Stratification of Colloidal Particles on a Surface: Study by a Colloidal

Subscriber access provided by Warwick University Library

B: Glasses, Colloids, Polymers, and Soft Matter

Stratification of colloidal particles on a surface: Study by a colloidal probe atomic force microscopy combined with a transform theory Ken-ichi Amano, Taira Ishihara, Kota Hashimoto, Naoyuki Ishida, Kazuhiro Fukami, Naoya Nishi, and Tetsuo Sakka J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.8b01082 • Publication Date (Web): 03 Apr 2018 Downloaded from http://pubs.acs.org on April 4, 2018

Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.

is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

Page 1 of 32 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

The Journal of Physical Chemistry

Stratification of colloidal particles on a surface: Study by a colloidal probe atomic force microscopy combined with a transform theory

Ken-ichi Amano,†* Taira Ishihara,† Kota Hashimoto, † Naoyuki Ishida, ‡ Kazuhiro Fukami,§ Naoya Nishi, † and Tetsuo Sakka †

†

Department of Energy and Hydrocarbon Chemistry, Graduate School of Engineering,

Kyoto University, Kyoto 615-8510. ‡

Division of Applied Chemistry, Graduate School of Natural Science and Technology, Okayama University, Okayama 700-8530, Japan.

§

Department of Materials Science and Engineering, Graduate School of Engineering, Kyoto University, Kyoto 606-8501, Japan.

1

ACS Paragon Plus Environment

The Journal of Physical Chemistry 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

ABSTRACT: Colloidal probe atomic force microscopy (CP-AFM) can be used for measuring force curves between the colloidal probe and the substrate in a colloidal suspension. In the experiment, an oscillatory force curve reflecting the layer structure of the colloidal particles on the substrate is usually obtained. However, the force curve is not equivalent to the interfacial structure of the colloidal particles. In this paper, the force curve is transformed into the number density distribution of the colloidal particles as a function of the distance from the substrate surface using our newly developed transform theory. It is found by the transform theory that the interfacial stratification is enhanced by an increase in an absolute value of the surface potential of the colloidal particle, despite a simultaneous increase in a repulsive electrostatic interaction between the substrate and the colloidal particle. To elucidate mechanism of the stratification, an integral equation theory is employed. It is found that crowding of the colloidal particles in the bulk due to the increase in the absolute value of the surface potential of the colloidal particle leads to pushing out of the some colloidal particles to the wall. The combined method of CP-AFM and the transform theory (the experimental-theoretical study of the interfacial stratification) is related to colloidal crystallization, glass transition, and aggregation on a surface. Thus, the combined method is important for developments of colloidal nanotechnologies.

2


Page 2 of 32

Page 3 of 32 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


■ INTRODUCTION Colloidal suspensions are ubiquitous in daily life. For example, several foods, pharmaceutical products, cosmetic products, coating materials, and printer inks contain colloidal particles. Structural analysis of colloidal suspensions is usually performed by small angle X-ray scattering (SAXS)1–4 or small angle neutron scattering (SANS), 5,6 as these techniques permit acquisition of the colloidal particle sizes and the bulk interparticle density distributions. Meanwhile, when it comes to the number density distribution of the colloidal particles as a function of the distance from the substrate surface (ρ WP ), such a structural analysis is not often experimented compared with the bulk analysis. 7–9 However, the measurement of ρ WP is important to understand the interplays among a substrate and colloidal particles in various conditions. In this paper, we report on the structural analysis of colloidal particles on a substrate in terms of experimental data and theoretical calculations. Measurement of ρ WP without drying or freezing the colloidal suspension is difficult when the colloidal particles are optically invisible. If ρ WP can be determined without drying or freezing, such data can be used for studies of epitaxial growth of a colloidal crystal, 10 surface induced colloidal crystallization, 11 glass transition, 12 and aggregation, 13 to name a few. The present study’s main concern, ρ WP, is related to structural information before the crystallization, glass transition, and aggregation. Recent nanotechnology is chasing more useful colloidal crystals, because such crystals can be used for quantum dots14 and photonic crystals. 15 They are expected as the future materials for memory devices, solar cells, optical fibers, coating materials with no sick house syndrome, and so on. Moreover, the information of ρ WP on surfaces of capsules and bottles are important for long-term preservation of a colloidal product. Small angle x-ray7 or neutron 8,9 reflectivity has been used to measure ρ WP . When scattering contrast exists between the solution and the colloidal particle, ρ WP can be 3



