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MATCH
Communications in Mathematical
and in Computer Chemistry
MATCH Commun. Math. Comput. Chem. 56 (2006) 291-316
ISSN 0340 - 6253
ON THE DEFINITION OF SCALES
IN VAN OSS-CHAUDHURY-GOOD ACID-BASE THEORY
by
S. Siboni(o ) and C. Della Volpe
Dept. of Materials Engineering and Industrial Technologies
University of Trento
Mesiano di Povo 38050 Povo, Trento - Italy
e-mail: [email protected] Fax: I-0461 881977
(o ) to whom correspondence should be addressed
(Received April 26, 2006)
Abstract
Acid-base components of van Oss-Chaudhury-Good theory for surface energetics are extratermodynamical parameters whose direct measurement cannot be performed. Estimates
of acid-base components are obtained by solving the model equations of the theory, which
involve the experimental assessment of liquid surface tensions and contact angles for appropriate liquid-solid pairs. Due to symmetries implicit in the mathematical form of such
model equations, theory admits a triple innity of formally equivalent scales, with the same
prediction ability. This entails, in particular, the impossibility of interpreting acid-base
components in a strictly direct way. In this note we illustrate a way to select an univocal
vOCG scale by an essentially conventional assignment of the component values for suitably
chosen reference materials. The extension of the same procedure is outlined for a more
general class of multicomponent models, of which vOCG theory is a very particular case.
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1. Introduction
Multicomponent theories play an important role in modeling the interfacial interactions of
many materials and more specically in the prediction of the solid-liquid adhesion work
and surface free energy of solid surfaces. The strategy of describing the surface interactions of two materials by means of an appropriate number of \components", pertaining to
contributions of dierent physico-chemical nature, is shared my a lot of models proposed
through the years. Van Oss-Chaudhury-Good (vOCG) acid-base theory 0 constitutes
one of the most known and successful multicomponent models; it expresses the work of
adhesion of a liquid on a solid as
[1
l
6]
s
W
adh
hq
=2
LW LW
s l
+
q
0+
s l
+
q
0
s l
+
i
(1 1)
:
while the surface tension of the liquid and the surface free energy of the solid are respectively written as
l
=
LW
l
q
+2
+
0
l l
s
=
LW
s
q
+2
s+s0 ;
(1 2)
:
on having denoted with the superscript the Lifshizt-van del Waals components of the
materials, related to dispersive interactions, and with + and 0 respectively the acidic and
the basic components, which take into account the acid-base interactions between electrondonor (basic) and electron-acceptor (acidic) sites of the molecules involved. Through the
geometrical means of basic components with acidic components, all the model equations
reect the intrinsic complementarity of acid-base interactions, acidity and basicity being
obviously understood in a wide Lewis' sense. Acid-base components cannot be measured
in a direct way and at the moment no signicant and convincing correlation with some
absolute scale of acid-base behaviour is recognized, although some indication has been given
in the literature that the acidic component of water is 3 2 to 5 5 times the basic one 0 .
LW
:
:
[7
9]
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It is known[7;8] that equations (1.1)-(1.2) are invariant under appropriate linear transformations of acid-base components. These linear transformations form a group with respect
to composition, which means that the corresponding representation matrices constitute a
nonabelian group when endowed with the usual matrix multiplication | see Theorem 1
in Section 2. Such a simple remark has important consequences, since it entails that the
only experimental measurement of surface tensions and adhesion works does not determine
vOCG components in a unique way. Model equations (1.1)-(1.2) are intrinsically compatible with a multiplicity of \scales" of acid-base components, each obtainable from each
other by a suitable transformation of the invariance group, and all in principle equivalent
as for prediction ability.
In the present note we address the problem of determining a
univocal vOCG acid-base scale, manipulating the acid-base components of some appropriately chosen reference compounds by means of suitable transformations of the invariance
group.
The plan of the paper is the following: the main theoretical results concerning the structure
of the invariance group transformations are stated and proved in Section 2; application to
the problem of scale selection in vOCG theory is faced up in Section 3; nally, the more
general category of quadratic multicomponent theories is introduced in Section 4, and the
related problem of scale selection addressed in Section 5.
