18TH
`
`EDITION
`
` fiigijj'ffij: eming1'On ’ S
`
`ALFQNSO R GENNARO
`Editor, and Chairman
`of the Editorial Board
`
`AKN 1014
`
`1
`
`

`

`
`
`Pharmaceutical
`
`Sciences
`
`1 990
`
`MACK PUBLISHING COMPANY
`
`Easton, Pennsylvania 18042
`
`2
`
`

`

`CHAPTER 19
`
`Disperse Systems
`
`
`George Zogrofi, PhD
`Professor
`School of Pharmacy, University of Wisconsin
`Madison, Wi 58706
`
`Hons Schott, PhD
`Professor of Pharmaceutics and Colloid Chemistry
`School of Pharmacy, Temple University
`Philadelphia, PA 19140
`
`James Sworbrick, DSc, PhD
`Professor and Chairman
`Division of Pharmaceutics
`School of Pharmacy, University of North Carolina at Chapel Hill
`Chapel Hill, NC 27599—7360
`
`Interfacial Phenomena
`
`Very often it is desirable or necessary in the development
`of pharmaceutical dosage forms to produce multiphasic dis—
`persions by mixing together two or more ingredients which
`are not mutually miscible and capable of forming homoge—
`neous solutions. Examples of such dispersions include sus-
`pensions (solid in liquid), emulsions (liquid in liquid) and
`foams (vapor in liquids). Because these systems are not
`homogeneous and thermodynamically stable, over time they
`will show some tendency to separate on standing to produce
`the minimum possible surface area of contact between
`phases. Thus, suspended particles agglomerate and sedi—
`ment, emulsified droplets cream and coalesce and the hub-
`bles dispersed in foams collapse, to produce unstable and
`nonuniform dosage forms.
`In this chapter the fundamental
`physical chemical properties of dispersed systems will be
`discussed, along with the principles of interfacial and colloi-
`dal physics and chemistry which underly these properties.
`
`Interfaciai Forces and Energetics
`
`In the bulk portion of each phase, molecules are attracted
`to each other equally in all directions, such that no resultant
`forces are acting on any one molecule. The strength of these
`forces determines whether a substance exists as a vapor,
`liquid or solid at a particular temperature and pressure.
`At the boundary between phases, however, molecules are
`acted upon unequally since they are in contact with other
`molecules exhibiting different forces of attraction. For ex-
`ample, the primary intermolecular forces in water are due to
`hydrogen bonds, whereas those responsible for intermolecu—
`lar bonding in hydrocarbon liquids, such as mineral oil, are
`due to London dispersion forces.
`Because of this, molecules situated at the interface con-
`tain potential forces of interaction which are not satisfied
`relative to the situation in each bulk phase.
`In liquid sys-
`tems such unbalanced forces can be satisfied by spontaneous
`movement of molecules from the interface into the bulk
`phase. This leaves fewer molecules per unit area at the
`interface (greater intermolecular distance) and reduces the
`actual contact area between dissimilar molecules.
`Any attempt to reverse this process by increasing the area
`of contact between phases, ie, bringing more molecules into
`the interface, causes the interface to resist expansion and to
`
`Dr Zografi authored the section on Interfacial Phenomena. Dr
`Schott authored the section on Colloidal Dispersions. Dr Swarbrick
`authored the section on Particle Phenomena and Coarse Dispersions.
`
`and therefore
`
`257
`3
`
`behave as though it is under a tension everywhere in a tan-
`gential direction. The force of this tension per unit length
`of interface generally is called the interfacial tension, except
`when dealing with the air—liquid interface, where the terms
`surface and surface tension are used.
`To illustrate the presence of a tension in the interface,
`consider an experiment where a circular metal frame, with a
`looped piece of thread loosely tied to it,
`is dipped into a
`liquid. When removed and exposed to the air, a film of
`liquid will be stretched entirely across the circular frame, as
`when one uses such a frame to blow soap bubbles. Under
`these conditions (Fig 19-1A), the thread will remain col—
`lapsed.
`If now a heated needle is used to puncture and
`remove the liquid film from within the loop (Fig 19-13), the
`loop will stretch spontaneously into a circular shape.