measured using model functions and appropriate parameters. The measured ρ WP is helpful for studying the layer structure if the surface potential and chemical structure of the substrate do not change considerably by the beam irradiation. 16 Such an analysis has revealed that the number of layers in ρ WP increases as the volume fraction of the colloidal particles increases. 8 Total internal reflection microscopy (TIRM)17–19 has also been used to analyze the interfacial structure. TIRM takes advantage of evanescent light generated on a translucent substrate, and the potential of mean force between a floating colloidal probe and the substrate is measured by detecting intensity of the evanescent light. Thus, TIRM can also measure the force curve as well as CP-AFM. However, as far as we see, reproducibility (precision) of the force curve obtained by TIRM is inferior to CP-AFM. Hence, in this paper, we focus on CP-AFM for the study of the transformation from the force curve into ρ WP. CP-AFM (see Fig. S1) has been used for studying the effects of pH, 20 ionic strength, 21 polyelectrolyte solutions, 22 and surface modifications of the colloidal probe and the substrate 23 on the force curve. When the colloidal suspension has a certain volume fraction, an oscillatory force curve can be measured, and hence it has also been used for studying the layer structure of the colloidal particles on the substrate. 23–25 However, the oscillatory force curve does not directly indicate ρ WP. (A surface force apparatus can also measure the oscillatory force curve, 26 but it cannot directly obtain ρ WP either.) To obtain ρ WP from CP-AFM, transformation from the force curve into ρWP is necessary (ρ WP has never been obtained from the measured force curve). Hence, the present work proposes a transform theory to obtain ρ WP from the force curve based on statistical mechanics of a simple liquid. The transform theory proposed here is an improved version of our previous theory for reconstruction of solvation structure. 27 Advantages of the transform theory are as follows: (1) The main inputs for the transform theory are just two force curves measured in the absence and presence of the

4


Page 4 of 32

Page 5 of 32 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


colloidal particles. (2) The theory can be used for various conditions, and hence the effects of pH, ionic strength, polyelectrolyte solutions, and surface modifications of the colloidal probe and the substrate to ρ WP can be studied. (3) Because of the compactness of the CP-AFM machine, combination of CP-AFM and the transform theory can be easily performed in a normal laboratory space. Simulations, 28,29 in which the solution and the colloidal particles respectively are treated as a continuous fluid and granular particles, may be useful for determination of ρ WP. However, the simulation requires several empirical parameters and model potentials. Furthermore, it is difficult to determine the coverage ratio of the surface modification and ionization degree of the colloidal particles. Moreover, it is difficult to treat polyelectrolytes dissolved in the colloidal suspension due to their complex behavior. As a model potential for the colloidal suspension, Derjaguin–Landau– Verwey–Overbeek (DLVO) potential is notable and useful. 30 Two types of DLVO potentials exist: constant surface potential model and constant surface charge density model. 31 To perform the simulation, either type of potential should be selected (or a combined potential of the two models with an arbitrary ratio should be used). To overcome this difficulty, use of an explicit molecular dynamics (MD) simulation is an attractive solution, because the effects in the DLVO potential are inherently included in the explicit MD simulation. However, the cost of such a calculation is considerably high due to the large number of molecules. Furthermore, it still requires some model potentials and empirical parameters. Meanwhile, the combined method of CP-AFM and transform theory requires fewer parameters than the above simulations and it is performed with lower calculation cost. This paper treats the force curve measured by CP-AFM and transforms it into ρ WP using the newly developed transform theory. Until now, visualization of ρ WP from the force curve has never been achieved, and hence we can say that our transform theory

