Article pubs.acs.org/Langmuir
Geometric Effects on Non-DLVO Forces: Relevance for Nanosystems Jeffery A. Wood* and Lars Rehmann Department of Chemical and Biochemical Engineering, University of Western Ontario, London, Ontario N6A 3K7, Canada S Supporting Information *
ABSTRACT: In this paper, the surface element integration (SEI) method was used derive analytical force/potential versus distance profiles for two non-DLVO forces: Lewis acid−base and solvation forces. These forces are highly relevant in a variety of systems, from bacterial adhesion, nanoparticle suspension stability to atomic force microscopy (AFM) profiles. The SEI-derived expressions were compared with the more commonly utilized Derjaguin approximations in order to assess the effect of curvature on the resulting interaction for the test cases of sphere-flat plate and equally sized spheres. For acid−base interactions, the deviation was found to be significant for particles up to 40 nm in diameter for the conventionally used decay length (λ = 1 nm) for water. The resulting expressions show that accounting in curvature for acid−base interactions is important even for simple smooth geometric shapes, recovering the Derjaguin expression at smaller values of λ/R. These results allow for correction of the acid−base force/potential versus distance from the Derjaguin-derived expressions using simple functions of λ/R. Conversely, for the solvation force the deviation was far less significant due to the oscillatory nature of the potential damping out effects and the smaller order of magnitude range of the solvation decay length, indicating that for solvation forces the Derjaguin approximation is suitable for most conceivable cases.
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INTRODUCTION The study of the interaction of colloids and surfaces is an area of great importance from an academic and industrial perspective, particularly with the more recent focus on nanoparticles and nanoscale systems. Quantum dots,1,2 nanoparticle self-assembly,3,4 interaction of atomic force microscope (AFM) tips with substrates5,6 and fouling of membranes with nanoparticles7−9 are all examples of nanometer scale colloidal systems, of which it is of vital importance to be able to predict and understand the underlying physics. For much of the 20th century, the primary theoretical framework for understanding these interactions was that of Derjaguin−Landau−Verwey−Overbeek (DLVO).10,11 In the DLVO framework, van der Waal interactions along with those of the electrical double layer are combined to give an overall interaction potential. From this potential, the stability against aggregation, adsorption characteristics, force versus distance profile and other of value can be determined.12,13 However, the DLVO framework is limited to systems where polar interactions are negligible; this excludes its utilization for a vast number of systems of interest, such as biological and environmental systems where strong hydrogen bonding or other polar forces are dominant.10,11,14 The so-called “extended” DLVO (XDLVO) interaction potential accounts for polar interactions via decomposing the interfacial tension, γ, into three components: Lifshitz-van der Waals (γLW), Lewis Acid (γ⊕) and Lewis Base γ⊖.15 The polar interactions are determined by the interaction of the acid and base components, with a substance requiring nonzero values of both acid and base to possess overall polarity while still allowing monopolar substances to interact with a media with a single © 2014 American Chemical Society
nonzero component. For materials with significant polar interactions in a given media, the DLVO framework can often be unable to quantitatively (or qualitatively) explain mechanisms of stability, adhesion and other colloidal characteristics, which the XDLVO approach is able to by accounting for polar forces.9,11,16−18 The XDLVO approach is particularly effective at regressing and predicting contact angle data for liquids on solid substrates, as long as care is chosen in the preparation of the solid and choice of test liquids.19,20 A simple exponential type decay, characterized by a single characteristic decay length, is the established functional form for decay vs distance of these interactions. This functional form was arrived at through analogy with hydration pressure expressions and lines up with similar expressions used for fitting AFM profiles for hydrophobic interactions.8,21 In particular, for AFM, the impact of curvature on another non-DLVO force, the solvation force, is of interest. Solvation forces arise from solvent molecules ordering around an interface, with the variation in density due to adsorption on a surface giving rise to an overall interaction potential. In AFM systems, the solvation force is common and similar to the XDLVO acid−base force treated mostly often using the Derjaguin approximation. Given the use of the importance of oscillations/peak-to-peak ratios for characterizing AFM systems experiencing solvation force,22,23 it is valuable to determine the deviation from the approximated and exact curvature on the force/potential versus distance. Received: February 18, 2014 Revised: April 3, 2014 Published: April 4, 2014 4623