`The result of this experiment demonstrates the spontane-
`ous reduction of interfacial contact between air and the
`liquid remaining and, indeed, that a tension causing the loop
`to remain extended exists parallel to the interface. The
`circular shape of the loop indicates that the tension in the
`plane of the interface exists at right angles or normal to every
`part of the looped thread. The total force on the entire loop
`divided by the circumference of the circle, therefore, repre-
`sents the tension per unit distance of surface, or the surface
`tension.
`Just as work is required to extend a spring under tension,
`work should be required to reverse the process seen in Figs
`19-1A and B, thus bringing more molecules to the interface.
`This may be seen quantitatively by considering an experi—
`ment where tension and work may be measured directly.
`Assume that we have a rectangular wire with one movable
`side (Fig 19—2). Assume further that by dipping this wire
`into a liquid, a film of liquid will form within the frame when
`it is removed and exposed to the air. As seen earlier in Fig
`19-1, since it comes in contact with air, the liquid surface will
`tend to contract with a force, F, as molecules leave the
`surface for the bulk. To keep the movable side in equilibri-
`um, an equal force must be applied to oppose this tension in
`the surface. We then may define the surface tension, 7, of
`the liquid as F/Zl, where 2l is the distance of surface over
`which F is operating (21 since there are two surfaces, top and
`bottom).
`If the surface is expanded by a very small dis—
`tance, Ax, one can then estimate that the work done is
`
`W = FAx
`
`W = yZle
`
`(1)
`
`(2)
`
`3
`
`

`

`
`
`
`
`_‘.2”;)—-I
`
`
`
`Table I—Surface Tension of Various Liquids at 20°
`Surface tension,
`
` Substance dynes/cm
`
`476
`Mercury
`72.8
`Water
`63.4
`Glycerin
`325
`Oleic acid
`28.9
`Benzene
`27.1
`Chloroform
`26.8
`Carbon tetrachloride
`26.5
`1-Octanol
`27.4
`Hexadecane
`25.4
`Dodecane
`23.9
`Decane
`21.8
`Octane
`19.7
`Heptane
`18.0
`Hexane
`11.0
`Perfluoroheptane
`
`Nitrogen (at 75°K) 9.4
`
`Table ll—lnterfacial Tension of Various Liquids against
`Water at 20°
`Interfacial tension.
`dynes/cm
`
` Substance
`
`Decane
`Octane
`Hexane
`Carbon tetrachloride
`Chloroform
`Benzene
`Mercury
`Oleic acid
`1-Octanol
`
`52.3
`51.7
`50.8
`45.0
`32.8
`35.0
`428
`15.6
`8.51
`
`exhibits a higher surface tension than the alkanes of compa—
`rable molecular weight, but increasing the molecular weight
`of the alkanes (and hence intermolecular attraction) in-
`creases their surface tension closer to that of benzene. The
`lower values for the more nonpolar substances, perfluoro-
`heptane and liquid nitrogen, demonstrate this point even
`more strongly.
`Values of interfacial tension should reflect the differences
`in chemical structure of the two phases involved; the greater
`the tendency to interact, the less the interfacial tension.
`The 20—dynes/cm difference between air—water tension and
`that at the octane—water interface reflects the small but
`significant interaction between octane molecules and water
`molecules at the interface. This is seen also in Table II, by
`comparing values for octane and octanol, oleic acid and the
`alkanes, or chloroform and carbon tetrachloride.
`In each case the presence of chemical groups capable of
`hydrogen bonding with water markedly reduces the interfa-
`cial tension, presumably by satisfying the unbalanced forces
`at the interface. These observations strongly suggest that
`molecules at an interface arrange themselves or orient so as
`to minimize differences between bulk phases.
`That this occurs even at the air—liquid interface is seen
`when one notes the relatively low surface—tension values of
`very different chemical structures such as the n-alkanes,
`octanol, oleic acid, benzene and chloroform. Presumably, in
`each case, the similar nonpolar groups are oriented toward
`the air with any polar groups oriented away toward the bulk
`phase. This tendency for molecules to orient at an interface
`is a basic factor in interfacial phenomena and will be dis-
`cussed more fully in succeeding sections.