2. Analytical results
VOCG theory states that for each liquid (or solid) a triplet of components,
0 (respectively, l
LW
s
lLW , l+
and
0
, , ), can be dened in such a way that the surface free energy of
+
s
s
the material can be written according to (1.2) and the corresponding liquid-solid adhesion
work takes the form (1.1). By Young-Dupr
e equation we obtain then
l (1 + cos )
2
= 2
q
q
lLW sLW
+
l+ s0 +
q
l0 s+
3
(2:1)
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where is the equilibrium contact angle of the pair. Let L and Sbe two sets of L 1 liquids
and S 1 solids, respectively; each liquid is denoted with a value of the index i = 1; : : : ; L,
whereas an index j = 1; : : : ; S distinguishes the dierent solids. The components of the
i-th liquid will be then written as
LW
l;i
+
l;i
;
0
l;i
;
and a similar notation will be introduced for the corresponding components of the j -th
solid
LW
s;j
+
s;j
;
0 :
s;j
;
Equations (1.2) and (1.1) trivially generalize to any liquid, solid, and liquid-solid pair:
l;i
=
LW
l;i
adh
Wi;j
q+
+2
=
0
l;i l;i
8 i = 1; : : : ; L
l;i (1 + cos i;j )
hq LW
=2
s;j
LW
l;i s;j
+
LW
s;j
=
q+
q+
+2
0
l;i s;j
+
0
s;j s;j
8 j = 1; : : : ; S
l;i s;j
8 i; j ;
q0 +i
provided that i;j and Wi;jadh are taken as the contact angle and the adhesion work of the
i-th liquid on the j -th solid. The previous equalities can be usefully put into a matrix form
l;i
=
XiT RXi
adh
Wi;j
by posing, for each i and j ,
8i = 1; : : : ; L
=
0 qLW 1
BB q l;i+ CC
Xi = B
@ ql;i CA
0
l;i
s;j
l;i (1 + cos i;j )
=
YjT RYj
8j = 1; : : : ; S
= 2XiT RYj 8i; j
0 qLW 1
BB q s;j+ CC
Yj = B
@ qs;j CA
0
s;j
01
R = @0
(2:2)
1
0 0
0 1A
0 1 0
(2:3)
and denoting with the superscript T the transpose of an arbitrary matrix | it is understood
that the dispersive, acidic and basic components of any material are always nonnegative. In
+ , 0 and s;j , LW , + ,
the set (2.2) of LS +L+S equations, the 3L+4S variables l;iLW , l;i
s;j
s;j
l;i
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0
s;j
must be regarded as unknowns. Only the total surface tensions l;i of liquids can be
measured independently and are taken as given parameters, together with the equilibrium
contact angles i;j . For L; S suciently large we obtain LS + L + S > 3L + 4S and (2.2)
turns out to be an overdetermined set of nonlinear equations, to be solved by means of
some best-t algorithm. The research of any best-t solution implies the optimization of
some merit function V dependent on the rests
1l;i = XiT RXi 0 l;i
1s;j = YjT RYj 0 s;j
1i;j = XiT RYj 0 12 (1 + cos i;j )l;i :
(2:4)
Existence of best-t solutions for V is highly nontrivial, since no general argument can
be easily invoked. In any case, uniqueness of such solutions is excluded by the following
theorem, whose proof is immediate ; .
[7 8]
Theorem 1
Let Ai , i = 1; : : : ; L, and Bj , j = 1; : : : ; S be some | real or complex | 3 2 3 nonsingular
matrices such that the substitutions Xi ! Ai Xi and Yj ! Bj Yj leave each of the rests
(2.4) invariant. Then:
(i)
Ai
= Bj = C 8 i = 1; : : : ; L ; j = 1; : : : ; S , where C is any matrix satisfying C T RC =
;
(ii) the set G of matrices C as above constitutes a group with respect to the usual
matrix product. Such a group is isomorphic to the orthogonal group O(2; 1; C ), and
its intersection with the the group Gl(3; R) of real 323 invertible matrices is isomorphic
to the orthogonal group O(2; 1; R).
R
3
The lack of uniqueness for the best-t solution easily follows because invariance of all the
rests implies invariance of V .
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Theorem 2
The group G 3 includes the 3-parameter family of nonsingular matrices
6 exp[!1E1 + !2E2 + !3E3 ]
where
00 1
E1 = @ 0 0
1
8 !1 ; !2 ; !3 2 C
0 0 0 11
E2 = @ 01 0 0 A
0
0A
01 0 0
0 0 0
;
00 0 0 1
E3 = @ 0 1 0 A :
0 0 01
Proof
Since C 2 G 3 is nonsingular, a complex 3 2 3 matrix E exists such that
C = exp(E ) =
1 1
X
n=0 n!
En
and, due to R2 = I, the condition C T RC = R assumes the equivalent form
exp(RE T R) = exp(0E ) :
(2:5)
Equation (2.5) is certainly veried if RE T R = 0E , i.e.
E T R = 0RE
()
(RE )T = 0RE ;
which implies that RE is a skew-symmetric matrix
0 0 ! ! 1
1
2
RE = = @ 0!1 0 0!3 A
0!2
!3
!1 ; !2 ; !3
0
2C
As a consequence, the matrix E is given by
01 0 010 0 ! ! 1 0 0
1
2
E = R
= @ 0 0 1 A @ 0!1 0 0!3 A = @ 0!2
0 1 0
0!2
!3
0
and can be rewritten as
E = !1 E1 + !2 E2 + !3 E3 ;
!1
!3
:
1
0 A
!2
0!1 0 0!3
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so that
C
E)
!E
= exp(
= exp[ 1 1 +
The nal result follows by observing that
! 2 E2 + !3 E3 ] :
C T RC = R implies (0C )T R(0C ) = R.
A useful by-product is the following corollary.
Corollary 2.1
For any ! 2 C there holds
0
exp(!E1 ) = @
!
1
0
1
0! 0!2=2
01 0
exp(!E3 ) = @ 0 e!
0
0
Proof
1
0A
01
exp(!E2 ) = @ 0!
0
0
0
e 0!
1
1
0
1
A:
1
!
0!2=2 A
0
0
1
!E1) is easily deduced from the denition, by taking into account
The expression for exp(
that
E1 is a nilpotent matrix:
exp(
!E1)
=
1 !n
X
n=0
The immediate equality
n!
E1n = I+ !
E2 = 0E1T
exp(
!E2)
0
@
0
0
1
0
1
0
0
0
0
0
1
00
2
!
A + 2 @0
0
0
0
1
0
0
0
0
1
A:
leads to
= exp(
0!E1T ) =
and provides the second expression. As for exp(
2exp(0!E )3T
1
!E3 ), the computation is straightforward
| it concerns the exponential of a diagonal matrix.
