Elastic Modulus of a Polymer Nanodroplet: Theory ... - ACS Publications

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Elastic Modulus of a Polymer Nanodroplet: Theory and Experiment Apostolos E. A. S. Evangelopoulos,*,† Emmanouil Glynos,‡ Frédéric Madani-Grasset,§ and Vasileios Koutsos*,† †

Institute for Materials and Processes, School of Engineering, The University of Edinburgh, Edinburgh EH3 9JL, United Kingdom Department of Materials Science and Engineering, University of Michigan, H. H. Dow Building, 2300 Hayward Street, Ann Arbor, Michigan 48109-4136, United States § Nanolane, Parc des Sittelles, 72450 Montfort le Gesnois, France ‡

ABSTRACT: We redevelop a theoretical model that, in conjunction with atomic force microscopy (AFM), can be used as a noninvasive method for determination of the elastic modulus of a polymer nanodroplet residing on a flat, rigid substrate. The model is a continuum theory that combines surface and elasticity theories for prediction of the droplet’s elastic modulus, given experimental measurement of its adsorbed height. Utilization of AFM-measured heights for relevant droplets reported in the literature and from our own experiments illustrated the following: the significance of both surface and elasticity effects in determining a polymer droplet’s spreading behavior; the extent of a continuum theory’s validity as one approaches the nanoscale; and a droplet size effect on the elastic modulus.

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INTRODUCTION The wetting versus dewetting behavior of a polymeric substance, be it in the form of a film or individual droplet comprising a single-chain or multichain aggregate, has raised great interest due to the wealth of practical applications such as adhesion,1 surface lubrication and friction modification,2 colloidal stabilization,3 chromatography,4 surface nanopatterning,5 drug delivery,6 and biocompatibility of artificial organs,7 stimulating in turn fundamental research. The present work concerns the spreading behavior of a nanoscopic polymer droplet on a flat, rigid substrate and addresses the following fundamental research topics: importance of the elastic nature of the material on spreading, a method of theoretically predicting an elastic modulus for the droplet from experimental measurements of its state of adsorption, and effect of droplet size on the elastic modulus. Whereas adsorption and deformation of isolated polymer droplets have attracted considerable attention through experimental techniques,8−20 ranging from light scattering to scanning electron microscopy (SEM) and atomic force microscopy (AFM), theoretical and computational approaches15,20−22 have fallen behind, with the exception of molecular dynamics23−25 (MD) and Monte Carlo26−31 (MC) simulations, which offer treatments from a microscopic viewpoint and are usually limited to short time scales for equilibration and to length scales in the 10 nm range.32 Structures usually examined in experiments can be larger by 1 order of magnitude or more, calling for an alternative approach. The first was developed by Lau et al.,15 employing surface physics and elasticity theory to construct a model that describes the spreading of polymer droplets. Lau et al.15 also scanned profiles of latex droplets adsorbed on silica using AFM. Recently, Araujo et al.22 developed a simple computer model © 2012 American Chemical Society

according to which a polymer droplet is conceptualized as a spherical spring matrix, with each spring representing a polymer chain between two knots of reticulation. Two types of forces were introduced in their algorithm, governing the droplet’s wetting behavior: elastic forces between the knots of reticulation, and attraction forces between each knot and the substrate. Araujo et al.22 tested their model with success against the experimental data of Lau et al.15 Engqvist et al.20 performed similar experiments to those of Lau et al.,15 investigating statics as well as dynamics of spreading. They borrowed the theoretical model of Lau et al.15 not for the purposes of testing it against their own experiments but rather for illustrating its mathematical consistency, unlike Lau et al.,15 who did compare their theory against experiment. The comparison essentially comprised the following steps: (1) Given the droplet size and AFM-measured height, the droplet’s contact angle with the substrate was deduced by geometry. (2) The contact angle was substituted into their theoretical model to produce a value for the spreading parameter as a function of the elastic modulus of the droplet. (3) Given the spreading parameter, Young’s law and the Dupré adhesion equation33 were combined to deduce the adhesion energy. (4) The adhesion energy was used in conjunction with the Johnson−Kendall−Roberts (JKR) theory34 to predict the droplet-substrate contact radius. (5) Ultimately, this prediction was compared to the AFM-measured contact radius, and the theoretical model was thus assessed. We have the following reservations regarding this method: (1) For the stage of recovering the spreading parameter, the droplet’s Young’s modulus was required. That was obtained from Received: June 12, 2011 Revised: January 20, 2012 Published: January 25, 2012 4754

dx.doi.org/10.1021/la2049037 | Langmuir 2012, 28, 4754−4767

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the consensus is currently led toward the view that line tension is more complicated than originally thought,52 and its measurement is extremely sensitive to details of the model employed52 and experimental procedure followed,47 even for a simple liquid. Few authors have yet attempted to study the three-phase contact line in polymeric systems,55,56 and they do not deal with pure polymers but chains in solution or colloidal suspension. We report the problems that arose with incorporating line tension into our model in the relevant discussion sections and say that, while we are not dismissive of the possibility of line tension effects altogether, treating them goes beyond the scope of this article. It is quite telling, having mentioned the above, that we get reasonable results from our theory down to ∼10 nm while ignoring line tension. Scale invariance does not absolutely hold regarding (intrinsic) properties of materials. Specifically, the elastic modulus, which concerns the present study, has been shown, not just for polymers, to become size-dependent in the vicinity of the nanoscopic scale, through both experiment57−66 and theory,67−74 supporting either stiffening58−61 or softening62−66 with decrease in size. As a result, nanoscale-sized materials do not exhibit the behavior anticipated through an analysis of their macroscopic counterparts. Polymers are a typical example.75,76 Whereas most investigations on polymers have focused on glass transition temperature76,77 and structural relaxation,78,79 their elastic propertiesespecially of compliant polymershave received less attention,80 owing to experimental challenges, such as the measuring instrument’s poor sensitivity,81,82 imaging resolution,83 and its interaction with the sample.84 Polymer molecules have an associated characteristic size, the root-mean-square (rms) end-to-end distance, which scales as a rational power of the polymerization index. This inherent length scale introduces the possibility of chain confinement effects as the system size becomes smaller than the unperturbed molecular size.75 Confinement effects, deformation-induced structure rearrangement effects,85 and substrate82,86 interactions are key mechanical interactions that cause the elastic modulus to deviate from that of the corresponding macroscopic system. Our model provides a step toward estimating the elastic modulus via a noninvasive experimental procedure, therefore avoiding a plethora of errors associated with mechanical testing and shedding light on the relationship between the material’s response and intrinsic effects. The benefits of nonmechanical testing are acknowledged by Torres et al.80 Examination of the elastic modulus of a purely polymeric substance of spherical geometry has not been previously undertaken, except by Ishikawa et al.,87 who used the direct peeling method with atomic force microscope tip (DPAT) to separate spherical polymer aggregates from a resist pattern and deduced the elastic modulus from the separation force, potentially involving the aforementioned drawbacks of mechanical testing, however. Thin films have received the most attention, and considerable work has also been done on small molecules in nanoporous structures.75,76 Glynos et al.88 measured the Young’s modulus of the shell of polymeric microspheres used as ultrasound contrast agents in medicine, revealing a decreasing Young’s modulus with increasing size, converging to the expected value for the macroscopic material. Very few authors have investigated individual nanofibers: Tan et al.18 examined nanostructural and elastic properties but of a size too large for size effects to be observed; Shin et al.60 examined the size dependency of elasticity and found that the elastic modulus increased exponentially with decreasing fiber diameter below 70