`Solid substances such as metals, metal oxides, silicates
`and salts, all containing polar groups exposed at their sur—
`face, may be classified as high-energy solids, whereas nonpo-
`
`4
`
`B
`
`258
`
`CHAPTER 19
`
` A
`
`A circular wire frame with a loop of thread loosely tied to
`Fig 19—1.
`it:
`(A) a liquid film on the wire frame with a loop in it; (B) the film
`inside the loop is broken.1
`
`
`
`Fig 19—2. A movable wire frame containing a film of liquid being
`expanded with a force, F.
`
`Since
`
`AA = 2le
`
`(3)
`
`where AA is the change in area due to the expansion of the
`surface, we may conclude that
`
`W = 7AA
`
`(4)
`
`Thus, the work required to create a unit area of surface,
`known as the surface free energy/unit area, is equivalent to
`the surface tension of a liquid system, and the greater the
`area of interfacial contact between phases, the greater the
`free—energy increase for the total system. Since a prime
`requisite for equilibrium is that the free energy of a system
`be at a minimum, it is not surprising to observe that phases
`in contact tend to reduce area of contact spontaneously.
`Liquids, being mobile, may assume spherical shapes
`(smallest interfacial area for a given volume), as when eject-
`ed from an orifice into air or when dispersed into another
`immiscible liquid.
`If a large number of drops are formed,
`further reduction in area can occur by having the drops
`coalesce, as when a foam collapses or when the liquid phases
`making up an emulsion separate.
`Surface tension is expressed in units of dynes/cm, while
`surface free energy is expressed in ergs/cmz. Since an erg is
`a dyne-cm, both sets of units are equivalent.
`Values for the surface tension of a variety of liquids are
`given in Table I, while interfacial tension values for various
`liquids against water are given in Table II. Other combina-
`tions of immiscible phases could be given but most heteroge—
`neous systems encountered in pharmacy usually contain wa—
`ter. Values for these tensions are expressed for a particular
`temperature. Since an increased temperature increases the
`thermal energy of molecules, the work required to bring
`molecules to the interface should be less, and thus the sur-
`face and interfacial tension will be reduced. For example,
`the surface tension of water at 0° is 76.5 dynes/cm and 63.5
`dynes/cm at 75°.
`As would be expected from the discussion so far, the rela—
`tive values for surface tension should reflect the nature of
`intermolecular forces present; hence, the relatively large val-
`ues for mercury (metallic bonds) and water (hydrogen
`bonds), and the lower values for benzene, chloroform, carbon
`tetrachloride and the n—alkanes. Benzene with 7r electrons
`
`4
`
`

`

`"B" FACE
`
`\
`
`Fig 19-3. Adipic acid crystal showing various faces.2
`
`Table ill—Values of 75,, for Solids of Varying Polarity
` Solid 75V (dynes/cm)
`
`
`19.0
`Teflon
`25.5
`Paraffin
`37.6
`Polyethylene
`45.4
`Polymethyl methacrylate
`50.8
`Nylon
`61.8
`Indomethacin
`62.2
`Griseofulvin
`68.7
`Hydrocortisone
`155
`Sodium Chloride
`
`Copper 1300
`
`lar solids such as carbon, sulfur, glyceryl tristearate, polyeth-
`ylene and polytetrafluoroethylene (Teflon) may be classi-
`fied as low-energy solids.
`It is of interest to measure the
`surface free energy of solids; however, the lack of mobility of
`molecules at the surface of solids prevents the observation
`and direct measurement of a surface tension.
`It is possible
`to measure the work required to create new solid surface by
`cleaving a crystal and measuring the work involved. How-
`ever, this work not only represents free energy due to ex-
`posed groups but also takes into account the mechanical
`energy associated with the crystal (is, plastic and elastic
`deformation and strain energies due to crystal structure and
`imperfections in that structure).
`Also contributing to the complexity of a solid surface is the
`heterogeneous behavior due to the exposure of different
`crystal faces, each having a different surface free energy/unit
`area. For example, adipic acid, HOOC(CH2)4COOH, crys—
`tallizes from water as thin hexagonal plates with three dif-
`ferent faces, as shown in Fig 19—8. Each unit cell of such a
`crystal contains adipic acid molecules oriented such that the
`hexagonal planes (faces) contain exposed carboxyl groups,
`while the sides and edges (A and B faces) represent the side
`View of the carboxyl and alkyl groups, and thus are quite
`nonpolar.
