THERMODYNAMIC THEORIES OF ADHESION

Adhesion is a steady or firm attachment of two bodies, and as such it can be char-acterized by the thermodynamic work of adhesion, i.e., by the work which is needed to separate two different bodies in contact with each other under equilib-rium conditions. The action of the molecular forces at an interface forms funda-mental reasons for the adhesive forces between a substrate and an adhesive.¹⁰

Initial premises for the thermodynamic description of adhesion are the characteristics of two surfaces: their surface tension and interfacial tension at the interface between the two bodies in contact.¹⁰In the simplest case of two liq-uids with surface tensionγ_l1andγ_l2, their surface tension at the interface (inter-facial tension) is always lower than the highest surface tension at the interface with saturated vapor:

γ_l1l2=γ_l1 −γ_l2 [2.1]

This empirical Eq 2.1 has been referred to as Antonov’s rule. The separa-tion of two surfaces (their breaking away from each other in the direcsepara-tion per-pendicular to the interface) requires work per unitary surface area, Wa:

W =a γl1+γl2 −γl1l2 [2.2]

Eq 2.2 is applicable in any case, including the case when one of the compo-nents is a solid body. Accordingly, the cohesion energy is the work of destruction of a body, or the work needed to form a unitary surface area in this body. If the only result of the isothermal process here is the formation of 2 cm³of new surface area of the body, having surface tension,γ, the thermodynamic work of cohesion can be expressed as:

W_c= 2γ [2.3]

On wetting, a droplet of liquid forms a definite contact angle on the solid.

The state of mechanical equilibrium of the droplet on the surface is determined by

γ_s =γ_sl + cosΘγ_l [2.4]

where index s refers to the surface tension of solid, l to liquid, and sl to the inter-facial tension at the interface.

The joint solution of Eqs 2.2 and 2.4 gives the thermodynamic work of adhe-sion between liquid and solid:

W = (1+ cos )a γl Θ [2.5]

which is Dupre-Young’s equation. AtΘ= 0, the behavior of droplet is determined by the condition

W = 2_a γ_l [2.6]

W_acan be greater than 2γ_slifγ_s >γ_ls +γ_l, and in this case, the droplet of liquid spreads on the surface. The spreading condition occurs when Wa> 2γ_l, i.e., the liquid begins to spread on the solid surface when its adhesion becomes greater than its cohesion.

Continuing our discussion, we would like to emphasize the erroneous na-ture of the existing viewpoint that good wetting is a condition for providing good adhesion. Thermodynamically, it is high adhesion that determines good wet-ting.

Under conditions of complete wetting, the difference in surface tensions causes the droplet start to spread, which is given by

S=γs −γls −γl [2.7]

The spreading coefficient, S, as follows from Eqs 2.4-2.7, is equal to the dif-ference in the work of adhesion and cohesion of liquid:

S= W_a − γ2 _l [2.8]

The energy required for cohesive failure of the liquid should be 2γl and be equal to the adhesion, ifΘ= 0, i.e., if liquid completely wets solid. With incom-plete wetting (Θ >0), failure of adhesion occurs more easily because Wa< 2γ_l.

In Eq 2.2 for solids, the surface tension of the solid is the energy of the solid-gas interface only after the liquid has been removed from contact with the solid. Only in this case is Eq 2.5 also valid. When the liquid is being removed from the surface, some part of it stays adsorbed and, consequently, liquid is not removed completely. Then, the interface solid-gas is composed of two areas: the unchanged initial solid and the solid with adsorbed layer. The latter possesses a lower surface energy and has an adhesion value corresponding to that obtained from Eq 2.2. A virgin surface is characterized by a higher energy,γ_s + . This un-π known value, taken into account when expressing the work of adhesion, as

W = (1+ cos ) +a γi Θ π [2.9]

can be obtained afterΘis measured for the interface of solid-gas, following the removal of the liquid, i.e.,Θcorresponding to receding contact angle should be taken.