mechanical testing of macroscopic polymer films made up of such close-packed droplets; however, it is not at all obvious (also noted by Lau et al.15 themselves) that the modulus of a film corresponds to that of a droplet. In assemblies of smallscale-sized objects or structures that introduce a large amount of interface (such as the aforementioned array of close-packed droplets comprising the polymer film), the surface-to-volume ratio becomes significant, causing the global properties to deviate from those of the constituents. Dingreville et al.35 and references therein offer material in support of this argument. (2) For the stage of deducing the adhesion energy, Lau et al.15 employed Young’s law and the Dupré equation, both of which ignore any effects of elasticity altogether. In a sense, this step defeats the original purpose of the model by treating the adhesion energy as independent of elastic effects (we do, however, recognize that the value attributed to the spreading parameter, from which the adhesion energy followed, incorporated both surface and elastic effects). (3) Our final reservation concerns the use of AFM-measured contact radii. Measurements in the lateral direction potentially bear considerable uncertainty due to AFM tip convolution effect; therefore, it would have been preferable to deduce the contact radii from geometry, given the droplet sizes and measured heights. The above motivated us to seek a way of utilizing such a theory more soundly. We redeveloped in detail the model introduced by Lau et al.15 to produce a free energy equation that differs somewhat from theirs due to minor algebraic discrepancies in calculations (resulting in different coefficients in the equation of the elastic contribution to the system’s free energy). For the first time in the literature, we employed such a theory to predict an effective elastic modulus for adsorbed droplets of a range of nanoscopic sizes (5−100 nm) by utilizing the experimental data of Lau et al.,15 Engqvist et al.,20 and our own. Through a continuum theory approach, as opposed to a microscopic theory, we do not confine ourselves to a particular type of polymeric material, as is, for example, the case with the work of Araujo et al.,22 where their computer code specifically models the cross-linked elastomeric structure. We offer an extensive discussion of our findings. These findings carry implications for the appropriateness of such a (continuum) theory for meso- and nanoscale systems and demonstrate its strengths and limitations as we approach nanodroplets of single chains. When it comes to nanoscale droplet spreading, line tension has been one the of the most popular correction terms to the classical Young’s law. A proper thermodynamic account involves not only bulk and surface contributions to the free energy of the system but also a contribution related to the three-phase contact region; and the smaller the length scale concerned, the higher the relevant contribution of the latter to the total free energy. One of the first documented references of line tension came with a footnote by Gibbs in one of his papers,36 where he introduced it as a lower-dimension analogue of surface tension, with the distinction that not only positive but also negative values are observable. Line tension has since attracted much theoretical,37−41 computational,38,46−49 and experimental42−45 attention, creating at the same time a lot of controversy, not just over its sign50,51 but also over its definition and what has been at one time or another perceived by the concept,52,53 including arguments, in the extreme case, for its redundancy,54 provided other effects and trends that influence spreading are properly accounted for. It seems that 4755


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nm; Burman et al.89 set the upper limit at 500 nm for observable size dependency but via a different mechanical testing method. The rest of this article is structured as follows: We begin with a detailed exposition of the theoretical model and arrive at a free-energy equation that describes the equilibrium state of adsorption. Two sections subsequently follow. The first, Polymer Droplets of ∼100 nm, sees the application of our theory on experimental work by Lau et al.15 and Engqvist et al.,20 who dealt with 50−100 nm droplets. The second, Polymer Droplets of 5−50 nm, repeats a similar process with our own experiments on 5−50 nm droplets. While discussion of the respective results is offered in both sections, a general Conclusions section closes the article, bearing on the validity and limitations of the theoretical model, its significance (in conjunction with the AFM technique) as part of a noninvasive method for elastic modulus estimation, and a size effect that this method has revealed.

Incompressibility means Vcap = V0, where V0 is the volume of the same droplet in the undeformed state, and leads to a few useful relationships [for which it is required that θ ∈ (0, π) so that infinities are avoided]: R3 =

R=

3h2

(3)

4R 03 − 2h3 4R 03 + h3

(4)

if we apply eq 3 to it. It is important to realize that eqs 2 and 3 are basically geometric relationships, bearing sphericality and incompressibility as the only physical assumptions. They relate a sphere and spherical cap of equal volumes. Incompressibility, viewed as a constraint in the context of the above geometrical construction, reduces the degrees of freedom associated with the adsorbed state by one: given an initial R0, a compressible droplet’s adsorbed shape and size can be fully specified given R and h (or θ, equivalently), which can be seen mathematically by eq 1, whereas an incompressible droplet’s adsorbed shape and size can be fully specified given only h (or θ, equivalently), seen by eq 4, which then becomes a measure of the deformation suffered upon adsorption. In the present paper, h is directly measured experimentally by AFM. Physical Considerations. For our system, the polymer droplet residing on the solid and surrounded by air, we write the condition for equilibrium in terms of the Gibbs free energy differential: dG = dUE + dUI = 0

(5)

dUE represents contributions from elastic deformation and dUI from interfaces, with fluctuations about the equilibrium state. For a simple liquid droplet, dUE = 0, and eq 5 becomes the classical Young’s law. Clearly, a polymer droplet behaves differently, for any elastic deformation induced by interfacial effects results in stress development across the bulk in opposition to that deformation, due to the material’s elastic nature. In what follows we construct expressions for UE and UI in terms of parameters that physically characterize the system and its interactions with the environment. To construct UE, we first consider the normal to the surface (z) component of the adsorption-induced deformation field of the droplet, uz(ρ), as a function of the radial distance, ρ, from the center of the (circular) area of contact between the droplet and the substrate:

droplet (not to scale) is superimposed on the deformed droplet, adsorbed on the substrate. R0 is the radius of curvature in the undeformed state (e.g., when suspended in the same medium), R is the radius of curvature in the adsorbed state (cap radius), ρ is the horizontal radius of the cap from the spherical symmetry axis, h is the height at the apex, and θ is the contact angle between the solid−polymer and polymer−air interface. h and θ can be used equivalently as a measure of deformation upon adsorption, during which the droplet’s volume remains constant by assumption of the polymer’s incompressibility.90 Also, we take the effect of gravity to be negligible compared to surface and elasticity effects in the given size range (≲100 nm); in other words, the adsorbed cap retains a spherical shape. The condition for this to be true is that h be less than the capillary length,91 which is fulfilled in our case. The volume, Vcap, of the adsorbed cap may be calculated by elementary calculus: πρ2(h) dh =

4R 03 + h3

cos θ =

Figure 1. Undeformed spherical polymer droplet (blue; not to scale) superimposed on deformed spherical polymer cap (black) adsorbed on substrate. R0 is the radius of curvature in undeformed state, R is the radius of curvature in adsorbed state, ρ is the horizontal radius of the cap, h is the height at apex, and θ is the contact angle between solid− polymer and polymer−air interfaces.

0

(2)

giving R as R(R0, h), from the geometric fact of cos θ = (R − h)/R, which, itself, may be re-expressed as

THEORETICAL MODEL Geometrical Considerations. The geometry of the system is illustrated in Figure 1, where the undeformed spherical

∫h

cos3 θ − 3 cos θ + 2

giving R as R(R0, θ) or

■

Vcap =

4R 03

⎡ uz(ρ) = δ − R 0 ⎢⎣1 −

⎛ ρ2 ⎞ ⎤ ⎟⎟ 1 − (ρ/R 0)2 ⎥⎦ ≃ δ⎜⎜1 − 2δR 0 ⎠ ⎝ (6)

δ  2R0 − h is the magnitude of deformation at the center. Equation 6 follows by geometry: multiplying out the middle section makes clear that this is the equation of a concave-down semicircle of radius R0 displaced parallel to the z axis by R0 − h