`Indeed,
`interactions involving these different
`faces reflect the differing surface free energies.2
`Other complexities associated with solid surfaces include
`surface roughness, porosity and the defects and contamina—
`tion produced during a recrystallization or comminution of
`the solid.
`In View of all these complications, surface free
`energy values for solids, when reported, should be regarded
`as average values, often dependent on the method used and
`not necessarily the same for other samples of the same sub-
`stance.
`In Table III are listed some approximate average values of
`73,, for a variety of solids, ranging in polarity from Teflon to
`copper, obtained by various indirect techniques.
`
`Adhesional and Cohesional Forces
`
`Of prime importance to those dealing with heterogeneous
`systems is the question of how two phases will behave when
`brought in contact with each other.
`It is well known, for
`instance, that some liquids, when placed in contact with
`other liquid or solid surfaces, will remain retracted in the
`form of a drop (known as a lens), while other liquids may
`
`DlSPERSE SYSTEMS
`
`259
`
`exhibit a tendency to spread and cover the surface of this
`liquid or solid.
`Based upon concepts developed to this point, it is appar-
`ent that the individual phases will exhibit a tendency to
`minimize the area of contact with other phases, thus leading
`to phase separation. On the other hand, the tendency for
`interaction between molecules at the new interface will off-
`set this to some extent and give rise to the spontaneous
`spreading of one substance over the other.
`In essence, therefore, phase affinity is increased as the
`forces of attraction between different phases (adhesional
`forces) become greater than the forces of attraction between
`molecules of the same phase (cohesional forces).
`If these
`adhesional forces become great enough, miscibility will oc-
`cur and the interface will disappear. The present discussion
`is concerned only with systems of limited phase affinity,
`where an interface still exists.
`A convenient approach used to express these forces quan-
`titatively involves the use of the terms work of adhesion and
`work of cohesion.
`The work of adhesion, W0, is defined as the energy per cm2
`required to separate two phases at their boundary and is
`equal but opposite in sign to the free energy/cm? released
`when the interface is formed.
`In an analogous manner the
`work of cohesion for a pure substance, We, is the work/cm2
`required to produce two new surfaces, as when separating
`different phases, but now both surfaces contain the same
`molecules. This is equal and opposite in sign to the free
`energy/cm2 released when the same two pure liquid surfaces
`are brought together and eliminated.
`By convention, when the work of adhesion between two
`substances, A and B, exceeds the work of cohesion for one
`substance, eg, B, spontaneous spreading of B over the sur-
`face of A should occur with a net loss of free energy equal to
`the difference between Wu and WC.
`If WC exceeds Wa, no
`spontaneous spreading of B over A can occur. The differ-
`ence between Wu and We is known as the spreading coeffi-
`cient, S; only when S is positive will spreading occur.
`The values for Wu and WC (and hence S) may be expressed
`in terms of surface and interfacial tensions, when one con—
`siders that upon separation of two phases, A and B, 7A3 ergs
`of interfacial free energy/cm2 (interfacial tension) are lost,
`but that ”m and 713 ergs/cm2 of energy (surface tensions of A
`and B) are gained; upon separation of bulk phase molecules
`in an analogous manner, 2%, or 273 ergs/cm2 will be gained.
`Thus
`
`and
`
`Wa = M + 73 " 7A8
`
`W, = 27A or 2'“;
`
`For B spreading on the surface of A, therefore
`
`SB = ’YA + We " 7A1; “ 2713
`
`or
`
`$13 = 7.4 _ (73 + 7.48)
`
`(5)
`
`(6)
`
`(7)
`
`(8)
`
`Utilizing Eq 8 and values of surface and interfacial tension
`given in Tables I and II, S can be calculated for three repre-
`sentative substances—decane, benzene, and oleic acid—on
`water at 20°.