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In order to accomplish this, the surface element integration (SEI) method, developed by Bhattacharjee and Elimelech (1997), is extremely useful. SEI has been used in numerous works to study the impact of geometry on derived DLVO and XDLVO forces.23−26 In particular, derivations for spheroidal and rough surfaces have been the focus of study. Hoek and Agarwal (2006) studied the influence of surface roughness effects on XDLVO predictions for stimulated interactions of a sphere with a roughened membrane. In that study, the values for sphere− sphere and sphere-plate acid−base interactions were calculated numerically but not the specific analytical expressions. Derivations of the van der Waals interaction potential along with electrostatic double layer forces have been carried out for a number of geometries, such as sphere−plate and sphere−sphere (equal and unequal radii), ellipsoid−sphere, sphere−cylinder, and rough surfaces.27−30 To date, the exact analytical expressions for the XDLVO acid−base interaction for even simple geometric configurations (sphere−plate, sphere−sphere) have not been derived and compared with the Derjaguin approximation. Consequently, the analytical expressions based on SEI for sphere-flat plate and sphere−sphere (equally sized) interaction potential and force were derived using the SEI method. As a comparison in the case of sphere-plate configuration, the recent surface integration approach (SIA) was also used as a comparison with the derived SEI expressions.31 The SEI method relies on the interaction potential being normal to the surface, SIA does not possess that restriction, but for the case of acid−base interactions, the interaction is presumed to be acting normal to the surface through the intervening suspending media and the SEI method is then suitable for use. Despite the importance of these interactions, relatively little work has been done on generally quantifying the impact of curvature/geometric effects on the suitability of use of the Derjaguin approximation for estimating the potential/force versus distance profiles for various geometries for acid−base interactions. Work has primarily focused on numerical evaluation of given test cases for different surface roughness, which demonstrated the importance of properly accounting for surface roughness on estimating acid− base decay with distance and the subsequent overall force balance.25,32−34 However, the exact analytical expressions for sphere-flat plate and between two spheres have, to the authors’ knowledge, not been derived or reported anywhere in the literature. Given the importance of these interactions to colloidal stability, adhesion and other related properties, it is of value to derive these expressions and compare their deviation from the generally used Derjaguin approximation versions. In addition to the effect of geometry/curvature on deviations from the Derjaguin approximation for the polar XDLVO force, their impact on deviations for the oscillatory solvation-type forces are also examined.
UAB,flat = U0e−(l − l0)/ λ
(1)
where U0 is the interaction potential at minimum contact (l = l0) determined from the interfacial tension between the plates in a given medium, l is the distance of closest approach, l0 is the equilibrium/minimum contact distance (0.157 Å), and λ is the decay length of the interaction potential, which correlates to the average size of molecular clusters in a liquid. The value of λ depends on the solvent, as well as the surface charge/ion concentration of the media for the case of hydration repulsion.35,36 For water, λ can range between 0.2 nm to slightly over 1 nm36 but has been reported in the range of 4−9 nm for oppositely charged hydrophilic colloids in high ionic strength aqueous solution37 and even larger values (up to 13 nm) have been reported.38,39 In the XDLVO framework, λ is most often treated as 1 nm, and in this work expressions are derived in general with specific plots of the “conventional” as well as longer decay lengths to illustrate the impact of the ratio of decay length to colloid size. For simplicity, the value of U0 is chosen as unity throughout the remainder of this work without the loss of generality. This factor can be reintroduced as necessary by multiplying each derived