πR3 [cos3 θ − 3 cos θ + 2] 3 (1) 4756


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working in polar coordinates, and using eq 6 and eq 8 (or eq 9) with coefficients from eq 10, eventually leads to

units. This makes sense because the deformation is basically the x−y-plane reflection of the concave-up section of the undeformed droplet that lies below the plane when the droplet is displaced by h units in the negative z direction. The righthand side of eq 6 is a first-order binomial approximation. Equation 3 allows us to express the contact radius, ρ = a, as a function of the undeformed radius, R0, and the height of the cap, h: 1/2 ⎛ 8R 3 2⎞ h 0 ⎟ a = R sin θ = ⎜⎜ − 3 ⎟⎠ ⎝ 3h

⎡ δ UE = KP⎢δ2a − 2 ⎢⎣

⇒UE ⎧ ⎪ ⎪ 1/2 ⎛ 8R 3 2⎞ EPG ⎪ h 0 2 ⎟ ⎨(2R 0 − h) ⎜⎜ = − 3 ⎟⎠ (1 − ν 2) ⎪ ⎝ 3h ⎪ ⎪ ⎪ ⎩

(7)

(8)

where ρ < a, on the surface of a semi-infinite half-space elastic medium, creates the displacement field uz(ρ) =

⎤ 1 πa ⎡ σ0 + σ1(1 − ρ2 /a2)⎥ ⎦ KP ⎢⎣ 2

− (9)

KP ⎛ δ a ⎞ ⎜ − ⎟ π ⎝a 2R 0 ⎠

σ1 =

KP ⎛ a ⎞ ⎜ ⎟ π ⎝ R0 ⎠

UI = γPA + (γSP − γS)O

1 kuz 2(x) = 2 D

∫

(10)

UI = −

(14)

⎛ h ⎞2 4π 2 ⎛ 2R 0 ⎞ 4π 2 ⎟ + R 0 S⎜ R 0 (3γP + S)⎜ ⎟ ⎝ h ⎠ 3 3 ⎝ 2R 0 ⎠ (15)

where S = γS − (γSP + γP)

∫D σz(x)uz(x)

(13)

A is the polymer−gas interfacial area with associated interfacial tension γP, and O is the polymer−substrate interfacial area with associated interfacial tension γSP. Elementary calculus gives A = 2πR2(1 − cos θ) and O = π(R sin θ)2, which can be expressed in terms of the undeformed radius, R0, and the cap height, h, by use of eq 3 and geometry (Figure 1), as A = [(8πR03/3h) + (2πh2/3)] and O = [(8πR03/3h) − (πh2/3)]. Substituting A and O back into eq 14 yields

∫D ∫strain σz(x)

1 = 2

R0

Equation 13 expresses the contribution to free energy due to the droplet experiencing deformation quantified by h, given R0. To construct UI, we first note that the excess free energy contribution due to all interfaces can be expressed in an additive manner:

From a generic definition of work, we form an expression for the elastic energy of deformation due to a stress field σz(x′) in the z-direction inducing a deformation field uz(x) over an area D. x represents any point in the z plane, and k is a proportionality constant (in keeping with linear elasticity): UE =

(2R 0 − h) 2

3 ⎡⎛ 1/2 ⎤ 3 2 ⎢⎜ 8R 0 − h ⎞⎟ ⎥ ⎢⎝ 3h 3⎠ ⎥ ⎦ ⎣

5 ⎡⎛ 1/2 ⎤ ⎫ 3 2⎞ 8 R h ⎢⎜ 0 − ⎟ ⎥ ⎪ ⎪ ⎢⎝ 3h 3⎠ ⎥ ⎪ ⎦ 1 ⎣ ⎬ + 10 ⎪ R 02 ⎪ ⎪ ⎪ ⎭

which is of the form of eq 6, where KP  ED/(1 − ν2) is a measure of stiffness, ED is the modulus of elasticity, and ν is Poisson’s ratio. An advantage of considering a theoretical stress field comprising terms of the form σ(ρ) = σi(1 − ρ2/a2)n is that it allows us to calculate analytic solutions for uz(ρ). The particular values of n = −1/2 and n = 1/2 correspond to uniform normal displacement of the loaded circle and Hertz pressure, respectively. The former is the pressure that would arise on the face of a flat-ended, frictionless cylindrical punch pressed squarely against an elastic half-space, while the latter is the pressure exerted between two frictionless elastic solids of revolution.92 Comparing the geometrical argument for the displacement field, eq 6, with the elasticity theory result, eq 9, in terms of powers of ρ allows us to determine values for the coefficients σ0 and σ1: σ0 =

(12)

Ultimately, we substitute eq 7, KP, and δ into the above to get UE in terms of R0, h, ED, and ν:

From contact mechanics,92 application of an external stress of the form σ(ρ) = σ0(1 − ρ2 /a2)−1/2 + σ1(1 − ρ2 /a2)1/2

a3 1 a5 ⎤⎥ + R0 10 R 02 ⎥⎦

(16)

is the spreading parameter or spreading coefficient,93 quantifying the energy difference between a completely dry and completely wet substrate (whereupon the adsorbed substance has completely spread) or, equivalently, the difference between the work of adhesion of the fluid on the substrate and the cohesion energy of the fluid. Equation 15 expresses the excess contribution to free energy due to the existence of

(11)

The above can be used as general formulas for the elastic energy required to attain uz(x) given k or σz(x). With D being the circular contact area between droplet and substrate, 4757


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adjustment of the glass transition temperature, Tg, of the latex, while the degree of cross-linking, quantified through the gel fraction GF, allowed for adjustment of the elastic modulus of the core, though the relation between GF and elastic modulus is not quantitative because the cross-linking reaction is accompanied by chain scission. Table 1 contains data provided by Lau et al.15 that we use. EM is the elastic modulus of a polymer film of macroscopic

interfaces. UI depends on the chemical nature of the interfaces involved, quantified through γP and S, and also, being an extensive quantity, on the interface sizes, quantified through R0 and h. Minimization. As the system approaches equilibrium, fluctuations of the droplet’s height, h, progressively decrease about an equilibrium value which minimizes free energy, expressed in terms of a set of parameters. We, therefore, minimize eq 5 with respect to h, after inserting eq 13 and eq 15, to recover an expression relating the equilibrium value of h to parameters R0, γP, S, ED, ν:

Table 1. Latexes in the Form SB-(Tg)-GF with Properties as Reported by Lau et al.15 a

⎡ 3 ⎢ (2R − h)2 ⎛⎜ − 2h − 8R 0 ⎞⎟ 0 3h2 ⎠ ⎝ 3 ED ⎢ dΦ ⎢ = 1/2 dh (1 − ν 2) ⎢ ⎛ h2 8R 03 ⎞ ⎢ 2⎜ − + ⎟ 3h ⎠ ⎢⎣ ⎝ 3 1/2 ⎛ h2 8R 03 ⎞ ⎟ − 2(2R 0 − h)⎜⎜ − + 3h ⎟⎠ ⎝ 3

−

+

GF (%)

Tg (°C)

h (nm)

R0 (nm)

γP (mJ·m−2)

EM (MPa)

SB-(−2)-75 SB-(11)-75 SB-(28)-75 SB-(−2)-92 SB-(−8)-43

75 75 75 92 43

−2 11 28 −2 −8

26 39 49.5 46 15.8

87.5 83.5 83.5 74.0 82.0

53 46 48 54 48

0.63 1.35 1.92 0.6 2.02

a

GF, gel fraction; Tg, glass transition temperature; h, maximum height of adsorbed droplet; R0, radius of droplet in undeformed state; γP, surface tension of polymer in air; EM, elastic modulus of macroscopic film.