`
`Decane:
`
`Benzene:
`
`S = 72.8 — (23.9 + 52.3)
`
`S = 728 ~ (28.9 + 35.0)
`
`Oleic acid: S = 72.8 — (82.5 + 15.6)
`
`= —3.4
`
`8.9
`24.7
`
`=
`
`=
`
`As expected, relatively nonpolar substances such as decane
`exhibit negative values of S, whereas the more polar materi-
`als yield positive values; the greater the polarity of the mole-
`
`5
`
`

`

`260
`
`CHAPTER 19
`
`cule, the more positive the value of S. The importance of
`the cohesive energy of the spreading liquid may be noted
`also by comparing the spreading coefficients for hexane on
`water and water on hexane:
`
`SH/W = 72.8 — (18.0 + 50.8) =
`
`4.0
`
`SW/H = 18.0 — (72.8 + 50.8) = —105.6
`
`Here, despite the fact that both liquids are the same, the
`high cohesion and air—liquid tension of water prevents
`spreading on the low-energy hexane surface, while the very
`low value for hexane allows spreading on the water surface.
`This also is seen when comparing the positive spreading
`coefficient of hexane to the negative value for decane on
`water.
`
`To see whether spreading does or does not occur, a powder
`such as talc or charcoal can be sprinkled over the surface of
`water such that it floats; then, a drop of each liquid is placed
`on this surface. As predicted, decane will remain as an
`intact drop, while hexane, benzene and oleic acid will spread
`out, as shown by the rapid movement of solid particles away
`from the point where the liquid drop was placed originally.
`An apparent contradiction to these observations may be
`noted for hexane, benzene and oleic acid when more of each
`substance is added, in that lenses now appear to form even
`though initial spreading occurred. Thus, in effect a sub-
`stance does not appear to spread over itself.
`It is now established that the spreading substance forms a
`monomolecular film which creates a new surface having a
`lower surface free energy than pure water. This arises be‘
`cause of the apparent orientation of the molecules in such a
`film so that their most hydrophobic portion is oriented to-
`wards the spreading phase.
`It is the lack of affinity between
`this exposed portion of the spread molecules and the polar
`portion of the remaining molecules which prevents further
`spreading.
`This may be seen by calculating a final spreading coeffi—
`cient where the new surface tension of water plus monomo-
`lecular film is used. For example, the presence of benzene
`reduces the surface tension of water to 62.2 dynes/cm so that
`the final spreading coefficient, SF, is
`
`SF = 62.2 — (28.9 + 35.0) = —1.’7
`
`The lack of spreading exhibited by oleic acid should be
`reflected in an even more negative final spreading coeffi—
`cient, since the very polar carboxyl groups should have very
`little affinity for the exposed alkyl chain of the oleic acid
`film. Spreading so as to form a second layer with polar
`groups exposed to the air would also seem very unlikely, thus
`leading to the formation of a lens.
`
`Wetting Phenomena
`
`In the experiment described above it was shown that talc
`or charcoal sprinkled onto the surface of water float despite
`the fact that their densities are much greater than that of
`water.
`In order for immersion of the solid to occur, the
`liquid must displace air and spread over the surface of the
`solid; when liquids cannot spread over a solid surface spon-
`taneously, and, therefore, 8, the spreading coefficient,
`is
`negative, we say that the solid is not wetted.
`An important parameter which reflects the degree of wet—
`ting is the angle which the liquid makes with the solid sur-
`face at the point of contact (Fig 19—4). By convention, when
`wetting is complete, the contact angle is zero; in nonwetting
`situations it theoretically can increase to a value of 180°,
`where a spherical droplet makes contact with solid at only
`one point.
`
`VAPOR
`
`
`
`LlQUlD
`
`
` S 0 LI D
`
`\ \ \ \
`
`Forces acting on a nonwetting liquid drop exhibiting a
`Fig 19—4.
`contact angle of 6.3
`
`In order to express contact angle in terms of solid—liquid—
`air equilibria, one can balance forces parallel to the solid
`surface at the point of contact between all three phases (Fig
`19-4), as expressed in
`
`’st = 'YSI. + TLv COS 0
`
`(9)
`
`where 75% 73L, and 'yLv represent the surface free enerv
`gy/unit area of the solid—air, solid—liquid, and liquid—air
`interfaces, respectively. Although difficult to use quantita-
`tively because of uncertainties with ’st and 75L measure-
`ments, conceptually the equation, known as the Young
`equation, is useful because it shows that the loss of free
`energy due to elimination of the air—solid interface by wet—
`ting is offset by the increased solid—liquid and liquid—air
`area of contact as the drop spreads out.