The difference between the works of adhesion of pure and covered surfaces cannot be measured directly and must be calculated from the adsorption iso-therm of vapor on the solid. IfΓis the amount of vapor sorbed at pressure, P, the Gibbs equation gives:

Γ= 1 Π

RT d dlnP



 

 [2.10]

and hence Π= RT ΓdP

0

∫

P ^[2.11]

The above discussion concerns an ideal system. The work of adhesion is di-rectly related to the strength of the molecular bonds at the interface and in-creases when the strength of the bond grows. In practice, one always seeks to attain maximum strength of the bond equal, in its limit, to the strength of the chemical bonds.

The thermodynamic approach to the description of adhesion has many ad-vantages as compared with some other theories. It does not require knowledge of the molecular mechanism of adhesion but considers only the equilibrium pro-cesses at the polymer-solid interface. The approach to the problem developed by Zisman^11,12is widely accepted. Zisman introduced the concept of the critical sur-face tension of wetting as a value which is found by extrapolation of the depend-ence of cosΘon γto cos =1, i.e., when liquid fully spreads on the surface. The valueγ_cfound by extrapolation is considered as the critical surface tension of a solid. If the valueγ_cis known, the equilibrium contact angle can be predicted for any liquid on any surface. Ifγ_l < , the contact angle equals zero and the liquidγ_c spreads on the surface.

According to Zisman, if liquid does not spread on a surface having high sur-face energy, it means that the sursur-face is covered by an adsorption layer of a sub-stance which decreases the surface energy of the solid to the level typical of a surface of low surface energy. Any liquid or gas may be adsorbed, forming a monolayer, and preventing the spread of another liquid. Such cases are of high importance in considering adhesion in PCM. According to Zisman any sub-stance adsorbing at the interface should not decreaseγ_c to a value lower than the surface tension of liquid adhesive.

The thermodynamic work of adhesion in the case of removal of a liquid from a surface covered by an adsorption monolayer may be found in the following way.According to Zisman,

cosΘ= a - Bγ_l [2.12]

Asγ_l approachesγ_c atΘ →0,

cosΘ = 1 + B (γ_c +γ_l) [2.13]

and

Wa= (2 + Bγ_c)γ_l - Bγ_l² [2.14]

The maximum value of Wacan be found, provided that

γ= 1/B + 1/2γ_c [2.15]

Then

Wa max= 1/B +γ_c + 1/4 Bγ_c² [2.16]

Sometimes the valueγ₂is considered equal toγ_c.^13,14However, from Eq 2.4, it tentatively follows that at cosΘ=1,γ₂=γ_conly when the interfacial tension is zero. The valueγ_c should be considered as an empirical value.

Another way to find the thermodynamic work of adhesion is to postulate a relationship betweenγ₂,γ_Landγ_sL(the Girifalko-Good equation):^15,16

γ_sL=γ_s +γ_L− Φ2 (γ γ_{2 L})¹² [2.17]

whereΦis an empirical parameter which can be calculated theoretically from molecular properties. Rewriting Eq 2.17 as

2 (Φ γ γ_{s L}) =¹² γ_s + γ_L−γ_sL [2.18]

and taking into account that the right-hand side of Eq 2.18 represents the ther-modynamic work of adhesion, Wa, (Eq 2.2), Eq 2.17 can be written in a more con-venient form:

W = 2 (_a Φ γ γ_{s L})¹² [2.19]

Combination of Eqs 2.4 and 2.17 gives:

1+ cos = 2 ^s

L

Θ Φ

 



γ

γ [2.20]

AtΘ →0, Eq 2.19 is transformed to:

γ γ γ

s L

2 c

= = 2

Φ Φ [2.21]

The value ofγ₂depends greatly on the accepted value ofΦ, which varies for different polymer-liquid pairs.^15,16Such measurements have been done only for liquid systems. It was also found that:

Φ = −1 00075. γ₂ [2.22]

which is the same for a great number of solid polymers and liquids. Substituting Eq 2.22 into Eq 2.20, we obtain

Φ −

= (0.015 2)( ) +− (0.015( ) 1)

sL s L 1/ 2

L

sL s L

12

γ γ γ γ

γ γ γ [2.23]

It can be seen thatγ_cis only a part of the surface tension of a solid polymer, because, according to the experimental data,Φ <1. The non-conformance of indi-vidual, experimentally found, values of γ_candγ_L and calculated according to Zisman can be explained by the difference of forces acting on the interface, and by the fact thatγ_c is determined by wetting which reflects only part of the free surface energy influenced only by dispersion or polar interactions, depending on the liquid applied.