1/2 ⎛ 2h 8R 03 ⎞⎛ h2 8R 03 ⎞ 3(2R 0 − h)⎜ − − ⎟ ⎟⎜ − + 3h ⎠ 3h2 ⎠⎝ 3 ⎝ 3

dimensions, formed by slow evaporation of the water of the latex suspension and exposing an array of close-packed droplets adhering together by their shells. Results. In order for our model, eq 17, to be employed for prediction of the elastic modulus, ED, of the polymer droplets, we also required values for the spreading parameter S and Poisson’s ratio ν. For the Poisson ratio, a good approximation is ν = 0.5 for rubbery polymers.94 As far as the spreading parameter is concerned, it can be seen from the definition in eq 16 that we were required to individually predict surface tension of the silicon substrate, γS, and the substrate−polymer interfacial tension, γSP. Classical liquid-drop-on-surface and immersion calorimetry experiments95−98 lead to an estimate of γS = 254 mJ·m−2 for the surface tension of a freshly prepared, clean silicon surface with a native oxide coating, always forming on bare silicon exposed to oxygen-containing environment. This value is also compatible with that of Portigliatti et al.,13 with reference to the same experimental system. To estimate the substrate−polymer interfacial tension, γSP, we require a mechanism that allows prediction of interfacial tension based on the respective surface tensions of the surfaces comprising the interface. There exist three basic semiempirical candidate models: the model of Girifalco and Good,99 the oldest and dating back to 1957; the model of Fowkes,100,101 which came only a few years later; and the model of van Oss, Chaudhury, and Good102 (vOCG), a more modern take of the late 1980s. All three are based on the assumption that surface tension is a direct measure of intermolecular interactions in a given material, while the last two further assume that surface tension consists of independent additive components, each corresponding to a different type of intermolecular interaction, for which reason they are known as surface tension component (STC) models. (For completeness, we ought to mention the most recent improvement of the vOCG model102 by Lee103 in 2001, whereby the basicity catastrophe is cured and the adsorbed vapor film pressure is taken into account.) Throughout this article we shall in turn use all three, depending on the available data for each set of experiments under

4R 0 3/2 ⎛ h2 8R 03 ⎞ ⎟ ⎜− + 3h ⎠ ⎝ 3

2R 0

3/2 ⎤ ⎛ 2h 8R 03 ⎞⎛ h2 8R 03 ⎞ ⎥ ⎟ ⎟⎜ − + ⎜− − ⎥ 8πR 3S 3h ⎠ 3h2 ⎠⎝ 3 ⎝ 3 0 ⎥+ + ⎥ 4R 02 3h2 ⎥ ⎥⎦ 2 + h π(S + 3γP) 3

=0

latex

(17)

As mentioned earlier, when some geometrical facts were established, given R0, h uniquely specifies the geometry of an adsorbed state. In the physical context, R0 may depend on the length and number of chains in a droplet, its density, and the solvent conditions, while h depends (according to the present framework) on the set of physical parameters γP, S, ED, and ν. For experimental situations where h was measured reliably (AFM-measured) and γP, S, and ν were known or estimated, we solved eq 17 for ED. We illustrate our findings in the following sections.

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POLYMER DROPLETS OF ∼100 nm Experiments of Lau et al. Lau et al.15 performed an experimental investigation of the spreading of ∼100 nm-sized isolated latex droplets on a silicon substrate at room temperature. By use of an AFM in contact mode, the equilibrium height of the adsorbed latex droplets was measured. The latexes used were formed by a soft core of partially crosslinked styrene−butadiene copolymer molecules, surrounded by a stiffer shell made of carboxylic co-monomers. The ratio of styrene to butadiene co-monomers in the core allowed for 4758


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examination (Lau et al.,15 Engqvist et al.,20 and our own). For the present case we employed the model of Girifalco and Good.99 It essentially sums up the independent work of many authors104−108 in the following statement: γ12 = γ1 + γ2 − 2Φ(γ1γ2)1/2

(18)

where Φ (precisely defined therein) is a ratio involving the energies of adhesion and cohesion for the two phases involved, equal to a constant and characteristic of the system. In simpler terms, Φ translates into a correction factor deviating from unity to account for mismatch in intermolecular interaction type between the two media. Carboxyl groups of the droplets’ shells carry both a basic (H-bond-accepting) and an acidic (H-bonddonating) group, while styrene−butadiene of the soft core is somewhat basic due to the π-bonds of the aromatic ring double bonds and of the carbon−carbon double bond of the alkene group. Its cohesion is then brought about by dispersive interactions as well as acid/base ones, which explains its fairly high surface tension compared to most polymers. External carboxyl groups will tend to bury themselves into the core so as to minimize the surface energy, but probably not all groups will have a chance to do that. The naturally oxidized silicon substrate carries SiO, Si2O3, and other oxides and is surrounded by a great many silanol (OH−) and siloxane (R2SiO−) groups that make it acidic and basic in the Lewis sense. As a result, the latex droplets will adhere well through H-bonds and dispersive and polar van der Waals forces, utilizing all of their intermolecular force mechanisms, and the same will be true for silica. This means that we can approximately set Φ = 1 in eq 18 and express that as γSP = γS + γP − 2(γSγP)1/2. Rewriting the spreading parameter, eq 16, in terms of the above gives S = −2γP + 2(γSγP)1/2, which was evaluated upon numerical substitution of γS and γP for each droplet type (Table 1). Having established all required parameters, we substituted h, R0, γP, S, and ν into eq 17 and solved with respect to ED for a prediction of the adsorbed droplet’s elastic modulus. Table 2 summarizes all calculated data.

Figure 2. Comparison between geometrical predictions, a geo (triangles), and AFM-measured contact radii of Lau et al.,15 a (squares), in support of spherical geometry assumption for the adsorbed latex droplets.

of that assumption. If sphericality is viewed as a necessary, but not sufficient, condition for equilibrium, the results of Figure 2 suggest that equilibrium is likely. From the EM and ED columns of Tables 1 and 2, respectively, Figure 3 can be generated. In comparing the two data sets, we

Table 2. Calculated Data from Our Theory Using Table 1 and Literature-Based Dataa

a

latex

S (mJ·m−2)

ED (MPa)

SB-(−2)-75 SB-(11)-75 SB-(28)-75 SB-(−2)-92 SB-(−8)-43

124.2 122.5 123.1 124.4 123.1

1.04 3.07 5.10 6.37 0.44

Figure 3. Comparison between EM (squares) and ED (triangles).

find that the elastic moduli of the macroscopic strips generally lie, in value, below those of the nanoscopic droplets. The structure of the single droplet is different from that of the film, which is made up of an array of close-packed droplets adhering together by their shells, and one should not anticipate a direct comparison between EM and ED. Both EM and ED lie within the expected range for rubbery materials in general, 10−10 000 kPa, to account for a range of chemical compositions, material formation processes, ambient conditions, etc. As far as ED is concerned, this is encouraging because it means our model returns reasonable estimates. Kan and Blackson109 have specifically measured elastic moduli for a series of carboxylated styrene−butadiene latex films and found it to be monotonically decreasing from 1 GPa to 0.1 MPa with a temperature increase from −10 to 150 °C, a range covering temperatures both below and above the glass transition temperature of the latex. The trend exhibited by EM as measured by Lau et al.15 is only generally supported by the gel fraction (GF) and glass transition temperature (Tg) data for the latexes, while the trend exhibited

S, spreading parameter; ED, elastic modulus of nanoscopic droplet.