`The ‘YLV cos 0 term arises as the horizontal vectorial com-
`ponent of the force acting along the surface of the drop, as
`represented by 7“]. Factors tending to reduce fly and 73L,
`therefore, will favor wetting, while the greater the value of
`'ysv the greater the chance for wetting to occur. This is seen
`in Table IV for the wetting of a low—energy surface, paraffin
`(hydrocarbon), and a higher energy surface, nylon, (polyhex—
`amethylene adipamide). Here, the lower the surface ten—
`sion of a liquid, the smaller the contact angle on a given solid,
`and the more polar the solid, the smaller the contact angle
`with the same liquid.
`With Eq 9 in mind and looking at Fig 19-5, it is now
`possible to understand how the forces acting at the solid-
`
`Table IV—Confacf Angle on Paraffin and Nylon for Various
`Liquids of Differing Surface Tension
`Surface tension,
`Contact angle
`dynes/cm
`Paraffin
`Nylon
`Substance
`
`
`70°
`105°
`72.8
`Water
`60°
`96°
`63.4
`Glycerin
`50°
`91°
`58.2
`Formamide
`41°
`66°
`50.8
`Methylene iodide
`16°
`47°
`44.6
`a-Bromonaphthalene
`spreads
`38°
`83.7
`tert—Butylnaphthalene
`“
`24°
`28.9
`Benzene
`“
`17°
`25.4
`Dodecane
`Decane
`23.9
`7°
`“
`
`Nonane
`22.9
`spreads
`“
`
`
`
`7’
`3L
`
`LIQUID
`
`Fig 19-5. Forces acting on a nonwetfable solid at the air+|iquid+so~
`lid interface: contact angle 0 greater than 90°.
`
`6
`
`

`

`
`
`Table V—Critical Surface Tensions of Various Polymeric
`Solids
`'Yc,
`Polymeric Solid Dynes/cm at 20°
`
`
`
`10.6
`Polymethacrylic ester of ¢’—octanol
`16.2
`Polyhexafluoropropylene
`18.5
`Polytetrafluoroethylene
`22
`Polytrifluoroethylene
`25
`Poly(vinylidene fluoride)
`28
`Poly(vinyl fluoride)
`31
`Polyethylene
`31
`Polytrifluorochloroethylene
`33
`Polystyrene
`37
`Poly(vinyl alcohol)
`39
`Poly(methyl methacrylate)
`39
`Poly(vinyl chloride)
`40
`Poly(vinylidene chloride)
`43
`Poly(ethylene terephthalate)
`
`Poly(hexamethylene adipamide) 46
`
`liquid—air interface can cause a dense nonwetted solid to
`float if 73L and my are large enough relative to 73v-
`The significance of reducing 7“; was first developed em-
`pirically by Zisman when he plotted cos 0 vs the surface
`tension of a series of liquids and found that a linear relation—
`ship, dependent on the solid, was obtained. When such
`plots are extrapolated to cos H equal to one or a zero contact
`angle, a value of surface tension required to just cause com—
`plete wetting is obtained. Doing this for a number of solids,
`it was shown that this surface tension (known as the critical
`surface tension, 'yp) parallels expected solid surface energy
`75V; the lower ”yr, the more nonpolar the surface.
`Table V indicates some of these 7c values for different
`surface groups, indicating such a trend. Thus, water with a
`surface tension of about ’7 2 dynes/cm will not wet polyethyl—
`ene (70 = 31 dynes/cm), but heptane with a surface tension
`of about 20 dynes/cm will. Likewise, Teflon (polytetrafluo-
`roethylene) (”yo = 19) is not wetted by heptane but is wetted
`by perfluoroheptane with a surface tension of 11 dynes/cm.
`One complication associated with the wetting of high-
`energy surfaces is the lack of wetting after the initial forma-
`tion of a monomolecular film by the spreading substance.
`As in the case of oleic acid spreading on the surface of water,
`the remaining liquid retracts because of the low—energy sur—
`face produced by the oriented film. This phenomenon, of—
`ten called autophobic behavior, is an important factor in
`many systems of pharmaceutical interest since many solids,
`expected to be wetted easily by water, may be rendered
`hydrophobic if other molecules dissolved in the water can
`form these monomolecular films at the solid surface.