expression for interaction potential/ force with the appropriate value of U0 for that system. Solvation Forces. Solvation forces arise from fluctuations in the bulk density of liquids near an interface, which leads to layers of solvent forming. These density fluctuations give rise to an exponentially decay oscillatory force, with the oscillations possessing a periodicity of around one molecular diameter.36 Quite frequently, solvation-type forces are extremely relevant when considering interaction of an AFM tip with a flat-substrate, with the Derjaguin approximation often employed to provide the force expression.23,40 Although the mechanism for the force is different versus acid−base type interactions, the form of the potential is very similar to an additional oscillating term factored in. To examine whether including these oscillatory terms has an addition effect on the resulting exact geometry-derived SEI terms even for simple nonrough geometries, the exact geometric expression derived using the SEI method is derived, starting from the oscillatory version of interaction for flat plates shown in eq 2. The primary interest is for the case of sphere−flat plate, which corresponds to AFM and adsorption-type behavior but for completeness is also derived for the case of two equal radius spheres. The equation for the force (and corresponding potential) is shown in eq 2:41 2πl −l / σ e σ f0 σ ⎛ 2πl ⎞ + φs⎟e−l / σ cos⎜ ⎝ σ ⎠ 1 + 4π 2
Fsolvation,flat = f0 cos Usolvation,flat =
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φs = tan−1 2π
THEORETICAL BACKGROUND Non-DLVO Forces. Acid−Base Forces. The Lewis acid− base-type interactions in the XDLVO framework account for all electron donor−electron acceptor interactions, with a single characteristic free energy and decay form. This decay is exponential, most frequently characterized by a single effective decay length (λ). This type of interaction encompasses hydrogen bonding, as well as the energy to remove adsorbed layers of molecules from a surface (hydration energy). The base expression for the overall acid−base interaction between two flat plates separated by distance l is shown in eq 1 below.
(2)
where f 0 is the force at minimum contact (l = l0), l is the distance of closest approach, and σ is the characteristic molecular diameter of the medium/characteristic scale of interaction. As can be seen, solvation potential is phase-shifted compared with the force (∼81° for the case where the exponential decay length and periodicity are equal). The interaction potential at minimum contact would then be U0 = f 0σ cos φs/(1 + 4π2)1/2. The factor f 0 can be set to unity without a loss of generality and reintroduced as needed with an appropriate value by multiplying each derived expression with the appropriate value of f 0 for that system. In practicality, estimating the value of solvation force at minimum 4624
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Figure 1. Normalized force versus distance profiles for sphere−flat plate acid−base interactions.
Surface Element Integration. The SEI expressions for sphere-plate, eq 4, and sphere−sphere for equal radius particles, eq 5 are given below. As a consistency check, the resulting expressions derived from SEI should reduce to their Derjaguin equivalents when curvature effects are minimized, i.e., large particle(s) interacting over short distances.
contact can be extremely challenging, as there are generally layers of solvent which are not penetrated for example when measuring with AFM.36 Additionally, the decay length of solvation interactions (λs) is not necessarily equal to the molecular diameter of the solvent (σ), nor is the periodicity necessarily exactly one molecular diameter or constant between layers of solvent. However, the deviation is often negligible,40,41 particularly in the case of nonpolar liquids, and the approximation of λs = σ can be employed, although this approximation would not hold in the case of more polar liquids.42 Geometry Effects on Interaction. Derjaguin Approximation. The Derjaguin approximation for sphere−plate and sphere−sphere interactions of the XDLVO acid−base force are well-known;43 however, those equivalent expressions for the oscillatory form are derived herein for comparison with their SEI equivalents. The Derjaguin approximation relies on the radii of curvature being of a much larger scale than the characteristic distance of interaction, which is often the case for acid−base type interactions of micrometer and submicrometer sized colloids (μm diameter vs nm scale decay length). However, for nanoparticles it has been established that geometric effects become significant for van der Waals forces (in addition to impact of electromagnetic retardation and continuum hypothesis breakdown).