Discussion. As a preliminary step in testing the correctness of our methods, we employed eq 7 to geometrically deduce the contact radii of the adsorbed droplets, given their corresponding undeformed radii R0 and heights h, as reported in Table 1. Our geometrical predictions, ageo, are compared against the AFM-measured contact radii, a, of Lau et al.15 in Figure 2. While we attribute the systematically negative (except one case, out of seven) differences in absolute value mainly to experimental uncertainty due to the AFM tip geometry requiring deconvolution from measurements,15 we feel that the clear qualitative agreement supports our assumption of the spherical geometry of the adsorbed cap-shaped droplet, because the function ageo = ageo(R0, h) (eq 7) is constructed on the basis 4759


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high predicted values, on the order of 10−8 J·m−1, 3−5 orders of magnitude larger than expected for similar systems.21,56 Experiments of Engqvist et al. Engqvist et al.20 investigated the effects of temperature, time, and preparation method on the adsorption of isolated styrene−acrylic latex droplets on a silicon substrate. Three types of styrene−acrylic latexes were used: L−20, L20, and L60, where the subscripts denote glass transition temperatures of −20, 20, and 60 °C, respectively. Following evaporation of the solvent, these latexes received different treatments during adsorption. One sample of L−20 was left for 1 week at room temperature (RT), another was additionally treated by ultrafiltration (UF), while a third underwent ultrafiltration and then was left for 7 weeks at room temperature. One sample of L20 was given 48 h to adsorb at room temperature, while a second spent a night at 105 °C. L60 was left at 70 °C for 60 h. AFM imaging was performed at 23 °C and 50% relative humidity (RH). Samples, treatments, and all data provided by Engqvist et al.20 that we use are summarized in Table 3. The elastic

by ED as calculated with our model is in complete agreement with GF and Tg. GF corresponds to the degree of cross-linking in the core and is defined as the ratio of insoluble species remaining after swelling of the droplets in a good solvent of styrene−butadiene copolymers.15 The larger the GF, the higher the degree of cross-linking and, hence, the stiffness of the material, however, not in a known quantitative fashion because the cross-linking reaction is accompanied by chain scission.15 A higher Tg also increases stiffness. SB-(−2)-75, SB-(11)-75, and SB-(28)-75 all share the same GF, and therefore we can say that the increasing Tg is in accord with both EM and ED increasing. A comparison between SB-(−2)-75 and SB-(−2)-92 should require the latter to exhibit a higher elastic modulus. This is not seen with the measured EM of Lau et al.15 but agrees with our prediction of ED. As far as SB-(−8)-43 is concerned, having the both the lowest Tg and GF, it is expected to be the least stiff. Indeed, this is observed with our smallest calculated modulus of ED = 0.44 MPa, but this does not agree with Lau et al.15 EM = 2.02 MPa. These inconsistencies regarding SB-(−2)92 and SB-(−8)-43 we attribute to the fact that EM refers to the corresponding film, which, by construction, contains an interface between the close-packed constituent droplets, while our ED, calculated directly from AFM data, is completely compatible with the materials’ properties. In Figure 4, we offer ED against Tg, where we have divided points into three classes on the basis of GF. In the presence of

Table 3. Latexes in the Form LTg and Treatment (Adsorption) Conditions with Properties as Reported by Engqvist et al.20 a latex L−20 RT, 1 week L−20 UF, RT, 1 week L−20 UF, RT, 7 weeks L20 RT, 12 h L20 RT, 48 h L20 105 °C, overnight L60 70 °C, 60 h

Tg (°C)

h (nm)

R0 (nm)

γP (mJ·m−2)

ν

−20 −20

39 38

98 98

40 40

0.5 0.5

0.2 0.2

−20

36

98

40

0.5

0.2

20 20 20

97b 86b 51

89 89 89

40 40 40

0.5 0.5 0.5

12 12 12

60

59

107

40

0.5

1243

EM (MPa) (RT)

a

RT, room temperature; Tg, glass transition temperature; h, maximum height of adsorbed droplet; R0, radius of droplet in the nonadsorbed state; γP, surface tension of polymer in air; ν, Poisson’s ratio of polymer material; EM, elastic modulus of macroscopic film. bFrom contact angles reported in Figure 2 of Engqvist et al.,20 converted to heights through our eq 4.

modulus, EM, of a polymer film of macroscopic dimensions, formed by slowly evaporating the water of the latex suspension and exposing an array of close-packed droplets adhering together by their shells, was measured exclusively at room temperature (as opposed to the droplet spreading experiments, performed at various temperatures). We note that the R0 values have come from dynamic light scattering measurements of the droplets in diluted dispersions, where a reported 5% swelling compared to the dry state occurs, resulting in R0 carrying a small positive error. Results. In order for our model, eq 17, to be employed for prediction of the modulus ED, we also required a value for the spreading parameter S. We employed the STC model of vOCG,102,110 which the available data in this case deemed as appropriate. This model combines London dispersion, Keesom (dipole−dipole), and Debye (dipole−induced dipole) forces into a single term, LW (Lifshitz−van der Waals), resulting in the surface tension component γLW. Remaining surface tension contributions are attributed to acid−base (AB) interactions, defined through γ = γLW + γAB and comprising two complementary kinds of behavior, from acidic (γ+) and basic

Figure 4. Predicted moduli for the adsorbed polymer droplets plotted against glass transition temperature for three structural classes: GF = 43% (square), GF = 75% (circles), and GF = 92% (triangle).

more data, we would expect GF = 43% points to generally lie in an area below GF = 75% points, and GF = 75% points to lie below GF = 92% points. This is generally observed with the few data at our disposal. Also, we would expect a monotonic increase of ED with increasing Tg within the same GF class, which is observed with the few data of GF = 75%. All in all, data support that the equilibrium shape of a polymer droplet does not depend only on surface forces, as for the simple liquid case, but is greatly affected by the elastic modulus of the material. Data pertaining to the polymeric droplets’ structure and material properties are in good numerical agreement with our model’s predictions. As a final remark following from our introductory note concerning line tension, incorporating that as a one-dimensional analogue of surface tension in our model36 while keeping the elastic modulus constant at its bulk value resulted in unreasonably 4760


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(γ−) groups, of which γAB is the geometric mean: γAB = 2(γ+γ−)1/2. Given that an interface composed of two adjacent interfacial regions will have an associated interfacial tension equal to the sum of the surface tensions of those regions as part of the interface,101 γ12 = γ1 + γ2 − 2 ⎡⎣(γ1LW γ2 LW )1/2 + (γ1+γ2−)1/2 + (γ1−γ2+)1/2 ⎤⎦ (19)

Regarding the silica substrate, we referred to the same literature95−98 as for treating the data of Lau et al.15 to arrive at the following values of surface tension components and total surface tension: γSLW = 39 mJ·m−2, γS+ = 172 mJ·m−2, γS− = 78 mJ·m−2, and γS = 254 mJ·m−2. Styrene−acrylic latex of the droplets, on the other hand, suggests almost exclusively basic behavior: the aromatic ring and double bond of styrene are slightly basic, whereas the carbonyl groups of acrylate are strongly basic. In this sense, the latex should resemble a mix of poly(methyl methacrylate) (PMMA), polystyrene (PS), or poly(tert-butylacrylate) (PtBA), and its γ− should be significant, while γ+ should be next to zero.111,112 Given the total surface tension20 and an average γ−/ γ ratio of like polymers,111 we have γPLW = 40 mJ·m−2, γP+ ≃ 0 mJ·m−2, γP− = 9 mJ·m−2, and γP = 40 mJ·m−2. Rewriting the spreading parameter, eq 16, in terms of the vOCG model, eq 19, gives S = −2γP + 2 [(γSLWγPLW)1/2 + (γS+γP−)1/2 + (γS−γP+)1/2], which can be evaluated after numerical substitution of the aforementioned tensions. Having established all required parameters, we substituted h, R0, γP, S, and ν into eq 17 and solved with respect to ED, for a prediction of the adsorbed droplet’s elastic modulus. Table 4 summarizes all calculated data.

Figure 5. Comparison between geometrical predictions, a geo (triangles), and AFM-measured contact radii, a (squares), of Engqvist et al.20 in support of spherical geometry assumption for the adsorbed latex droplets.