5



expanded capability of CP-AFM. In this paper, several ρ WP s are obtained, and a mechanism of the stratification is proposed. It is expected that the combined method can support development of colloidal materials (e.g., colloidal crystals for quantum dots 14 and photonic crystals, 15 colloidal amorphous for structural color materials32 random lasers, 33 and artificial opals for accessories) from fundamental aspects.

■ METHODS Preparation of the particle-induced force curve. We extracted plot data of the force curves from the paper24 by Piech and Walz using the freely available software GSYS2.4. The plot data were fitted using a power series of s, where s represents separatioin between the substrate surface and the center of the colloidal probe. For the fitting, the number of power series terms (Nterm ) was tested from 1 to the number of the plots in the extracted force curve. N term was determined by AICc (Akaike’s Information Criterion, corrected). 34,35 A fitted function with the smallest AICc value was selected. Then, a smooth step function (exp[−(s/s tail )10 ]) was multiplied by the fitted function in order to make the tail of the fitted function almost completely zero, where s tail represents the separation at which the original force curve is converged to almost zero. Finally, the particle-induced force curve (f P) is obtained by subtracting f abs from f pre , where f abs and f pre represent the fitted force curves obtained in the absence and presence of the colloidal particles, respectively (see Fig. 1). That is, an equation f p = f pre – fabs is used for acquisition of the particle-induced force curve. In the acquisition, the ionic strength and pH in f pre and f abs are assumed to be the same. Fortunately, in the force curve measurement by Piech and Walz, 24 the ionic strength and pH had been adjusted by using sufficient amounts of the chemical agents. Hence, 6


Page 6 of 32

Page 7 of 32 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


the equation f p = f pre – f abs can be used for preparation of the particle-induced force curve.

Transform theory. Transformation from f P into ρ WP is performed by using an improved method of our previous method. 27 We explain the new calculation process for obtention of g WP (≡ ρ WP /ρ 0 ), where ρ 0 (constant) is bulk number density of the colloidal particle. Since g WP is normalized by ρ 0 , it is generally called the normalized number density distribution. When g WP is equal to 1, the particle number density is the same as that in the bulk, and gWP never takes negative value. In theory, it is assumed that the surface properties (materials) of the colloidal probe and the substrate are identical (f P s shown in Fig. 1 are all meeting the assumption). Considering a monodisperse colloidal dispersion, a flat substrate, and a spherical colloidal probe (see Fig. S1), f P can be expressed as, 27 =

− − , 1 2π

where π, s, and rCP represent the circular constant, the separation between the substrate surface and the center of the colloidal probe, and the radius of the colloidal probe, respectively. l is vertical distance between the centers of the colloidal particles contacting at the substrate and the colloidal probe, and reff is effective radius of the colloidal particle contacting on the colloidal probe. P represents the pressure between flat surfaces (see Fig. S2), where the surface properties (materials) of the flat surfaces are the same as that of the colloidal probe and the substrate shown in Fig. S1. Firstly, value of 2reff is estimated from the local minimum position in ∂f P /∂s near the position 7