USEI,sphere − plate = 2π
∫D
Eplate(h) dh
R
[Uflat(l + R − R 1 − r 2/R2 )
− Uflat(l + R + R 1 − r 2/R2 )]r dr
(4)
USEI,sphere − sphere = U AA + U A ′ A ′ − 2U A ′ A U AA = 2π
∫0
R
Uflat(l + 2R − 2R 1 − r 2/R2 )r
1 − r 2/R2 dr U A ′ A = 2π
∫0
R
Uflat(l + 2R + 2R 1 − r 2/R2 )r
1 − r 2/R2 dr U A ′ A = 2π
∫0
R
Uflat(l + 2R )r 1 − r 2/R2 dr
(5)
where R is the sphere radius, l is the distance of closest approach and r is the radial distance from the center of the sphere. To determine the force profile the potential is differentiate with respect to the separation distance (F = −(dU/dl)) or in the case
∞
Uderj. = f (geometry)
∫0
(3) 4625
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Figure 2. Normalized force versus distance profiles for sphere−sphere acid−base interactions.
⎛ ⎞ ⎡ λ λ⎤ + ⎢1 + ⎥e−2R / λ⎟Fderj.,sphere − plate FSEI,sphere − flat = ⎜1 − ⎣ ⎝ ⎠ R R⎦
of solvation forces, the force can be integrated to determine the potential.
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(9)
RESULTS AND DISCUSSION
Comparing eqs 8 and 9 with their Derjaguin counterparts, an additional number of terms arise from curvature effects. These effects, however, only become significant at given combinations of particle size (R) and characteristic decay length (λ), i.e., depend only on the ratio λ/R. In the limit of large R and small λ, the SEI derived equations simplify to those derived from the Derjaguin approximation. Figure 1 illustrates the impact of particle size and decay length on the deviation of the derived acid−base force, using test cases of 10 and 100 nm radius spherical particles interacting with a flat plate for decay lengths of 1 and 13 nm. The decay lengths were chosen based on reported values for water, where 1 nm is the typical effective value chosen for water while values up to 13 nm have been reported for certain surfaces.15,38 The plots are normalized against the geometric correction factor for the Derjaguin approximation (2πR) and particle size (R) for the force and distance, respectively. As can be seen, the radius and decay length both have a significant impact on the curvature correction for the force versus distance. In the case of 100 nm radius particles in a solvent with a decay length of 1 nm (Figure 1a), the SEI-derived expression is identical to that of the Derjaguin approximation. For smaller particles or for larger decay lengths, the deviation increases and can be fairly significant. The percent deviation between the magnitude of force estimated by the SEI method and Derjaguin is ∼15% for R = 100 nm and λ
Acid−Base Polar Force. Sphere-Flat Plate. Using the SEI method, which accounts for the exact geometry of a configuration, the general equations for acid−base type interaction decay versus distance and force versus distance were derived for the case of a sphere interacting with an infinite flat plate. These equations were also derived based on the SIA method and found to be identical. The interaction potential (eq 8) and force (eq 9) are given below, along with the corresponding equations derived from using the Derjaguin approximation (eqs 6 and 7). Detailed derivations of the SEIbased interaction potential and force for this and other test cases are provided in the Supporting Information. Uderj. = 2πRλe−(l − l0)/ λ
(6)
Fderj. = 2π Re−(l − l0)/ λ
(7)
⎛ ⎞ ⎡ λ λ⎤ + ⎢1 + ⎥e−2R / λ⎟Uderj.,sphere − plate USEI,sphere − flat = ⎜1 − ⎣ ⎦ ⎝ ⎠ R R (8) 4626
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= 13 nm (Figure 1b) and for a radius of 10 nm with λ = 13 nm (Figure 1d) the Derjaguin approximation of force is approximately 5× larger. Even with the more typical value for λ of 1 nm, for a radius of 10 nm the deviation is ∼10% (Figure 1c), which covers a wide size range for colloids and nanosystems of interest, such as the adsorption of quantum dots on surfaces, interaction of atomic and scanning force microscopy with substrates, and so on. Sphere−Sphere: Equal Radii. Similarly, the SEI method was used to derive the analytical expression for the XDLVO Lewis acid−base-type interaction for the case of two equally sized spheres of radius R. The interaction potential (eq 12) and force (eq 13) are given below, along with the corresponding equations derived from using the Derjaguin approximation (eqs 10 and 11).