Table 5. Collected Data from Engqvist et al.20 and Our Calculationsa

S (mJ·m−2)

ED (MPa)

L−20 RT, 1 week L−20 UF, RT, 1 week L−20 UF, RT, 7 weeks L20 RT, 12 h L20 RT, 48 h L20 105 °C, overnight L60 70 °C, 60 h

78 78 78 78 78 78 78

1.15 1.08 0.95 6.62 5.72 2.89 2.22

h (nm)

R0 (nm)

EM (MPa) (RT)

ED (MPa)

39 38 36 97 86 51 59

98 98 98 89 89 89 107

0.2 0.2 0.2 12 12 12 1234

1.15 1.08 0.95 6.62 5.72 2.89 2.22

a

h, maximum height of adsorbed droplet; R0, radius of droplet in the nonadsorbed state; EM, elastic modulus of macroscopic film at RT; ED,. elastic modulus of nanoscopic droplet at treatment temperature. Note the distinction that EM refers to RT, whereas ED refers to the treatment temperature and time scale.

Table 4. Calculated Data from Our Theory Using Table 3 and Literature-Based Dataa latex

latex L−20 RT, 1 week L−20 UF, RT, 1 week L−20 UF, RT, 7 weeks L20 RT, 12 h L20 RT, 48 h L20 105 °C, overnight L60 70 °C 60 h

resulting in R0 carrying a small positive error. This means that for a given measured h the calculated ED carries a small negative error, as the calculations assume larger strain under given surface forces. We therefore expect the droplets to be slightly stiffer than what we report. Specifically, for the present range of sizes, +5% systematic error in R0 will result to about −5% to −10% systematic error in our calculated ED. Bearing that in mind, all calculated moduli fall within the expected range for rubbery materials in general, 10−10 000 kPa. At this point it ought to be stressed that polymeric systems can never be purely elastic or purely viscous, and their modulus will in fact be associated with a time scale.113 Some observations discussed in the following paragraph relate to that fact. Also to be noted is that whereas the actual AFM measurements were performed at RT, the adsorbed conformations and all AFM data for that matter correspond to the (higher) treatment temperatures, at which the conformations “froze” upon return to RT in a time frame much shorter than the material’s relaxation time. In the L−20 series of 1 week, the ultrafiltration process leads to lower moduli, that is, samples filtered from surfactants will adsorb more spontaneously, seen from a slightly reduced ED. Comparison of the heights of the two ultrafiltration L−20 samples indicates the existence of viscous effects, though weak, even for as long as 7 weeks after initial deposition of the sample, bringing about a very slow change of conformation toward a state of increased wetting and portraying the

a

S, spreading parameter; ED, elastic modulus of nanoscopic droplet at treatment temperature (refers to treatment temperature and time scale).

Discussion. In the same spirit as with the analysis of experiments of Lau et al.,15 we employed eq 7 to geometrically deduce the contact radii of the adsorbed droplets, given their corresponding undeformed radii R0 and heights h, reported in Table 3. Our geometrical predictions, ageo, are compared against the AFM-measured contact radii, a, of Engqvist et al.20 in Figure 5 (we omit the L20 RT 12 h sample, for which no measured contact radius is provided). As was the case with the experimental data of Lau et al.,15 the clear qualitative agreement supports our assumption of spherical geometry of the adsorbed cap-shaped droplets. We collect R0, h, EM, and ED in Table 5 for convenience of comparison. We recall our previous mention that a reported 5% swelling compared to the dry state occurs in dispersions, 4761


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Figure 6. Representative topography (left) and phase (right) tapping-mode AFM images of linear PB droplets on mica, which was freshly cleaved and immediately exposed to the PB solution. Top, Mw = 78.8 kg/mol; bottom, Mw = 962 kg/mol.

difference in ED is also manifested through their difference in h, with L−20 experiencing more spreading. Regarding a comparison between EM measured by Engqvist et al.20 and ED calculated from our model, such can only be made for the L−20 sample, since the relevant experiments for the derivation of both quantities have been performed at equal temperatures. We see from Table 5 that the droplet appears stiffer than the film, in consistency with the trend observed with Lau et al.15 experiments, where almost all droplets appeared stiffer than their corresponding films.

association of larger time scales to lower elastic moduli. The effect is more prominent with the L20 sample at RT, pertaining to much shorter time scales of 12 and 48 h, which resulted in the calculated 6.62 and 5.72 MPa elastic moduli, respectively. Investigating the effect of temperature, we may compare the L20 of 12 h at RT to that of overnight exposure at 105. The substantial impact of exposing a sample to an environment of 85 above its Tg is only evident. Within a time scale on the order of 10 h, the elastic modulus is brought down to 2.89 MPa, compared to the 6.62 MPa of the RT treatment. Comparisons can also be made between droplets across different samples. For example, comparing L20 to L−20 on the basis of them having similar R0, we find the higher ED of L20 is in agreement with the higher measured h. In a similar spirit, we may compare L20 at 105 °C to L60 on the basis of their similar h. L60 being of greater R0 implies more spreading, which is in agreement with our calculated lower ED. These comparisons illustrate the mathematical consistency of our model. A comparison incorporating both temperature and time effects could be made between L−20 and L60. The former is treated at over 40 °C above its Tg, compared to 10 °C for the latter, and for much longer periods, justifying its lower ED. Given that the two samples are not very dissimilar in size, their

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POLYMER DROPLETS OF 5−50 nm Experimental Setup. We used linear polybutadiene (PB) polymer chains of two different molecular masses, 78.8 and 962 kg/mol, dissolved in an appropriate volume of toluene in order for the solutions to maintain concentrations well below the critical overlap concentration, c* (c/c* ≃ 0.3). Toluene was used as received (Fisher Scientific, Loughborough, U.K.). In a typical experiment, a freshly cleaved mica surface (Agar Scientific, Essex, U.K.) was incubated in a polymer solution for several hours. The mica surface was removed from the solution and placed in a 100 mL toluene bath for 24 h and then rinsed exhaustively with 100 mL of toluene to ensure the 4762


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can be stated as γP ≃ γPd. Specifically, we have γP = 32 mJ·m−2,115,116 a value typical of hydrocarbons. For mica (an extensive account of its surface properties can be found in the work of Hu et al.117) we have γSd = 30 mJ·m−2, γSp = 90 mJ·m−2, and γS = 120 mJ·m−2, with γS simply being the sum of the dispersion118−120 and polar121 contributions, in fair agreement with independently reported values of mica’s total surface tension.119,120,122 The spreading parameter, eq 16, written in terms of Fowkes’s model, eq 20, and the previous assumption γP ≃ γPd, illustrates that the interaction between mica and PB is governed by dispersion forces: S = −2γPd + 2(γSdγPd)1/2. A similar conclusion would be the product of an analysis from the perspective of the vOCG102 model (developed later in this article): the occasional double bond of PB would render it with an ever-so-slightly basic character, while siloxane and other groups make the mica surface mostly basic,123 meaning that any acid−base interaction cross terms between the two will be negligible compared to the dominant Lifshitz−van der Waals terms. This fact also makes the value of S more reliable in the context of contamination, which is hard to precisely take into account: build-up of contaminants on a freshly cleaved mica surface would initially mainly screen acid−base interactions, making the present analysis more tolerant of contamination uncertainties. S can be evaluated following substitution of the aforementioned surface tensions and components to give S ≃ 0 mJ·m−2. Experimental data are graphically presented in Figure 7, while calculated moduli are shown in Figure 8. Multichain