`
`Capillarity
`
`Because water shows a strong tendency to spread out over
`a polar surface such as clean glass (contact angle 0°), one
`would expect to observe the meniscus which forms when
`water is contained in a glass vessel such as a pipet or buret.
`This behavior is accentuated dramatically if a fine-bore caps
`illary tube is placed into the liquid (Fig 19—6); not only will
`the wetting of the glass produce a more highly curved menis—
`cus, but the level of the liquid in the tube will be appreciably
`higher than the level of the water in the beaker.
`The spontaneous movement of a liquid into a capillary or
`narrow tube due to surface forces is defined as capillarity
`and is responsible for a number of important processes in-
`volving the penetration of liquids into porous solids.
`In
`contrast to water in contact with glass, if the same capillary
`is placed into mercury (contact angle on glass: 130°), not
`
`DISPERSE SYSTEMS
`
`261
`
`Fig 19—6. Capillary rise for a liquid exhibiting zero contact angle.1
`
`
`
`
`Fig 19-7. Capillary fall for a liquid exhibiting a contact angle,
`which is greater than 90°.1
`
`(9,
`
`only will the meniscus be inverted (see Fig 19—7), but the
`level of the mercury in the capillary will be lower than in the
`beaker.
`In this case one does not expect mercury or other
`nonwetting liquids to easily penetrate pores unless external
`forces are applied.
`To quantitate the factors giving rise to the phenomenon of
`capillarity, let us consider the case of a liquid which rises to a
`height, h, above the bulk liquid in a capillary having a radius,
`r.
`If (as shown in Fig 19-6) the contact angle of water on
`glass is zero, a force, F, will act upward and vertically along
`the circle of liquid-glass contact. Based upon the definition
`of surface tension this force will be equal to the surface
`tension, 7, multiplied by the circumference of the circle, 27rr.
`Thus
`
`F = 'y27rr
`
`(10)
`
`This force upward must support the column of water, and
`since the mass, m, of the column is equal to the density, d,
`multiplied by the volume of the column, 7rr2h, the force W
`opposing the movement upward will be
`
`W = mg = 7rr2dgh
`
`where g is the gravity constant.
`Equating the two forces at equilibrium gives
`
`so that
`
`7rr2dgh = 72w
`
`2
`h = Big
`
`(11)
`
`(12)
`
`(13)
`
`Thus, the greater the surface tension and the finer the capil—
`lary radius, the greater the rise of liquid in the capillary.
`If the contact angle of liquid is not zero (as shown in Fig
`19—8), the same relationship may be developed, except the
`
`
`
`Fig 19—8. Capillary rise for a liquid exhibiting a contact angle, 0,
`which is greater than zero but less than 90°.1
`
`7
`
`

`

`262
`
`CHAPTER 19
`
`vertical component of F which opposes the weight of the
`column is F cos 6 and, therefore
`
`Table VI—Ratio of Observed Vapor Pressure to Expected
`Vapor Pressure of Water at 25° with Varying Droplet Size
`
`h:
`
`27 cos 6
`rdg
`
`(14)
`
`This indicates the very important fact that if 6 is less than
`90°, but greater than 0°, the value of h will decrease with
`increasing contact angle until at 90° (cos 6 = 0), h = 0.
`Above 90°, values of 11 will be negative, as indicated in Fig
`19-7 for mercury. Thus, based on these equations we may
`conclude that capillarity will occur spontaneously in a cylin-
`drical pore even if the contact angle is greater than zero, but
`it will not occur at all if the contact angle becomes 90° or
`more.
`In solids with irregularly shaped pores the relation—
`ships between parameters in Eq 14 will be the same, but they
`will be more difficult to quantitate because of nonuniform
`changes in pore radius throughout the porous structure.
`
`Pressure Differences across Curved Surfaces
`
`From the preceding discussion of capillarity another im-
`portant concept follows.
`In order for the liquid in a capil-
`lary to rise spontaneously it must develop a higher pressure
`than the lower level of the liquid in the beaker. However,
`since the system is open to the atmosphere, both surfaces are
`in equilibrium with the atmospheric pressure.