Theorem 2 can be applied to prove the statement below, about the possibility of redening
acid-base components for a particular liquid or solid.
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Theorem 3
Let 0LW , 0+, 00 be the nonnegative acid-base components of a given material, with
LW + 2 + 0 > 0. Consider the new set of components LW , + , 0 , dened by
q q
0
0
0
0 p LW 1 0 pq LW 1
@ pp A = C BB@ q CCA
means of
0
+
0
+
0
C T RC = R :
with
00
Then, 8 2 [0; 1) and 2 R a matrix C exists such that
q q 3
2
q q 03 1 0 2
= LW + 2 e
2
q
q
3
2
1 0 0
0 = LW + 2 0
e :
LW = 0LW + 2 0+ 00 +
0
+
0
0
0
+
0
0
(2:6)
2
Moreover, a suitable choice of C leads to either + = 0 or 0 = 0 .
Proof
We rstly show that the dispersive and polar contributions to surface free energy, LW
and 2 + 0 respectively, can take the values
p
q q 3
2
p p
LW = 0LW + 2 0+ 00
q q 3
2
2 + 0 = (1 0 ) 0LW + 2 0+ 00
(2:7)
for any 2 [0; 1]. To this end, we subdivide the proof in three parts and distinguish the
cases where + and 0 are both zero or not.
0
0
Suppose that 0+ > 0 and apply Corollary 2.1 to obtain, 8 !1 2 R,
0 p LW 1 0
0 p LW 1
B q CC = BBB
@p
A
! E )B
=
exp(
@q A @
p
0
+
0
1 1
+
0
0
0
P (!1 ) = 0!12
0+
2
1
0
+
0
+
0
0
q
The polynomial
p LW + ! q
q
q 0 p LW q
0! 0! q
1
0 !1 0LW +
0
q0
0
2
1
+
0 =2
1
CCC :
A
(2:8)
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q
p
has a maximum in !1 = 0 0LW = 0+ , where it takes the value
q LW q 1 LW + 2q q0
0
q
P 0 = =
+
0
0
+
0
0
0
> 0:
0+
2
The zeroes of P are the solutions of the equation
0!12
which provides
q
0+
2
!1 =
&
q LW q 0
0
0 = 0
+
pLW + rLW + 2q q0
q
0
r
q q
p
0 LW + LW + 2 0
q
0
%
0 !1
0
+
0
0
+
0
0
=
+
0
0
0
p0 0 in (2.8), we deduce that !
1
<0
q0
1
= 2q
!?
0
0+
and since
0!1?
0+
1
>0
must be chosen as follows
q0
2 q !1? :
0!1? !1
0
0+
1
pLW 0 in (2.8) requires ! 0pLW =q , so that
q0
pLW
(2:9)
0 q ! < 2 q !1? ;
p q
because of the trivial inequality ! ? 2 LW = . In the interval (2.9) the square root
p0 can assume any value from 0 to P 00pLW =q 1 = 2LW + 2q q03=2q .
The analogous condition
1
0
1
+
0
0
1
+
0
0
+
0
1
+
0
0
+
0
0
0
More precisely, for any given 2 [0; 1] there exists a unique !1 such that
q q
p0 = (1 0 ) LW + q2 0 ;
+
0
0
2
0+
0
+
0
0
+
0
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correspondingly,
p+ = q+ and
0
2
whereas
p+p 0 = (1 0 )2 LW + 2q+q 03 ;
0
q q
0
(2:10)
0
q q 03
2
0 :
(2:11)
q q
This proves (2.7) for 0+ > 0. Notice that [0; 0LW + 2 0+ 00 ] is the widest interval of
q q
p
p p
denition for both LW and 2 + 0 , since LW + 2 + 0 = LW + 2 + 0 and
p p
LW = 0LW + 2 0+ 00 0 2 + 0 = 0LW + 2 0+
0
0
0
the square roots of the new components must be nonnegative anyway.
The case 00 > 0 can be treated in a similar way by means of the transformation
1
p LW + ! q 0
0 p LW 1 0
0 p LW 1
2
0
0
0
q
@ pp+ A = exp(!2 E2) BB@ q0+ CCA = BBB@ q0+ 0 !2p0LW 0 !22q00 =2 CCCA ;
q0
0
0
0
with !2
(2:12)
0
2 R. If both 0+ and 00 vanish, we simply have to observe that
exp(!1 E1 + !2 E2 ) =
I+ !1 E1 + !2E2 + o(
q
!12 + !22 )
(!1 ; !2 )
2 R2 ! (0; 0)
and introduce the transformation
0 p LW 1
0 p LW 1
q0
@ pp + A = exp(!1E1 + !2E2) BB@ q0+ CCA =
0
00
1 0 p LW 1 q
q LW 0 1 1 q 2 2
0
2
2
A + o( !1 + !2 ) = 0 @ 0!2 A + o( !1 + !2 ) :
1
0 A@
0
0!1 0 1
0!1
0
p + and
The choice of !1 ; !2 < 0 suciently close to zero yields positive square roots
p 0, so that we are led to the previous cases | both acidic and basic components are
0 1
= @ 0!2
!1 !2
strictly positive.