removal of nonadsorbed PB chains. Subsequently, the samples were dried gently with a stream of nitrogen. The resulting polymer structures were investigated by atomic force microscopy. The PicoSPM (Agilent Technologies) was operated in tapping mode in air. Commercially available Si3N4 (MikroMasch, Talinn, Estonia) rectangular cantilevers, with a spring constant of 1.75 N/m and resonance frequency of 130−160 kHz, were used. The cantilevers were oscillated 5% below their natural resonant frequency and during imaging moved in a raster fashion. Each sample was imaged at several different areas. The size of the polymeric islands was determined by using the grain analysis of the software Scanning Probe Image Processor (SPIP, Image Metrology, Hørsholm, Denmark). It is well-known that the AFM tip can make an object lying on a surface to look wider due to convolution of the geometry of the tip and the shape of the object being imaged. In a previous publication,114 we extensively discussed the influence of tip geometry on the apparent dimensions of an object imaged with AFM. By use of similar geometrical arguments and assuming again that the object is a spherical cap (substantiated through AFM measurement of its profile), the real volume of the object imaged was estimated from its apparent height, apparent volume, and radius of the probe tip. We repeatedly imaged the samples at room temperature over the course of weeks and did not observe significant changes in the adsorbed state; that is, the contact angle did not change. Figure 6 contains representative AFM images. Results. The PB individual chains of molar masses Mw = 78.8 kg/mol and Mw = 962 kg/mol corresponded to estimated volumes of V = 147 nm3 and V = 1795 nm3, respectively (density ∼0.89 g·cm−3). The range of heights of adsorbed PB droplets was measured to be approximately 1−25 nm, corresponding to a range of volumes 1000−1 000 000 nm3 and a range of undeformed radii 5−50 nm, spanned through polydispersity and a variable number of chains in the aggregates. We converted volumes into undeformed radii (V = 4/3πR03), which, combined with AFM-measured adsorbed heights, returned the ED moduli through our model, eq 17, given also the surface tension of the polymer in air γP, the spreading parameter S, and Poisson’s ratio ν. As far as the latter is concerned, a good approximation is 0.5 for rubbery polymers.94 We obtain γP from the literature115,116 and estimate S through Fowkes’s model.100,101 This is similar in spirit to the vOCG102 model yet considers an alternative force repartition, specifically γ = γd + γp where γd refers to the dispersion interaction contribution (“apolar”) and γp collectively to hydrogen-bond (γh), dipolar, etc. contributions (“polar”). As an alternative to the vOCG model, eq 19, we now have

Figure 7. Height versus deconvoluted volume for 78.8 kg/mol multichain droplets (solid circles) and 962 kg/mol multichain droplets (open circles).

aggregates of Mw = 78.8 kg/mol are represented by solid circles, while multichain aggregates of Mw = 962 kg/mol are represented by open circles. Discussion. Overall, the multichain aggregates returned reasonable predictions for the elastic modulus of PB. Points lie in the range from a few megapascals to a few tens of kilopascals, the majority being within 100 kPa−1 MPa, exhibiting a decreasing trend with increasing size. Dossin et al.124 report a stress relaxation plateau modulus of 1.16 MPa at T = 298K corresponding to a time-scale of a few days. Similarly, Hvidt et al.125 report 0.66 MPa; Colby et al.,126 1.20 MPa; Fetters et al.,127 1.1 MPa; Carella and Graessley128 and Aranguren and Macosko,129 ∼ 1 MPa; and Byutner and Smith,130 a modulus in the range 10−106 Pa for time scales in the range 100−0.1 s for the melt state. We expect to recover the PB modulus bulk value

γ12 = γ1 + γ2 − 2(γ1dγ2d)1/2 − 2(γ1pγ2 p)1/2 = γ1 + γ2 − 2(γ1dγ2d)1/2 − 2(γ1hγ2 h)1/2 − ... (20)

PB is formed by polymerization of the monomer 1,3butadiene, a hydrocarbon diene molecule. The derived polymer contains single and occasional double covalent bonds along the backbone, and consists exclusively of C and H atoms. Due to the small difference in electronegativity between C and H (0.35 in the Pauling scale) we consider the polymer approximately nonpolar and therefore interacting mainly via dispersion forces with the environment. In the context of Fowkes’s theory this 4763


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thicknesses from 276 nm to 13 nm revealed an increase of stiffness by over 2 orders of magnitude at the smallest thickness, recovering the bulk value at around 200−300 nm. A smooth transition toward the bulk value was observed with experiments of Tweedie et al.,137 in agreement with the trend in Figure 8. Tweedie et al.137 applied contact loads over 5−200 nm from the surface of amorphous polystyrene, poly(methyl methacrylate), and polycarbonate. They observed that the apparent stiffness exceeded that of the bulk by up to 200% independent of processing scheme, macromolecular structural characteristics, and relative humidity, and they attributed that to contact-stress-induced formation of a mechanically confined phase at the probe−polymer interface. More specifically, attraction toward and repulsion from the probe material would restrict molecular mobility in the vicinity of mechanical contact with the probe. Balazs et al.138 analyze this effect in terms of enthalpic (intermolecular interactions) and entropic (stretching/alignment of chains with respect to the contact surface) components. Recognizing that there was no probeinduced deformation in the experiments we performed, we argue that the deformation due to adsorption on a substrate induces a very similar mechanism to the aforementioned. Finally, we provide two examples of experiments indirectly related to the elastic modulus size effect: Priestley et al.79 analyzed the structural relaxation of polymers near surfaces and interfaces. Relative to that of the bulk, the rate of structural relaxation at the substrate interface was reduced by a factor of 15, exhibiting a nearly complete arresting of relaxation and implying an increase of the effective stiffness. Moreover, Napolitano et al.139 have shown slow relaxation of polymer chains adsorbed on a substrate, relative to that of the bulk, even at temperatures above Tg, indicating that interaction of the macromolecules with the substrate can affect significantly the physical properties of the adsorbed chains. Referring the reader back to Figure 6, we point out a relationship between droplet size and phase signal. Larger droplets appear darker in phase image than smaller droplets. The phase signal is dependent on a number of factors, including viscoelasticity, adhesion, long-range forces, and contact area;140,141 however, the observation is compatible with our experimental finding of a decreasing elastic modulus with increasing size, through the contribution of (visco)elastic properties to the phase signal.142 For single-chain droplets ≤5 nm (data not shown), our continuum theory fails. Structures obtain a pancakelike shape.143 Monomers then become trapped/pinned against the surface, and while the “modulus” (if we assume for a moment that it is legitimate to talk about modulus at that length scale) ought to, therefore, be high, the theory predicts nonsensical values. It appears that the presence of several, as opposed to one, chains in a droplet makes a difference in its adsorption behavior, as discrepancy is observed between multichain and single-chain droplets of similar size (coexisting in a small range about 5 nm). We speculate that the penalty of repeated bending of the single chain arranging itself in multiple layers makes this a less favorable conformation than the flat, pancakelike arrangement. For completeness, we make reference to the effect of droplet size on surface tension, introduced theoretically by Tolman.144 The surface tension of a droplet of a single pure substance can be expected to decrease with decreasing droplet size. We have decided not to take this idea on board for the following reasons: First, a decrease of the fluid−air interface tension

Figure 8. Elastic modulus versus deconvoluted volume for 78.8 kg/ mol multichain droplets (solid circles) and 962 kg/mol multichain droplets (open circles).