Uderj. = πRλe−(l − l0)/ λ
(10)
Fderj. = π Re−(l − l0)/ λ
(11)
type configurations. Figure 3 provides an illustration of the magnitude of the correction factor between the expressions
⎛ λ λ2 4R −2R / λ + − e USEI,sphere − sphere = ⎜1 − 2 R 3λ 2 R ⎝ ⎡ λ λ 2 ⎤ − 4R / λ ⎞ ⎟Uderj.,sphere − sphere − ⎢1 + + ⎥e R ⎣ 2R2 ⎦ ⎠ FSEI,sphere − sphere
Figure 3. Correction factor versus radius between SEI and Derjaguin for sphere−plate and sphere−sphere configurations, λ = 1 nm. (12)
derived based on the exact geometry and the simplified versions resulting from the Derjaguin approximation for a test case of λ = 1 nm. Due to the importance of acid−base forces in dispersion stability, adhesion, and so on, this emphasizes the importance of properly accounting for the inadequacies of the Derjaguin approximation when calculating forces for nanosystems. A large amount of literature has been devoted to evaluating the effects of surface roughness on colloidal interaction, with a poplar methodology being to use the SEI technique to determine the interaction for a given system via numerical integration. While surface roughness is undoubtedly an important factor to account for, the resulting expressions for even simple nonrough geometric shapes show a significant geometric effect for smalllength scale colloids in water. As the decay length for other systems is not clearly established, much larger decay lengths and therefore much larger minimum particle radii beyond which these curvature effects cease to be significant are possible. Even for water, values as large as 13 nm have been reported depending on the nature of the surface,38,39 and for these systems 100 nm radius spherical colloids show a significant deviation between exact and Derjaguin based expressions independent of any surface roughness or other shape factors. For example, accounting for geometric effects in the case of membrane fouling by spherical nanoparticles44 or protein adsorption onto adsorbent beds45,46 assessed by the XDLVO method would be highly relevant as there are significant deviations expected at the λ/R values present in these cases. For example, in the latter papers the adsorption of proteins with hydrodynamic radii in the range of ∼2.2 to 5.5 nm were examined using an XDLVO approach. For the chosen decay length of 0.6 nm for water, this would give a ∼10 to 30% deviation between the SEI and Derjaguin calculated values. This deviation, while large, is relevant only when the AB force dominates versus van der Waals interactions and does not change the qualitative interpretation of the adsorption mechanism. Sphere−Sphere: Unequal Radii. The similarity of the functional dependence of the correction factors between
⎛ λ λ2 4R −2R / λ = ⎜1 − + − e 2 R 3λ 2R ⎝
⎡ λ λ 2 ⎤ − 4R / λ ⎞ ⎟Fderj.,sphere − sphere − ⎢1 + + ⎥e R ⎣ 2R2 ⎦ ⎠
(13)
Similar to the case for sphere-flat plate interactions, eqs 12 and 13 are composed of their Derjaguin counterparts with an additional number of terms arising from geometry. For sphere− sphere interactions, the correction factors play a larger role and show a quadratic dependence on the ratio of λ to R. Once again, the impact of these corrections only becomes signature at higher values of λ/R, and, while the functional form is overall more complicated, the correction factor is slightly decreased versus λ/ R when compared with that for sphere−flat plate geometry. As is expected in the limit of large R and small λ, the SEI derived equations simplify to those derived based on the Derjaguin approximation. Figure 2 illustrates the impact of particle size and decay length on the deviation of the derived acid−base force, with the same test cases of 10 and 100 nm radius spheres and 1 and 13 nm decay lengths. The plots are normalized against the geometric correction factor for the Derjaguin approximation (πR in this case) and particle size (R) for the force and distance, respectively. The resulting normalized force versus distance profiles are similar to those for sphere-flat plate (Figure 1), with large deviations for 10 nm radius particles with a 13 nm decay length (Figure 2d), moderate deviations (∼10%) for R = 10 nm, λ = 1 nm and R = 100 nm, λ = 1 nm (Figure 2c,b) and no significant deviation for 100 nm and a decay length of 1 nm (Figure 2a). For spheres in water with the established decay length value of 1 nm, the geometric correction terms only become relevant (>5%) for particles with a radius smaller than 20 nm. This places the Derjaguin approximation as being a suitable choice for much of the colloidal domain, but shows that even relatively larger nanoparticles (dP < 40 nm) can have moderate deviations from the acid−base force/potential profile predicted by this approximation for both sphere−sphere and sphere−plate 4627
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Figure 4. Normalized force versus distance and deviation of SEI with Derjaguin for sphere−flat plate solvation force.