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Corollary 2.1 allows now to vary the polar components by means of the substitution
0 p LW 1
0 p LW 1 0 p LW 1
p
p
@ p + A 0000! exp(!3 E3) @ p
p + A = @ e+! p + A
0
3
e0!3
0
0
!3
2R:
For instance, in the case 0+ > 0 the nal transformation assumes the form
0
0 p LW 1 B
B
@p
p +0 A = BB@
r
1
q +q 0
0 0
C
C
q+
C
+
!
C
e
0
A
q
q
q
3
2
+
+
0
e0! (1 0 ) 0LW + 2 0 0 =2 0
and coincides with (2.6) by posing
p
0LW
+2
3
3
q q #
"
1 0 0LW + 2 0+ 00
= 2!3 0 ln
2
0+
whenever 6= 1. If = 1 we get 0 = 0, as requested. An analogous result can be
established by starting from equation (2.12), with the only dierence that + = 0 for an
appropriate value of the parameter !2 .
Remark
p
q q
q q
For any given vector ( 0LW 0+ 00)T satisfying 0LW + 2 0+ 00 > 0 it is always
possible to perform a linear transformation C = exp(!1 E1 + !2 E2 + !3E3 ), !1, !2 , !3 2 R,
so that the new acidic and base components vanish:
0 p LW 1 0 r
q +q 0 1
0
q
LW
0 0 C
+
2
B + CC = B 0
exp(!1 E1 + !2E2 + !3E3 ) B
A:
@q0 A @
0
00
(2:13)
0
As explicitly written in the previous equation, the new dispersive component must be set
q q
p p
to 0LW + 2 0+ 00, owing to the invariance of LW + 2 + 0 . Condition (2.13) is
equivalent to
r
0LW
0 p LW 1
0 1
BB q0 + CC = exp(0! E 0 ! E 0 ! E ) @ 10 A =
1
1 1
2 2
3 3
q +q 0 @ q 0 A
0
+2
0
0
00
(2:14)
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1
0
2 ChR 0 1
CC 0 1
BB ChR 0 !3 R2
q
C
B
Sh
Ch
1
0
R
R
=B
BB !2 R 0 !3 R2 CCC @ A ; with R !32 0 2!1!2 2 R [ iR
@ A
01
!1 ShRR + !3 ChR
R2
and ChR = (eR + e0R)=2, ShR = (eR 0 e0R )=2 8 R 2 C . We have to show that any vector
( )T satisfying ; ; 0 and 2 + 2 = 1 can be obtained from (2.14) for a suitable
choice of !1; !2 ; !3 2 R. Actually, it can be proved even more: that the following map
8
9
(!1; !2; !3) 2 (x1 ; x2; x3 ) 2 R3 : x23 0 2x1 x2 0 000000!
0
1
2 ChR 0 1
BB ChR 0 !3 R2
C 80 1
9
BB ! ShR 0 ! ChR 0 1 CCC 2 <@ A : 2 + 2 = 1 ; ; ; 2 R=
BB 2 R 3 R2 CC : ;
@ A
01
!1 ShRR + !3 ChR
R2
is onto | although not one-to-one. Therefore, we can conne ourselves to consider R 0
and the entire functions ChR, (ChR 0 1)=R2, ShR=R will never vanish. Our rst goal is
to choose R and !3 in such a way that = 0, any preassigned real value. We preliminary
take R in such a way that ChR > 0 and then we determine an !3 2 R which satises
ChR 0 !32 ChRR20 1 = 0 :
Existence of the above !3 | and its uniqueness up to a change of sign | is obvious since
both ChR and (ChR 0 1)=R2 are strictly positive. The product !1!2 is now uniquely xed
and for the components , we have = (1 0 20)=2. A very tedious but rather simple
algebraic calculation shows that the parameters !1, !2, can be always chosen in such a
way that (; ) = (0 ; 0 ) for any given (0 ; 0) satisfying 0 0 = (1 0 20)=2.
This remark implies that Theorem 3 can be put in a stronger | although not particularly
useful, as a rule | form, by extending the validity of (2.6) also to the case = 1.
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The previous result gives us the possibility of assigning the vOCG components of a particular liquid almost arbitrarily. A natural question is whether after this assignation a certain
amount of indeterminacy is still available. An investigation of the spectral properties of
matrices in G 3 provides a satisfactory answer to the problem.
Theorem 4
Let C 2 G 3 . Then:
(i) there holds det(C 0 I) = det(C 01 0 I), so that if is an eigenvalue, 01 also is.
The spectrum of C always includes +1 or 01;
(ii) the eigenvalues of C = exp(!1 E1 + !2 E2 + !3E3), !1; !2 ; !3 2 C , are
1
p
e
!32 02!1 !2
e0
p
!32 02!1 !2
for !32 0 2!1!2 6= 0, whereas in the opposite case there is a unique eigenvalue +1 with
geometric multiplicity 1. In any case, for (!1 ; !2; !3 ) 6= (0; 0; 0) the eigenspace of +1
| kernel of C 0 I | coincides with the following set
8
9
Ker(C 0 I) = (!3 !2 0 !1)T ; 2 C n f0g :
Proof
From the denition of G 3 we have that C T R = RC 01 and therefore
C T R 0 R = RC 01 0 R
()
(C T 0 I)R = R(C 01 0 I) ;
which implies
det(C T 0 I) detR = detR det(C 01 0 I) () det(C T 0 I) = det(C 01 0 I)
so that C and C 01 have the same characteristic polynomial of order three. This proves
item (i). As for item (ii), we preliminary observe that if (x1 x2 x3 )T is an eigenvector
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!1E1 + !2E2 + !3E3 with eigenvalue , then the same (x1 x2 x3)T provides an
eigenvector of exp E with eigenvalue e :
0x 1 1
0x 1 1 0x 1
0x 1
1
1
1
n
X
X
1
n
@ x2 A = e @ x12 A :
exp E @ x2 A =
E @ x2 A =
!