for large enough droplet sizes where the surface-to-volume ratio becomes negligible and the effective properties are governed by classical bulk elastic strain energy.131 The reported elastic moduli of Ishikawa et al.,87 in the range 10−25 MPa for polymer aggregates of a size range 15−20 nm, are also an indication of what one should anticipate in the relevant size range; however, these are not to be compared directly against our calculated values, because their polymer consists of hydroxystyrene as base and its modulus has been measured by mechanical testing. It should be noted that some values of the physical quantities used in our model (e.g., surface tension) come from literature without mention of the corresponding uncertainty. Also, we have made assumptions (e.g., perfectly spherical caps) of which the uncertainty is not obvious to quantify. Therefore, placing error bars in our figures is rather unreliable. However, judging from our results, we anticipate the accuracy of our predicted elastic moduli to be at least within an order of magnitude. In this context, we do not attribute much significance to the local maximum of the 962 kg/mol data. Several sources in the literature are in agreement with our observed trend of an increasing modulus with decreasing size. Miyake et al.82 evaluated the surface and bulk elastic modulus of thick and thin polystyrene films by AFM indentation. Having eliminated the influence from the substrate material by indenting thick films, they measured a surface modulus smaller than the bulk, one possible explanation of which can be the existence of free space presented to the polymer segments at the surface. For thin films, they found that the surface modulus increased considerably with increasing deformation, suggesting an increased influence from the substrate. This was explained by potential trapping/pinning of monomers against the substrate, affecting the rest of the thin film above, which is a good candidate explanation for our observations also, where trapping would occur due to the adhesion process itself. This pinning effect was also identified in the experiments of Ge et al.132 O’Connell and McKenna133 took the bubble inflation of thin membranes biaxial test method134 and scaled it so that films of nanometer thickness can be tested. The important aspect of what they did was the use of the imaging capability of AFM to perform the measurement of deformation but not the indentation itself, thus avoiding contact mechanics problems when the AFM is used as an indentation machine.135 That work and a follow-up investigation136 of supported polymer films of 4764


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It was explained in the Introduction that consideration of a line tension contribution in our model would go beyond the work of this article. However, it is interesting to note at this point that we have shown that the experimental nanoscale results can be interpreted by a moderate increase of modulus (as expected for these nanometer-scale sizes) without the incorporation of line tension.

means that spreading becomes more energetically favorable, opposing our observed trend of increased dewetting with decreasing size (explained with an increased elastic modulus). Inclusion of a size-dependent surface tension would, therefore, demand an even higher predicted elastic modulus or some other effect to compensate for an increased wetting tendency. Second, the effect of droplet size on surface tension becomes significant only when the droplet size approaches or reaches the order of 10−9 m and the decreasing surface tension function becomes steep only below 5 nm,145 which is our lower limit in terms of the droplet sizes that we investigate. We thus consider it justifiable to presently ignore the effect of a varying surface tension. Perhaps, once the role of line tension is disambiguated, it could be used in conjunction with a variable surface tension as a future refinement. As a final remark following from our introductory note concerning line tension, incorporating that as a one-dimensional analogue36 of surface tension in our model while keeping the elastic modulus constant at its bulk value resulted in a sizedependent line tension (increasing with decreasing size), a fact nowhere reported in the literature to the best of our knowledge, except in one case146 where the opposite trend is observed (decreasing with decreasing size).

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AUTHOR INFORMATION

Corresponding Author

*E-mail [email protected] or [email protected]. uk; phone +44 (0)131 650 8704; fax +44 (0)131 650 6551. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS

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REFERENCES

A.E.A.S.E. acknowledges financial support from the Alexander S. Onassis Public Benefit Foundation, Athens, Greece.

(1) Lee, L.-H. Adhesion and adsorption of polymers; Plenum Press: New York, 1980. (2) Brown, H. R. Science 1994, 263, 1411−1413. (3) Napper, D. Polymeric stabilization of colloidal dispersions; Academic Press: London, 1983. (4) Svec, F.; Frechet, J. M. J. Anal. Chem. 1992, 64, 820−822. (5) Cox, J. K.; Eisenber, A.; Lennox, R. B. Curr. Opin. Colloid Interface Sci. 1999, 4, 52−59. (6) Imanishi, Y.; Yoshihiro, I. Pure Appl. Chem. 1995, 67, 2015−2021. (7) Ruckenstein, E.; Chung, D. J. Colloid Interface Sci. 1988, 123, 170−185. (8) Granier, V.; Sartre, A.; Joanicot, M. J. Adhes. 1993, 42, 255−263. (9) Granier, V.; Sartre, A. Langmuir 1995, 11, 2179−2186. (10) Sommer, F.; Duc, T. M.; Pirri, R.; Meunier, G.; Quet, C. Langmuir 1995, 11, 440−448. (11) Johnson, C. A.; Lenhoff, A. M. J. Colloid Interface Sci. 1996, 179, 587−599. (12) Unertl, W. N. Langmuir 1998, 14, 2001−2207. (13) Portigliatti, M.; Koutsos, V.; Hervet, H.; Léger, L. Langmuir 2000, 16, 6374−6376. (14) Kirsch, S.; Pfau, A.; Hädicke, E.; Leuninger, J. Prog. Org. Coat. 2005, 45, 193−204. (15) Lau, A. W. C.; Portigliatti, M.; Raphaël, E.; Léger, L. Europhys. Lett. 2002, 60, 717−723. (16) Kugge, C.; Craig, V. S. J.; Daicic, J. Colloids Surf., A 2004, 238, 1−11. (17) Yamamoto, T.; Fukushima, T.; Kanda, Y.; Higashitani, K. J. Colloid Interface Sci. 2005, 292, 392−396. (18) Tan, S.; Sherman, R. L. J.; Qin, D.; Ford, W. T. Langmuir 2005, 21, 43−49. (19) Lafon, M.-O.; Trannoy, M. Influence of Drying Temperature on the Morphology of Latex at the Surface of Pigments: Consequences on the Paper Properties. 9th TAPPI Advanced Coating Fundamentals Symposium, Turku, Finland, February 8−10, 2006 (20) Engqvist, C.; Forsberg, S.; Norgren, M.; Edlund, H.; Andreassohn, B.; Karlssohn, O. Colloids Surf., A 2007, 302, 197−203. (21) Seemann, R.; Jacobs, K.; Blossey, R. J. Phys.: Condens. Matter 2001, 13, 4915−4923. (22) Araujo, O.; Brauge, L.; Findlay, A. D. Macromol. Symp. 2008, 271, 75−82. (23) Kramarenko, E. Y.; Winkler, R. G.; Khalatur, P. G.; Kokhlov, A. R.; Reineker, P. J. Chem. Phys. 1996, 104, 4086−4813. (24) Michel, A.; Kreitmeier, S. J. Polym. Sci., Part B: Polym. Phys. 2001, 39, 2333−2339. (25) Liu, H.; Chakrabarti, A. Langmuir 2008, 24, 7923−7930.

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CONCLUSIONS We have considered through theory and experiment the spreading of a polymer droplet on a substrate. We have illustrated how the model presented can be used as a noninvasive method of extracting an elastic modulus for the droplet when an experimental measurement of its adsorbed height is available. As an advantage, one avoids errors and complications involved in direct mechanical testing. We have established the significance of both surface and elasticity effects in the adsorption behavior of a polymer droplet and illustrated the model’s limitations. All experimental data at our disposal supported the theoretical predictions. Specifically, the experiments of Lau et al.15 dealt with styrene−butadiene latex droplets of various gel fractions and glass transition temperatures in the ∼100 nm regime adsorbed on silica. The AFMmeasured adsorbed heights confirmed the assumed spherical shape and led to predicted elastic moduli of the anticipated magnitude, given the material. Furthermore, these predictions and their trend were in complete agreement with the gel fraction and the glass transition temperature properties. The experiments of Engqvist et al.20 took into account different glass transition and treatment temperatures, as well as treatment times in the spreading of their mesoscopic styrene−acrylic latex droplets. All three parameters came into agreement with our theoretically predicted moduli. The predicted moduli pertaining to droplets of the work of both Lau et al.15 and Engqvist et al.20 were, on the whole, of higher value than the moduli of macroscopic strips of corresponding consistency. This was attributed to the difference in structure, the strips being made up of an array of close-packed droplets adhering together by their shells. As far as our own experiments are concerned, AFM-measured adsorbed heights of PB on mica revealed a trend of decreasing elastic modulus with increasing droplet size, progressively tending toward the bulk material value. This size effect is supported by the literature for related systems, though a consensus has not yet been reached.147 Our experiments also illustrated the theory’s limitations, namely, its unsuitability for single-chain droplets of radius ∼5 nm or less. 4765


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