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Removal of some approximations in calculation of the effect of a block copolymer on the interfacial tension in polymer blends

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Abstract

Block and graft copolymers with blocks that are identical, miscible with, or adhere to related components of a polymer blend can serve as compatibilizers. In the previously used models established by Leibler and Noolandi, simple solutions were published for special cases called dry brush and wet brush regime. We have tried to find a more general solution which is still relatively uncomplicated. The solution found is consistent with the dry brush regime for higher copolymer concentrations at the interface and shorter copolymer, and with the wet brush regime for lower copolymer concentrations and longer copolymer.

Graphical abstract

Block and graft copolymers with blocks that are identical, miscible with or adhere to related components of a polymer blend can serve as compatibilizers. In the previously used models established by Leibler and Noolandi, simple solutions were published for special cases called dry brush and wet brush regime. We have tried to find a more general solution which is still relatively uncomplicated. A solution has been found which is consistent with the dry brush regime for higher copolymer concentrations at the interface and shorter copolymer, and with the wet brush regime for lower copolymer concentrations and longer copolymer.

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Supplementary electronic material is available as an OriginLab file.

Code availability

Program is written in FreePascal. The code is available on reasonable request. If used, this work should be cited as its source.

Abbreviations

A, B:

Homopolymers to be compatibilized

a :

Segment length

C, D:

Components constituting the compatibilizing block or graft copolymer

d:

Differentiation operator

F f,L :

Gibbs energy of the interfacial film in the Leibler model

F f,N :

Gibbs energy of the interfacial film in the Noolandi model

g C, g D :

Gibbs energy per C and D copolymer chain brushes in phases A and B

k :

Boltzmann constant

L i :

Thickness of the interfacial layer in phase i

N :

Total segment count in the copolymer molecule

N j :

Segment j count in the copolymer block (j can be C or D)

P i :

Segment i count in the homopolymer chain (i can be A or B)

Q :

Total number of copolymer molecules in the system divided by the system volume, with Qk in the k phase

S :

Interfacial area in the system divided by the system volume

T :

Temperature

\(\mathit{\Gamma}\) :

Auxiliary parameter \(\mathit{\Gamma}\) =aS/φA (normalized interfacial area to volume ratio)

\(\mathit{\gamma}\) :

Interfacial tension

\({\mathit{\gamma} }_{0}\) :

Interfacial tension between the A and B phases

∂:

Partial derivative operator

\({\mathit{\eta} }_{{i}}\) :

Average volume fraction of segments of the ith block in the interfacial layer

Λ i :

Thickness of the interfacial layer in phase Li i divided by the segment length a, Λi = Li/a

\({\mathit{\mu} }_{k}\) :

Chemical potential of the copolymer in the k phase, where k is the A or B homopolymer phase or interface I

π :

Ludolf number

\(\mathit{\Sigma}\) :

Interfacial area per copolymer joint, Σ= S/QI

\({\mathit{\varphi} }_{\mathrm{A}}\), \({\mathit{\varphi} }_{\mathrm{B}}\), \({\mathit{\varphi} }_{\mathrm{CD}}\) :

Volume fractions of homopolymers A and B and of copolymer CD in the system, respectively

\({\mathit{\varphi} }_{\mathrm{CD}}^{\left(k\right)}\) :

Copolymer volume fraction in the k phase, where k is the A or B homopolymer phase or interface I

\({\widetilde{\mathit{\varphi} }}_{i}\) :

Reduced (relative) volume fraction of the i component \(\widetilde\varphi\)i=φi/φA

\({\mathit{\chi} }_{ij}\) :

Partial Flory–Huggins interaction parameter between the i and j segments, where i and j represent the A and B homopolymer segments and the C and D copolymer segments (χii = 0), respectively; χ without subscripts is the same as χAB

Ψ :

Copolymer joint fraction within segments at the interface; Ψ = a2/ \(\mathit{\Sigma}\) = \({\widetilde{\mathit{\varphi} }}_{\text{CD}}^{\left(\mathrm{I}\right)}\)/NΓ

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Authors and Affiliations

  1. Institute of Macromolecular Chemistry of the Czech Academy of Sciences, Heyrovského náměstí 2, CZ, 162 06, Praha 6, Czechia

    Josef Jůza & Ivan Fortelný

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  1. Josef Jůza
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  2. Ivan Fortelný
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Contributions

Ivan Fortelný: the idea and principles including basic equations from which the derivation was started. Josef Jůza: formulae derivation, implementation and calculations with particular data, image production, the first version of the article text, and auxiliary sections. Both authors: further text modifications and text finalization.

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Correspondence to Josef Jůza.

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Jůza, J., Fortelný, I. Removal of some approximations in calculation of the effect of a block copolymer on the interfacial tension in polymer blends. Colloid Polym Sci 300, 21–40 (2022). https://doi.org/10.1007/s00396-021-04904-8

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