!
n
n
n=0
n=0
x3
x3
x3
x3
of
E
=
E is readily determined by means of the characteristic equation
0 !
!2 1
= 03 + (!32 0 2!1 !2) = 0
det(E 0 I) = det 0!2 !3 0 0
0!1 0 0!3 0 The spectrum of
which provides
p!2 0 2! ! , 0p!2 0 2! ! .
= 0 , +
1 2
3
1 2
3
Consequently, whenever
!32 0
2!1 !2 =
6 0 the matrix exp(E ) admits three simple eigenvalues:
e+
1
p!2 02!1!2
0 I]
and the eigenspace Ker[exp(E )
3
p!202!1 !2
e0
3
= Ker(E ) is specied by the nontrivial solutions
(x1 x2 x3 )T of
0 0 ! ! 10x 1
( +!1 x2 + !2 x3 = 0
1
2
1
@
A
@
A
0 = 0!2 !3
0
x2
() 0!2 x1 + !3 x2 = 0
0!1
0!3 x3
0!1 x1 0 !3 x3 = 0
that is by (x1 x2 x3 )T = t(!3 !2 0 !1 )T , t 2 C n f0g. In the case !32 0 2!1 !2 = 0 the only
eigenvalue of
0
E is 0, with geometric multiplicity 1.
0
Jordan form
0
E = Q01 @ 0
0
1 0
0 1
0 0
in terms of an appropriate nonsingular 3 2 3 matrix
exponential
1 1 00
X
1
0
@0
exp(E ) = Q
n=0
n!
01
= Q 01 @ 0
1 0
0 1
0 0 0
1
1
0 0
1=2
1
1
The matrix can be then put into the
1
AQ
Q. We deduce the expression for the
1n
2 00
1
0
A Q = Q 4I+ @ 0
1
A Q Q01 TQ ;
1 0
0 1
0 0 0
1 00
A+ 1 @0
01
T @0
2
0 1
0 0
0 0 0
1 1=2
1
1
0 0
1
1
A;
13
A5 Q =
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whose eigenvectors coincide with those of the triangular matrix T , up to the transformation
Q. The eigenspace KerT 0 I is easily obtained from
0 0 1 1=2 1 0 x 1 0 x + x =2 1
1
2
3
0 = @ 0 0 1 A @ x2 A = @ x3 A =)
0 0
0
0
x3
x2 = 0
x3 = 0
and we conclude that the geometric multiplicity of the only eigenvalue +1 is also 1. The
eigenspace Ker[exp(E ) 0 I] is therefore the same obtained in the previous case.
An immediate consequence of Theorem 4 is the corollary below.
Corollary 4.1
A nonzero vector (x1 x2 x3 )T 2 C 3 is invariant through the linear transformation dened
by C = exp(!1 E1 + !2E2 + !3E3), (!1 ; !2; !3 ) 2 C 3 n f(0; 0; 0)g, if and only if
C
2
= exp (0x3 E1 + x2E2 + x1 E3)
3
for some 2 C n f0g.
3. Determination of scales in vOCG theory
We are now ready to apply the previous results to the problem of scale denition in vOCG
+
0
theory. Suppose that the acid-base components 1LW
;0 , 1;0 , 1;0 of a material | referred to
as \primary" reference chemical, from now on | have been estimated in some way. Under
q
the general assumption that 0LW + 2 0+00 > 0, we can use Theorem 3 to redene the
components according to
2
q q 3
+
0
= 1LW
;0 + 2 1;0 1;0 2 + 2q + q 0 3 1 0 e
1+ = 1LW
;0
1;0
1;0
2
q
q
3
2
0
1
0
+
LW
0
0
1 = 1;0 + 2 1;0 1;0
2 e
1LW
(3:1)
-306-
with 2 [0; 1) and 2 R arbitrarily chosen. By applying the linear transformation (3.1)
to all the chemicals of the set, we get a preliminary denition of the acid-base scale. Such
a denition is obviously not yet complete, since Corollary 4.1 states that there exists a
one-parameter group of linear transformations C 3 ! C 3 :
0p 1
0 p LW 1
h 0 q 0 q + q LW 1i p LW
p
S () : @
p + A 0000! exp 0 1 E1 + 1 E2 + 1 E3 @ p + A ; (3:2)
0
0
q q
p
with 2 C , through which ( 1LW 1+ 10)T is invariant.
