kT ) χAβNA + χBβNB -
3 1 N χ 2 A Aβ 2
(
2
)
(A1)
In the present consideration we calculate the free energy of the copolymer block (11) adsorbed to the interface using the expression (8) for the polymer-solvent mixing free energy. Rewriting the latter in the form
fimix ) kTNi
(1 - φic) ln(1 - φic) φic
(A2)
we see that the zero level of the mixing free energy corresponds to φic ) 1 (the diblock copolymer molecules form “dry brush” at the interface). The extremely dilute brush (φic f 0) that can be related to the coil chain conformation corresponds to fimix f -Ni. To find the free (16) Lifshitz, I. M.; Grosberg, A. Yu.; Khokhlov, A. R. Rev. Mod. Phys. 1978, 50, 683.
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energy excess for the adsorbed copolymer molecule at the equilibrium state over the adsorbed molecule in the coil conformation we should estimate
f*/kT ) Nc + fAbrush(Σ*,(φAc)*) + fBbrush(Σ*,(φBc)*)
(A3)
where the brush free energy per one chain fibrush(Σ*,(φic)*) is given by (11) and (φic)* is the equilibrium volume fraction of the copolymer i blocks in the interfacial layer. At last it is necessary to determine the free energy excess, ∆fjp, originating from a confinement of the junction point between copolymer blocks to the interfacial volume (∼aS, S ) mcΣ) while the volume of the β bulk phase is Vβ ≈ φβV ) (1 - φ)V, then ∆fjp ) kT ln(Vβ/aS). Now we can formulate the criteria when the complete diblock copolymer chain prefers to be adsorbed to the interface rather than dissolved in the phase β: fbulk - fads . kT, fads ) ∆fjp + f*. In the full form
Nc(1 - φ) 3 12 - ln NA χAβ - Nc 2 2 φcopΣ ˜ (fAbrush(Σ*,(φAc)*) + fBbrush(Σ*,(φBc)*))/kT . 1 (A4)
χAβNA + χBβNB -
(
)
In particular, substituting into (A4) the simple analytical expression (19) for the Cantor model and using the expression (20) for the equilibrium area per chain, we get the conditions for R and β bulk phases
3 12 NB χBR 2 2 Ncφ 3 3 1/5 (G1Nc)3/5γ˜ 2/5 . 1 (A5) ln φcopΣ ˜ 24
(χBR - χBβ)NB -
(
) ()
3 12 N χ 2 A Aβ 2 Nc(1 - φ) 3 3 1/5 ln (G1Nc)3/5γ˜ 2/5 . 1 (A6) φcopΣ ˜ 24
(χAβ - χAR)NA -
(
) ()
One can see that the conditions A5 and A6 are satisfied for the typical values of the Flory-Huggins parameters if the copolymer blocks are long enough. LA990428F