case of a spherical colloid of radius R and a flat plate using the SEI method are shown in eqs 16 and 17.
sphere−plate and sphere−sphere geometries also allows the utilization of either of these correction factors as the “effective” correction factor for the interaction of two unequal radius spheres, with the expressions forming the lower (sphere−plate) and upper (sphere−sphere) bounds. Since these bounds have minimal deviation between one another, either would serve as a fairly accurate estimate of the deviation between the Derjaguin expression and the exact SEI expression. The Derjaguin expression for unequally sized spheres of radius R1 and R2 is identical to that for equally sized spheres, with an effective radius given as Reff = 2R1R2/(R1 + R2). Using this effective radius, the ratio of decay length to radius can be calculated using this effective radius and then the upper and lower bounds for the correction factor found from the sphere−sphere and sphere−flat plate cases and either value chosen as a very close approximation to the specific value that would be found through numerical integration. Solvation Forces. Sphere−Flat Plate. Similar to the acid− base force/potential expressions, the SEI method was used to determine the interaction potential and force versus distance profiles for an oscillatory solvation force of the form shown in eq 2. The Derjaguin approximation expressions, derived using eq 3, are shown in eq 14 and 15. As can be seen, the Derjaguin derived force and potential are phase-shifted from the flat plate expression as well from each other. As was the case between the potential and force for flat plate, the phase-shift between the force and potential is tan−1 2π. The resulting equations for the
Uderj. =
⎛ 2πl ⎞ 2πσ 2R + 2φs ,1⎟e−l / σ cos⎜ 2 ⎝ σ ⎠ 1 + 4π
φs ,1 = tan−1 2π Fderj. =
USEI =
(14)
⎛ 2πl ⎞ + φs ,1⎟e−l / σ cos⎜ ⎝ ⎠ σ 1 + 4π 2πσR
2
(15)
2πσ 2R [g + g2 + g3]e−l / σ 1 + 4π 2 1
⎛ 2πl ⎞ + 2φs ,1⎟ g1 = cos⎜ ⎝ σ ⎠ ⎛ 2π (l + 2R ) ⎞ + 2φs ,1⎟e−2R / σ g2 = cos⎜ ⎝ ⎠ σ ⎡ ⎛ 2πl ⎞ + φs ,1 + φs ,2⎟ ⎢cos⎜ ⎠ R 1 + 4π 2 ⎣ ⎝ σ ⎛ 2π (l + 2R ) ⎞ − 2R / σ ⎤ − cos⎜ + φs ,1 + φs ,2⎟e ⎥ ⎝ ⎠ σ ⎦
g3 = −
φs ,2 = tan−1 4628
σ
4π 1 − 4π 2
(16)
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Figure 5. Normalized force versus distance and deviation of SEI with Derjaguin for sphere−sphere solvation force.
FSEI =
2πσR 1 + 4π
2
terms would be negligible. The solvation force, normalized against 2πσR/(1 + 4π2)1/2, is shown for two different sized spheres (25 and 5 nm) for a decay length of 1 nm (Figure 4a,b, respectively), which is the approximate value for a number of experimentally observed systems.35,41 Comparing the SEI and Derjaguin approximations, it is clear they provide very similar results. The normalized deviation of the SEI versus Derjaguin approximations is shown in Figure 4c. The relative impact of accounting for the exact geometry only becomes relevant at large values of σ/R and is likely negligible when compared with that of surface roughness effects. This contrasts with the sphere-flat plate interactions for the acid−base force, where relatively larger impacts are more likely to be physically plausible due to the order of magnitude of the decay length in that case. Sphere−Sphere: Equal Radius. Finally, the oscillatory solvation force for two equal radius spheres. The Derjaguin approximation expressions are shown in eq 18 and 19. Once again the Derjaguin derived force and potential are phase-shifted from the flat plate expression, as well from each other. As was the case between the potential and force for flat plate, the phase-shift between the force and potential is tan−1 2π. The resulting equations for the case of two spherical colloids of radius R using the SEI method are shown in eqs 20 and 21.