+
0
If 2LW
;0 , 2;0 , 2;0 are now the components of another material | the \secondary" reference
chemical |, the linear transformation (3.2) allows us to write
0 q LW 1
0 p LW 1
2
q
q
q
q
q2+;0 C
BB + CC = exph00 0E + +E + LW E 1i B
B
C
2;0 C
3 B
1 1
1 2
1
@q2 A
@
A
q
0
0
2
2;0
in such a way that the residual parameter can be possibly xed by assigning one of the
q
q
p
numbers 2LW , 2+ or 20 . The choice is not arbitrary, since:
p
(a) the vectors ( 1LW ;
dependent, otherwise
q
(
2LW
;0 ;
q
2+;0 ;
q
q
1+ ;
20;0
q
10
q
) and (
2LW
;0 ;
q
2+;0 ;
q
20;0
) must be linearly in-
h0 q
q
q
1i 0 I
0 0 E + + E + LW E
) 2 Ker exp
1
1
1
2
1
3
and therefore
0 q LW 1 0 q LW 1
0 p LW 1
2
B
q2+;0 C
q2+;0 C
BB q + CC = exph00q 0E + q +E + q LW E 1i B
B
C
B
C
2;0 C = B
2;0 C
3 B
1 1
1 2
1
@q2 A
@
A
@
A
q
q
0
0
0
2
2;0
2;0
8 2 C . Components 2LW , 2+ , 20 do not depend on and no full denition of the
scale is possible;
-307-
(b) the transformation S () can be explicitly written as
q LW
q
q0
= C11
2
+
2 = C21
2 = C31
with
q LW
q LW;
q LW;
2 0 + C12
2 0 + C22
2;0 + C32
q
q ;
q ;
+
2 0 + C13
+
2 0 + C23
+
2;0 + C33
q0
q 0;
q 0;
20
20
2;0
0 1LW Ch(rr2) 0 1
q 0q LW Ch(r) 0 1 q 0 Sh(r)
0 =0 C11 = Ch(r)
C12
q
q
1
q r q LW
0 r
q q LW
r
0
qr 0q
r 0
1
C13 =
1
0
C21 =
C23 =
0
C31 =
0
+
1
+ Sh(
1
C22 = Ch(r)
0
)
01
q 0q LW
1
1
0 Ch(r) 0 1
C32 = 0
q 0q r
1
r
and r =
C33 =
0
1
r
2
+
1
r
2
01
01
Ch(r)
q LW
r
1
+
2
1
Sh(r)
r
0 1 + q 0 Sh(r)
1
r
0 1 + Ch(r) 0 q LW Sh(r)
2
+ Ch(r)
1
2
r
1
q q0
1LW + 2
1
+ Ch( )
1
2
r
Ch(r)
r
Ch(r)
1
+
1
1
+ Ch(r)
1
2
r
1
2
+ Sh(r)
r
1 . All the square roots of the new components are given
by expressions of the form
r + ce0r
a + be
0
0
LW
for suitable constants a, b, c dependent on 2LW
;0 , 2;0 , 2;0 , 1 , 1 , 1 and on the
+
pLW q q0
component. Constraints on the admissible values of that each
2
,
+
2,
2C
+
come from the requirement
2 be real and nonnegative. Typically, will belong to
some real interval.
We expect that whenever the secondary reference chemical is chosen appropriately, so as to
accomplish conditions (a) and (b) for some , the parameter takes a uniquely determined
-308-
value = ? by assigning one of the components 2LW , 2+ , 20 . A mapping through S (? )
of the whole component set will provide the requested scale.
It is noticeable that the self-consistency of the method imposes all the square roots of the
nal components to be nonnegative, thus not any choice will yield satisfactory results. In
this sense, both primary and secondary reference material must be selected \judiciously",
and the conventional values of the corresponding components assigned in a proper way.
4. Quadratic multicomponent theories
In the late 60s a prototype of multicomponent theory was already proposed by Owens and
Wendt[10] (OW), in order to determine the specic contribution of dispersive and \polar"
interactions to the whole work of adhesion. This is accomplished by the denition of
suitable dispersive and polar contributions for each material, in such a way that
l = ld + lp
s = sd + sp
for liquids and solids respectively. The work of adhesion becomes, accordingly,
hq
W adh = 2
q
i
sd ld + sp lp ;
a sort of geometric-mean rule being separately applied to dispersive and polar components.
The theory can be easily extended, for instance to take specically into account hydrogenbonding interactions
l = ld + lp + lH
hq
W adh = 2
sd ld +
q
s = sd + sp + sH
q
splp + sH lH
i
by introducing further appropriate components, labelled by H . A recent revision of vOCG
theory led Qin and Chang[11013] (QC) to propose a new, in principle more general threeparameter model whose equations reduce to
1
l = (Pld )2 0 Pla Plb
2
0
1
2
s = (Psd )2 0 Psa Psb
1
W adh = Psd Pld 0 Psa Plb + Psb Pla ;
-309-
the superscripts
d, a
and
b
LW -dispersive, Lewis-acidic
corresponding to
and Lewis-basic
components respectively. Although not developed to predict adhesion work or surface free
energy, other models show a very similar structure.
[14
theory
0
16]
A well-known example is Drago's
which distinguishes "acidic" and "basic" solvents (electron acceptors and
donors) and writes the enthalpy of adduct formation for any acceptor-donor pair as
01H
=
CA C B + E A E B
where the subscripts A and B indicate acceptor and donor and E and C represent electrostatic and covalent contributions.