`In order to
`be raised above the level of liquid in the beaker and produce
`a hydrostatic pressure equal to hgd, the pressure just below
`the liquid meniscus, in the capillary, P1, must be less than
`that just below the flat liquid surface, P0, by hgd, and there—
`fore
`
`P0 — P1 = hgd
`
`(15)
`
`Since, according to Eq 14
`
`then
`
`27 cos 6
`I = —
`rgd
`I
`
`9
`Po _. p1 = L056
`r
`
`(16)
`
`For a contact angle of zero, where the radius of the capillary
`is the radius of the hemisphere making up the meniscus,
`9
`m—fi=fl-
`r
`
`no
`
`The consequences of this relationship (known as the Laplace
`equation) are important for any curved surface when r be—
`comes very small and 'y is relatively significant. For exam-
`ple, a spherical droplet of air formed in a bulk liquid and
`having a radius, r, will have a greater pressure on the inner
`concave surface than on the convex side, as expressed in Eq
`17.
`Another direct consequence of what Eq 17 expresses is the
`fact that very small droplets of liquid, having highly curved
`surfaces, will exhibit a higher vapor pressure, P, than that
`observed over a flat surface of the same liquid at P’. The
`equation (Eq. 18) expressing the ratio of P/P’ to droplet
`radius, r, and surface tension, 7, is called the Kelvin equa—
`tion where
`
`log P/P/ =
`
`27M
`2.303RTpr
`
`(18)
`
`and M is the molecular weight, R the gas constant in ergs per
`mole per degree, T is temperature and p is the density in
`g/cmi‘. Values for the ratio of vapor pressures are given in
`Table VI for water droplets of varying size. Such ratios
`indicate why it is possible for very fine water droplets in
`
`P/P’ ‘3
`
`1.001
`1.01
`1.1
`2.0
`3.0
`4.2
`5.2
`
`Droplet size, pm
`
`1
`0.1
`0.01
`0.005
`0.001
`0.00065
`0.00060
`
`" P is the observed vapor pressure and P’ is the expected value for ”bulk"
`water.
`
`clouds to remain uncondensed despite their close proximity
`to one another.
`
`This same behavior may be seen when measuring the
`solubility of very fine solid particles since both vapor pres
`sure and solubility are measures of the escaping tendency of
`molecules from a surface.
`Indeed, the equilibrium solubili-
`ty of extremely small particles has been shown to be greater
`than the usual value noted for coarser particles; the greater
`the surface energy and smaller the particles, the greater this
`effect.
`
`Adsorption
`
`Vapor Adsorption on Solid Surfaces
`
`It was suggested earlier that a high surface or interfacial
`free energy may exist at a solid surface if the unbalanced
`forces at the surface and the area of exposed groups are quite
`great.
`Substances such as metals, metal oxides, silicates, and
`salts—all containing exposed polar groups—may be classi~
`fied as high-energy or hydrophilic solids; nonpolar solids
`such as carbon, sulfur, polyethylene, or Teflon (polytetraflu-
`oroethylene) may be classified as low—energy or hydrophobic
`solids (Table III). Whereas liquids satisfy their unbalanced
`surface forces by changes in shape, pure solids (which exhib-
`it negligible surface mobility) must rely on reaction with
`molecules either in the vapor state or in a solution which
`comes in contact with the solid surface to accomplish this.
`Vapor adsorption is the simplest model demonstrating
`how solids reduce their surface free energy in this manner.
`Depending on the chemical nature of the adsorbent (solid)
`and the adsorbate (vapor), the strength of interaction be-
`tween the two species may vary from strong specific chemi—
`cal bonding to interactions produced by the weaker more
`nonspecific London dispersion forces. Ordinarily, these lat-
`ter forces are those responsible for the condensation of rela-
`tively nonpolar substances such as N2, 02, C02 or hydrocar—
`bons.
`When chemical reaction occurs, the process is called che-
`misorption; when dispersion forces predominate, the term
`physisorption is used. Physisorption occurs at tempera-
`tures approaching the liquefaction temperature of the va-
`por, whereas, for chemisorption, temperatures depend on
`the particular reaction involved. Water~vapor adsorption
`to various polar solids can occur at room temperature
`through hydrogen-bonding, with binding energies interme-
`diate to physisorption and chemisorption.
`In order to study the