Behind this lies the Fowkes theory,¹⁷in accordance with which the total surface free energy is due to the action of dispersion and polar forces. The contri-bution of the latter in the free energy can be distinguished. It can be seen that the free surface energy of a liquid is given by

γ_L=γ_L^d +γ_L^h [2.24]

where d relates to dispersion and h to polar interactions, for example, hydrogen bonding. Fowkes assumed that interfacial tension between two liquids, such as hydrocarbon and mercury, is described by the relationship:

( )

γ_{L1 L2}=γ_L1+γ_L2 −2 γ γ^d_{L1 L2}^{d 12} [2.25]

Eq 2.25 takes into account only the dispersive interactions of components, which in accordance with Reference 15 can be expressed as the geometric mean of dispersion components of both liquids. Eq 2.25 for a solid-liquid interphase has the form:

( )

γ_sL=γ_s +γ_L−2 γ γ_s^d _L^{d 12} [2.26]

Eq 2.26 allows one to calculate the dispersive component for a liquid having free surface energy dependent on dispersive and polar forces based on values ob-tained for liquid determined only by dispersive forces. For example, for hydro-carbonsγ_L=γ_L^d, and then for waterγ_L^d = 0.0218 N/m.

Using the above relationships and Young’s equation (2.4) for contact an-gles, one can predict the valueγ_sLfrom:

( )

γ_Lcos =Θ −γ_s + 2 γ γ_L^d _s^{d 12} [2.27]

For liquids havingγ_L>γ_s, it follows from Eq 2.27 that:

( )

cos = 1+ 2 sd Ld L

12 12

Θ − 

 



γ γ

γ [2.28]

From the dependence of^{cos onΘ}

( )

^{γLd 12}, one can find the dispersive compo-nent of the surface tension of a solid,γ_s^d, measuring its contact angle. It is thus possible to determineγ_L^d for many liquids. Consequently, the values ofγ_c found by Zisman for non-polar liquids correspond to their dispersion component, rather than to the full surface energy of a solid. Indeed, according to Zisman, at cos = 1Θ ,γ_L= . For a non-polar liquidγ_c γ_L=γ_L^d, which modifies Eq 2.28 to the fol-lowing form:

( )

γ_L= γ γ_L^d _s^{d 12} [2.29]

and thenγ_s^d =γ_L= . If dispersion and polar forces are operative on the surface,γ_c

where γ^h is determined by hydrogen bonding and dipole-dipole interactions.

Eq 2.28 can similarly be written as:

1+ cos = 2( )_L^{d Ld} + 2( ) sum gives the approximate full surface energy of a solid,γ_s.¹⁸

For the analysis, it is convenient to combine the Zisman equation:

γ_c=γ_s −γ_sL(atΘ= 0 ) with Eq 2.32:

The term in parentheses is interfacial tension,γ_sL,equal toγ_s-γ_c. Breaking into constituents, it serves as means of approximate evaluation of the surface layer structure.

The Fowkes theory played an important role in the evaluation of not only the free surface energy of polymers and its relation toγ_c, but also in the case of in-terfacial energy and determination of the relationship between adhesion of poly-mer solids and their surface energy. In this respect, it is essential to take into account, apart from dispersive forces, polar forces operating at the interface.

De-veloping the approach considered above, some other theories have been pro-posed.^19-22

Thermodynamic relationships describing adhesion can be used to substan-tiate the criterion of maximum of adhesion.²³ The maximum of adhesion strength is observed in these systems which have very low or equal zero interfa-cial tension,γ_sL. It means that such systems have equal surface tensions of adhe-sive and substrate. If these two values are not equal, the decrease in the adhesion strength is lower in systems having surface energy of substrate higher than the surface tension of adhesive. In this case the parameterΦ, can be pre-sented as follows:

ParameterΦdepends on the contact angle and dimensionless ratio of surface tensions. Thus the valueΦ cannot be considered as a constant. If liquid fully spreads on the surface, then cosΘ= 1,γ_L=γ_c andΦ=Φ₀. In this case the

may be considered as a constant for a given system. From the equations given above, it follows that:

Because valueΦenters Eq 2.18 for thermodynamic work of adhesion, the latter is dependent on the factors influencingΦ. Eq 2.36 is valid for the case whenΦis constant and thus this value cannot correctly describe the situation when using Eq 2.35. In many cases, each point in the dependence of

cos = f( , )Θ Φ γL corresponds to another value ofΦ. The critical surface tension of wetting, as shown by Gutowski,²³is not a constant value which probably should be connected with various contributions in valueγ_Lof polar and dispersion con-stituents,Φ, which depend on them. Therefore, it is very important to find the correct form of the dependence of cosΘandγ_L.

To establish the physico-chemical criterion of maximum of adhesion, Gutowski²³used the following relationships for linear dependence of cosΘand γ_L:

Denotingγ_s/γ_L= a (a is the energetic modulus of the system), the following equations have been obtained for a linear dependence:

γ

Correspondingly for the first case the relative work of adhesion is:

Wa /γL= 3 (1/ a)(1/− Φo2) [2.41]

and for the second:

Maximum of adhesion, determined from the mechanical strength of an ad-hesion joint, corresponds to the minimum of interfacial energy:

a_min = ^s = 1

Atγ_sLapproaching minimum, we have:

γ

From these equations, the author concludes that the zero contact angle does not meet the condition of minimum interfacial energy and maximum of strength except for systems whereΦ_o= 1.0. The interfacial energy at the point where cosΘ= 1 is always greater than at the minimum point. Consequently, the expected strength at contact angle equals zero will be lower than the maximum strength attainable for a given system. The optimum contact angle correspond-ing to the minimum of the interfacial tension may be found from the equation:

( )

Θ_opt = cosΘ^-1 2 (1/− Φ_o [2.45]

The zero contact angle corresponding to full spreading of liquid meets the condition:

a =_s ^s = 1

L o2

γ

γ Φ [2.46]

Gutowski’s approach is based on the Girifalko-Good approximation, which postulates interrelations between valuesγ_s,γ_Landγ_sLnot proven theoretically.

This is a common shortcoming of all approaches to the description of adhesion based on the use of the above-mentioned postulate.

A new approach to the problem is based on the concept of acid-base interac-tions.^24-26The theory takes into account the existence of acid or base properties of the filler surface and polymer adhesives.^27-29For example, poly(vinyl chloride) or other chlorinated polymers have acid properties and are capable of interac-tion with fillers or polymers with basic properties (SiO2, CaCO3, polyesters etc).

The enthalpy of adsorption of polymer with base properties B from one neutral solution on acid surfaces A is really the enthalpy of acid-base interaction∆H^AB:

- H = C C + E E∆ ^AB _{A B A B} [2.47]

where C and E are constants which may be found from measurements of the in-teraction between organic acids and bases.

Using the Fowkes concept regarding contribution of dispersion, polar com-ponents, and hydrogen bonds to adhesion, the adhesion between two substances 1 and 2 may be expressed as:

W = W + W12 12d

12ab [2.48]

γ₁=γ₁^d +γ₁^ab [2.49]

∆H = H + H12 ∆ 12d ∆

12ab [2.50]

It is evident that in this case the dipole-dipole interactions are neglected.

However, the acid-base interactions cannot be found from the geometric mean value equal to 2(γ γ₁^AB ₂^AB)¹². The theory is based on the assumption that all inter-actions, which are determined by polar dipole-dipole interactions and by hydro-gen bonding, may be quantitatively described as acid-base interactions. The value W may be estimated from the equation:

W = W₁₂^ab ₁₂−W = (1+ cos ) 2(₁₂^d γ_L Θ − γ γ_L^d _s^d)¹² [2.51]

The adhesion, according to Fowkes, depends on adsorption, determined by acid-base interactions. This statement is confirmed by the observation that polymers with basic properties have higher adhesion to acidic surfaces as ex-pected from the adsorption data. The interactions between filler and matrix can be regulated by their acid-base properties - easily achieved by surface modifica-tion.