[f1 + f2 + f3 ]e−l / σ
⎛ 2πl ⎞ + φs ,1⎟ f1 = cos⎜ ⎝ σ ⎠ ⎛ 2π (l + 2R ) ⎞ + φs ,1⎟e−2R / σ f2 = cos⎜ ⎝ ⎠ σ ⎡ ⎛ 2πl ⎞ + φs ,2⎟ ⎢cos⎜ 2⎣ ⎝ ⎠ σ R 1 + 4π ⎤ ⎛ 2π (l + 2R ) ⎞ − cos⎜ + φs ,2⎟e−2R / σ ⎥ ⎝ ⎠ σ ⎦
f3 = −
φs ,2 = tan−1
σ
4π 1 − 4π 2
(17)
Unlike the nonoscillatory acid−base/hydration energy expressions, a simple function based on ratio of decay length (σ) to radius does not exist. Accounting for geometry shows there is an additional phase shift term arising, as well as additional shifted periodic terms. However, these terms are of relatively small magnitudes except at larger values of σ/R. The primary contributing term to both the force and potential versus distance is in fact the Derjaguin-derived expression (g1 and f1) and for larger values of σ/R the Derjaguin expression becomes exact. For larger ratios though, the shift in the magnitude becomes relatively more important, but for many practical systems any correction
Uderj. =
⎛ 2πl ⎞ πσ 2R + 2φs ,1⎟e−l / σ cos⎜ 2 ⎝ ⎠ σ 1 + 4π
φs ,1 = tan−1 2π 4629
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Fderj. =
⎛ 2πl ⎞ + φs ,1⎟e−l / σ cos⎜ ⎝ σ ⎠ 1 + 4π 2
USEI =
πσ 2R [g + g2 + g3 + g4 ]e−l / σ 2 1 1 + 4π
deviation of the SEI versus Derjaguin approximations is shown in Figure 5c, which shows that even for a particle radius as small as 5 nm the expressions line up fairly well with each other. This contrasts with the sphere-flat plate interactions for the acid−base force, where relatively larger impacts are more likely to be physically plausible due to the order of magnitude of the decay length in that case (1 to 13 nm vs ∼ < 1 nm for acid−base and solvation, respectively).
πσR
(19)
⎛ 2πl ⎞ + 2φs ,1⎟ g1 = cos⎜ ⎝ σ ⎠ g2 =
φs ,2 = tan−1 g3 = g4 =
■
⎡ ⎛ 2πl ⎞ ⎛σ ⎞ 1 ⎟ + φs ,1 − φs ,2⎟ ⎢cos⎜ ⎝ R ⎠ 1 + 4π 2 ⎣ ⎝ σ ⎠ ⎤ ⎛ 2π (l + 4R ) ⎞ + cos⎜ + φs ,1 − φs ,2⎟e−4R / σ ⎥ ⎝ ⎠ σ ⎦
CONCLUSIONS In this paper, the surface element integration method was used to assess the impact of the force/potential vs distance profiles for two important non-DLVO forces: Lewis acid−base and solvation forces. From an functional dependence point of view, these interactions are similar to the solvation force possessing an additional oscillating contribution relating to density fluctuations near interfaces. Analytical expressions for two different geometric cases were derived for each force, sphere-flat plate and sphere− sphere, and compared with their Derjaguin approximation equivalents. For the case of acid−base interactions, the deviation from the Derjaguin approximation was found to be significant over a wide range of physically plausible ratios of the decay length, λ, to the particle radius, R. The ratio of SEI to Derjaguin values is solely a function of this ratio, and the functions derived for sphere-flat plate and sphere−sphere form the lower and upper bounds for this ratio. The difference in functional dependence with respect to σ/R is small, meaning that either the sphere-flat plate or sphere−sphere function could be used for the other geometry without a significant error arising. This allows for using the results of this work to also characterize unequally sized-spheres, which no direct analytical solution existed for, by calculating an effective radius, determining the upper and lower bound for the correction factor and then averaging in some manner. The resulting expressions show that accounting in curvature for acid− base interactions is important even for simple nonrough geometric shapes, despite the rapid exponential decay. As the decay is exponential, at larger particle sizes relative to the decay length the deviation from the Derjaguin approximation indeed becomes negligible, as was expected. In the case of solvation force, the deviation was found to be much less pronounced. This was a result of the oscillatory component damping out changes in magnitude of the force/ potential, except once again at very large values of the ratio of decay length (σ in this case) to particle radius. As the solvation force has a smaller range of physically plausible decay lengths compared to the acid−base/hydration force, this means the impact of accounting for the exact geometry is significantly less pronounced. When compared with the likely deviations from surface roughness, it is less likely that the direct geometric effect for the simple geometries tested in this work would be significant.