Analogous empirical two-parameters relationships
for free energy have also been established by electrochemical techniques by Edwards[17] ,
Mulliken[18] and Foss[19] . Drago's theory may be readily extended to take into account the
eventual co-existence of acidic and basic sites in the same molecule by an equation of the
form[20]
01H
=
0 + EB E 0 :
CA CB0 + CB CA0 + EA EB
A
(4:1)
In recent papers[7;21] it has been recognized that from a mathematical point of view the
GvOC theory should be classied in the realm of Linear Free Energy or Solvation Energy
Relationships (LFER or LSER)[22;23] , where a thermodynamical quantity
Q,
pertaining in
this case to Lewis acid-base properties of two materials X and Y, is expressed as a sum of
pairwise products of some material coecients
Q
The index
i
=
X
i
Xi Yi :
refers to the class of the coecient: dispersive, acidic, basic and so forth,
while symbols
X
and
Y
pertain to the interacting materials. Such a kind of relations is
widely applied in physical organic chemistry[22;23] . All the above multicomponent models
involve
bilinear and quadratic forms of components, for adhesion work and surface free
-310-
energy respectively: they will be called Quadratic Multicomponent Models. Not all the
multicomponent theories have the same structure; an interesting counterexample is the
model by Wu[24;25] , who distinguishes dispersive and \polar" contributions to adhesion
work, as in WO, but adopting a harmonic-mean combining rule instead of a geometricmean approximation
W
adh
= 4
d
l
+
d
l
+
d
s
+ :
p
d
s
p
l
p
s
l
p
s
The general mathematical form of quadratic multicomponent models is immediately recognized. The model equations appear to be the same of vOCG theory in matrix notation
l
=
X RX
T
s
=
Y RY
T
W
adh
= 2
X RY
:
T
(4 2)
or the only third equation as in the case of Drago's theory, depending on whether it
does make sense to consider or it is of interest to model interactions of a material with
itself (in theories like vOCG, QC or OW, self-interaction describes surface free energy or
surface tension). Now, however,
X and Y denote c-dimensional column vectors of material
components (e.g. for each liquid and solid) or quantities simply related to them (square
R is any real, symmetrical
nonsingular c 2 c \structure" matrix characteristic of the model. If d ;:::;d denote the
(real and nonvanishing) eigenvalues of R and sgn(x) the sign function, by an appropriate
linear transformation of vectors X and Y it is always possible to reduce the structure
roots of components, suitably scaled components, etc.), while
1
matrix to the standard form
R
0
B
= @
sgn(
d)
O
..
1
O
1
.
sgn(
c
d)
C
A
:
c
Multicomponent models of the form (4.2) are thus properly dierent only if their structure
f
matrices do not share the same signature sgn(
d );:::; sgn(d )g, otherwise they should be
1
c
regarded as mathematically equivalent up to a linear redenition of components[20] | i.e.
up to a isomorphism.
-311-
5. Scale denition in quadratic multicomponent theories
In a way similar to the proof of Theorem 1 it is easy to check that the model equations
X ! CX , Y
T
real matrix C satises C RC = R. The set of matrices C
(4.2) are invariant through any linear transformation
G
=
fC
real
c 2 c nonsingular matrix : C T RC
! CY
=
Rg
forms a nonabelian group with respect to the common product, since
C RC
=
C
C T RC
=
R ; DT RD = R
T
01 )T RC 01 = R and nally
implies (C
)
=
(
, whenever the
:
(5 1)
IT RI =
R, while
CD)T RCD = DT C T RCD = DT RD = R :
As in the case of vOCG theory, scale multiplicity due to the invariance transformation
group (5.1) has the important consequence that the material parameters (components)
of a quadratic multicomponent model do not necessarily admit a direct interpretation,
unless they are susceptible of a direct measurement. Due to the \extrathermodynamical"
character of material components, such a constraint is quite typical.
vOCG acid-base components are calculated
only
As an example,
by surface tension and contact angle
data, solving the same model equations of the theory, and cannot be estimated in a direct
way. The unique alternative would be to establish a correspondence of a suitable vOCG
scale with some other scales of acid-base strength admitting a direct measurement of model
parameters, a correspondence which is missing at the present state of the art.
One can also immediately prove that matrices
since if
2C
is an eigenvalue of
C2G
C , so is 01 .
Matrices
as describing changes of bases in the linear space
metric of matrix representation
G
have very special spectral properties,
Rc
C2G
may also be interpreted
endowed with a bilinear symmetric
R, provided that the change of bases preserves the metric.
can then be visualized as a metric-preserving group and identied with the (generalized)
-312-
O(c+ ; c0; R), on having denoted with c+ and c0 the number of positive
and negative eigenvalues of R, respectively[26] . Once again, we have a conrmation that
orthogonal group
the multicomponent model and its invariance group of transformations depend uniquely
on the signature of the structure matrix
interchanging
R.
In this respect, it is worthy of note that
c+ with c0 is equivalent to replacing the metric by its negative and thus
gives the same group.
As an analytic manifold the group O(c+ ; c0 ; R) has dimension c(c 0 1)=2, with c = c+ + c0 .
This means that locally the elements of the group are completely specied by c(c 0 1)=2 real
parameters through an analytical parametrization dened in a suitable neghborhood of 0
in Rc(c01)=2. From a topological point of view this Lie group is not connected, consisting of
4 connected topological components, but it is always possibile | and physically reasonable
| to conne ourselves to the only connected component containing the unity, that is the
c2c unit matrix I. Such a choice seems anyway satisfactory, since the connected topological
component containing unity constitutes a subgroup[26], often denoted as SO+ (c+ ; c0 ; R).