There is still another approach to the determination of the thermodynamic work of adhesion. The validity of Antonov’s rule (Eq 2.10) is theoretically con-firmed,³⁰ forming the base for thermodynamic calculations. Thermodynamic work of adhesion can be found from a value of thermodynamic work of cohesion, Wc. If the surface tension of polymer isγ_p, then W = 2γ_p.

Using Eq 2.1, one obtains:

Wa= 2γ_p=Wc [2.52]

This relation characterizes the thermodynamic state of a two-phase sys-tem with minimum free energy and is valid for the impermeable phase border between two immiscible bodies. Eq 2.52 is a condition of the minimization of the free energy of the system. A minimization of the work of cohesion of two phases Wcsand Wcpmay be considered as an initial condition. From Eq 2.52, it follows that when any specific interaction at the phase border is absent, the thermody-namic work of adhesion is determined by the thermodythermody-namic work of cohesion of the phase with lower cohesion energy. In this case, the adhesion may be en-hanced by increasing cohesion strength of polymer.

To apply Eq 2.52, one needs to know the surface tension of adhesive. There are methods which allow one to estimate this value.^31,32The methods are based on the measurement of contact angles. To find the surface tension of a polymer γ_p, using various liquids with surface tension,γ_L, the following equation is used:

2γ_s=γ_L(1 + cosΘ) [2.53]

It should be noted that the surface tension of a solid polymer is dependent on the surface tension of a wetting liquid. The experimental data, schematically

presented in Figure 2.1, show that, at a contact angle close to 90^o, a zone of invari-able values of γ_p exists, whereas at other angles there is a linear dependence ofγ_pon γ_L. The slope depends on the nature of poly-mer. This effect is determined by polarizability of molecules of both phases in the interphase region due to the changes in the surface force field of wetting of solid polymer by liquids having various surface tensions. The polarization, P, changes with field intensity, E:

εE = E + 4πP [2.54]

whereε is a dielectric constant.

The intensity of the surface force field is determined by the surface tension, and the polarization will not change as long as the surface tension does not change during the process of wetting of solid polymer by liquid. From Eq 2.4, it follows that this condition is fulfilled when the contact angle is close to 0 when

γ_s=γ_sL. In this case, the surface force field does not change essentially, and as a result, a zone of values ofγ_pindependent onγ_Lexists. However, ifγ_s>>γ_sL, at low contact angles, then the surface field is also lower, leading to depolarization of molecules of solid polymer and to diminishing of its surface tension (depolariza-tion zone). There is also a zone of addi(depolariza-tional polariza(depolariza-tion whenγ_s< γ_sL.³³

Let us now consider some consequences of the conditionγ_p= f(γ_L). The sur-face tension is an integral characteristic of the interphase layer:³⁴

γ= (P^δ P )dz

0

N T

∫

^{− [2.55]}

whereδis the thickness of the interphase layer, PNand PTare normal and tan-gential parts of pressure, z is the direction normal to the plane of an interface.

Figure 2.1. Schematic representation of the dependence of polymer surface tensionγ2on the surface tensionγ1of wetting liquid: 1-depolarization zone, 2-unperturbed zone, 3-zone of addi-tional polarization.

From this equation, it follows that by changing γ_p, other properties connected withγ_pare changed as well. In particular, from Eq 2.52, it follows that Wafor any polymer is not a constant value but changes in accordance with the dependence shown in Figure 2.1. Whenγ_pandγ_Lare almost identical, the adhesion work is diminished, and, when the difference between these two values is relatively high, the adhesion increases. In the first case, Wc> Wciand in the second, Wc<Wci

(index i denotes the interphase region).

From the analysis presented above, it becomes pertinent that when a poly-mer is in contact with a solid having higher surface tension, the increase in the surface tension of a polymer will be observed due to polarization. It is also

From the analysis presented above, it becomes pertinent that when a poly-mer is in contact with a solid having higher surface tension, the increase in the surface tension of a polymer will be observed due to polarization. It is also

In document Copyright <9 1995 by ChemTec Publishing ISBN 1-895198-08-9 (pagina 72-88)