⎜
4π 1 − 4π 2
⎛ 2π (l + 2R ) ⎞ −4 1 + 4π 2 + φs ,1⎟e−2R / σ cos⎜ ⎝ ⎠ σ /R σ ⎛ σ ⎞2 160π 4 − 48π 2 + 10 ⎟ ⎝R⎠ (1 + 4π 2)5/2 ⎛ 2π (l + 4R ) + cos⎜ + φs ,1 + ⎝ σ ⎜
φs ,3 = tan−1
⎡ ⎛ 2πl ⎞ + φs ,1 − φs ,3⎟ ⎢cos⎜ ⎝ ⎠ σ ⎣ ⎤ ⎞ φs ,3⎟e−4R / σ ⎥ ⎠ ⎦
4π 2 − 3 1 − 12π 2 (20)
FSEI =
g1′ = g2′ =
g3′ =
πσ 2R [(g + g2 + g3 + g4 )/σ − g1′ − g2′ − g3′ 1 + 4π 2 1 − g4′]e−l / σ
⎞ −2π ⎛ 2πl + 2φs ,1⎟ sin⎜ ⎝ ⎠ σ σ ⎡ ⎛ 2πl ⎞ + φs ,1 − φs ,2⎟ ⎢sin⎜ ⎠ R 1 + 4π 2 ⎣ ⎝ σ ⎛ 2π (l + 4R ) ⎞ − 4R / σ ⎤ + sin⎜ + φs ,1 − φs ,2⎟e ⎥ ⎝ ⎠ σ ⎦ −2π
⎞ 8πR 1 + 4π 2 ⎛ 2π (l + 2R ) + φs ,1⎟e−2R / σ sin⎜ 2 ⎝ ⎠ σ σ ⎞ 160π 4 − 48π 2 + 10 ⎡ ⎛ 2πl + φs ,1 − φs ,3⎟ ⎢sin⎜ 2 5/2 ⎠ ⎣ ⎝ σ (1 + 4π ) ⎛ 2π (l + 4R ) ⎞ − 4R / σ ⎤ + sin⎜ + φs ,1 + φs ,3⎟e ⎥ ⎝ ⎠ σ ⎦
g4′ = −
σ R2
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(21)
ASSOCIATED CONTENT
S Supporting Information *
Once again, the accounting for geometry results in a number of additional terms in both the force and potential expressions, but, similar to the case of a sphere and flat plate, these terms are negligible unless σ/R is large. The solvation force, normalized against πσR/(1 + 4π2)1/2, is shown for two different sized spheres (25 and 5 nm) for a decay length of 1 nm (Figure 5a,b, respectively), as was previously compared for sphere-flat plate. Once again, very similar results are yielded between the Derjaguin- and SEI-derived expressions. The normalized
Detailed derivations of the equations used in this work are given in the provided Supporting Information. This material is available free of charge via the Internet at http://pubs.acs.org/.
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. 4630
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Notes
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The authors declare no competing financial interest.
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