The structure of the group in a neighborhood of unity can be detailed described in an
elementary way. Indeed any matrix close to unity can be written into the form
I+ D + o()
in terms of the scalar
( ! 0)
(5:2)
2 R and of any c 2 c real matrix D satisfying
[I+ DT + o()] R [I+ D + o()] =
R
( ! 0)
or, equivalently,
R D + DT R = 0 :
By using the symmetry of the structure matrix
R, equation (5.3) reduces to
R D + (R D )T
= 0
(5:3)
-313-
and therefore the most general form of
D must be
D = R01 on having denoted with any real skewsymmetric
:
(5 4)
c 2 c matrix, which depends on c(c 0 1)=2
parameters. Vice versa, a simple calculation shows that whenever (5.3) is satised the
exponential of
D
D
exp( ) =
X1
n=0
D
1 n
= lim I+
n!+1
!
nD
n
n
belongs to G . Matrices in the connected component of G containing I can be expressed as
exponentials of some generator
D and used to chose scales.
In order to remove multiplicity and specify a unique well-dened scale, a conventional
assignment of component values to some reference compounds is needed. The most general matrix
C2G
c c 0 1)=2 parameters, thus the reference
is completely determined by (
components must be chosen accordingly. Selection of a scale for a specic quadratic multicomponent model requires a detailed analysis, similar to that previously illustrated for
vOCG theory. Nevertheless the general strategy does not change in an appreciable way
when the group
O(2; 1; R) involved in vOCG analysis is replaced by a generic O(c+;c0 ; R).
For instance, the selection of component scales for the extended Drago model (4.1) can be
accomplished by transformation matrices of the form
C
= exp
0 0!
B@ 0!
!4 !5
0
!1 !2 !3
0
4 0!5 0!6
0!2 0!1 0 !6
0
1
1
CA
8 !i 2 R ; i = 1; 2;::: ; 6
obtained from (5.4) by the idempotent structure matrix
R=
0
B@
0 1 0 0
1 0 0 0
0 0 0 1
0 0 1 0
1
CA
-314-
and the most general 4 2 4 skew-symmetric matrix . The most important consequence
is that, since the dimensionality of the invariance group involved does not depend on
the detailed signature of the model structure matrix, the number of material components
which is necessary to assign in order to select a specic scale is typically related only to
the number c of components per material assumed by the theory.
-315-
References
[1] C.J. van Oss, R.J. Good, M.K. Chaudhury, J. Protein Chem.
5
, 385-402 (1986)
[2] C.J. van Oss, M.K. Chaudhury, R.J. Good, Adv. Coll. Interf. Sci.
[3] C.J. van Oss, R.J. Good, M.K. Chaudhury, Langmuir
28
, 35-60 (1987)
4
, 884-891 (1988)
[4] R.J. Good, M.K. Chaudhury, in Fundamentals of Adhesion, L.H. Lee Ed., Plenum
Press, New York (1991), Chapt. 3
[5] R.J. Good, C.J. van Oss, in Modern Approach to Wettability: Theory and Application, M.E. Schrader, G Loed Eds., Plenum Press, New York (1991), Chapt. 1
[6] C.J. van Oss. Interfacial Forces in Aqueous Media, Marcel Dekker, New York (1994)
[7] C. Della Volpe, S. Siboni, J. Coll. Interf. Sci.
195
, 121-136 (1997)
[8] C. Della Volpe, S. Siboni, Acid-base surface free energies of solids and the denition
of scales in van Oss-Good theory, J. Adhesion Sci.
14
Technol.
, 235-272 (2000).
Reprinted in Apparent and Microscopic Contact Angles, J. Drelich, J.S. Laskowsky
and K.L. Mittal Eds., VSP, New York (2000)
[9] I.W. Kuo, C.J. Mundy, Science
303
, No. 5658, 658-660 (2004)
[10] D.K. Owens, R.C. Wendt, J. Appl. Polym. Sci.
[11] X. Qin, W.V. Chang, J. Adhesion Sci. Technol.
[12] X. Qin, W.V. Chang, J. Adhesion Sci. Technol.
[13] W.V. Chang, X. Qin, Acid-base Interactions:
Technology, K.L. Mittal Ed.,
[14] R.S. Drago, Struct. Bond.
13
9
10
, 1741 (1969)
, 823-841 (1995)
, 963-989 (1996)
Relevance to Adhesion Science and
2
, 3-54, VSP, Utrecht (2000)
15
, 73-139 (1973)
[15] R.S. Drago, G.C. Vogel, T.E. Needham, J. Amer. Chem. Soc.
93
, 6014-6026 (1971)
-316-
[16] R.S. Drago, L.B. Parr, C.S. Chamberlain, J. Amer. Chem. Soc.
[17] J.O. Edwards, J. Amer. Chem. Soc.
1, 8-31 (1947)
[20] I.R. Peterson, Surf. Coat. Int.
[21] L.H. Lee, Langmuir
76, 1540-1547 (1954)
56, 801-822 (1952)
[18] R.S. Mulliken, J. Phys. Chem
[19] A. Foss, Acta Chem. Scand.
99, 3203-3209 (1977)
88(B), 1-8 (2005)
12, 1681-1687 (1996)
[22] M.J. Kamlet, J.M. Abboud, M.H. Abraham R.W. Taft, J. Org. Chem.
48, 2877-2891
(1983)
[23] M.H. Abraham, Chem. Soc. Rev.
[24] S. Wu, J. Polym. Sci. Part C
22, 73-83 (1993)
34, 19-30 (1971)
[25] S. Wu, Polymer Interface and Adhesion, Marcel Dekker, New York (1982)
[26] R. Gilmore, Lie Groups, Lie Algebras and Some of Their Applications, Wiley New
York, N.Y. (1974)