of the fist peak in f P . The final value of 2r eff is determined self-consistently in the calculation. Generally, 2reff is greater than the core diameter (d P ) of the colloidal particle and is less than (1/ρ0 )1/3 . The value of r eff does not significantly affect the calculation result, because r CP is much larger than reff in the CP-AFM system. To obtain P from f P , the local maximum position in ∂f P /∂s near the fist peak position in f P is estimated, and the values of f P from the local maximum position to a sufficiently far position are substituted into F * in Eq. (2): ∗ = , 2 where F * corresponds to the left-hand side of Eq. (1), P and H correspond to P(l) and the other parts of Eq. (1), respectively. H is a square matrix whose variables are l and s. A detailed explanation of H has been provided in our previous paper. 27 P can be obtained numerically using the inverse matrix of H. Subsequently, P is converted to P(l). (We note that use of Derjaguin approximation 36–38 is also applicable when r CP is sufficiently larger than r eff for calculation of P(l). In this case, the approximation is given by f P = 2π(r CP + reff)W, where W is the particle-induced potential of mean force between flat surfaces per unit area. P can be calculated from W as follows: P = – ∂W/∂s.) Next, value of the normalized number density of the colloidal particles on the colloidal probe at the first peak (g c ) is calculated by:

!

= − "#$ /&' ()* , 3

where k B and T are the Boltzmann constant and the absolute temperature, respectively. Pmin is the minimum value of P. Kirkwood superposition approximation 39–41 is not used to calculate g c , which is one of the improved features compared with our previous

8


Page 8 of 32

Page 9 of 32 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


technique. 42,43 The technique is improved, however, Eq. (3) still contains an approximation of the rigidization of the colloidal probe surface. We note that when reliability of f P near the wall is low, g c should be calculated by substituting the maximum value of P (P max ) into an equation below 42,43

!

1 -&. ( )* + 4&. ()* "01 = + , 4. 2 2&. ()*

because a position l max corresponding to Pmax is comparatively far from the wall in comparison with a position l min corresponding to P min . In Eq. (4), Kirkwood superposition approximation 39–41 is used. Finally, g WP is obtained as follows: 3 /2

3 /2

+ − "#$ = 0 for < "#$ , 5a

+ − "#$ =

&' ()*

!

+ 1 for "#$ ≤ ≤ * , 5b @

>

+ 1? for * < , 5c 3 /2 + − "#$ = = &' ()* ! >

where l 0 corresponds to a position P(l) = 0 near l min . When Eq. (4) is used for calculation of g c , l min is determined so as to g c s estimated from Eqs. (3) and (4) are the same. If a negative value is obtained in the parenthesis on the right-hand side of Eq. (5c) due to low accuracy of the input data, g WP(d P /2 + l – l min ) is calculated as follows: 27,44 {exp[P(l)/(k B Tρ 0 gc α )]} (1/α) when P(l) < 0 and [P(l)/(k B Tρ 0 g c α ) + 1] (1/α) when P(l) ≥ 0. This compensation method works well in most of the cases. If a position where the wall-wall steric repulsion becomes infinity (l str ) is known, the first peak position in g WP is moved in parallel to (l max – l str)/2. 42 This movement comes from an analogy from the wall-wall rigid body system. α is a parameter for modified Kirkwood 9



superposition approximation:

33 B; ℎ

= E

3 B 3 ℎ

>

− B F , 6

where g WPW represents the normalized number density of the colloidal particle within two flat walls (see Fig. S2). h and x are the surface separation between two flat walls and an arbitrary position in the confined space, respectively. The value of α is calculated as follows:

1 + L1 + 4 "01 /&' ()* H = ln K M /ln !. 7 2

Integral equation theory. To check the qualitative validity of the results obtained by the combined method of CP-AFM and the transform theory, a pure theoretical calculation is performed. In the calculation, an integral equation theory (Ornstein-Zernike equation coupled with hypernetted-chain closure: OZ-HNC)45,46 is used. In the calculation, small spherical particles with certain volumes are used as the colloidal particles and the solution is treated as the continuous fluid. For the pair potentials between colloidal particle-colloidal particle and between colloidal particle-substrate, DLVO’s electric double layer potentials with constant surface charge density are used. 31 The pair potential between the colloidal particles (V PP ) is given by OPP Q = −RS* ST UP lnE1 − exp−Q/Y F, 8 10


Page 10 of 32

Page 11 of 32 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


where ε 0 and ε r are the electric permittivity of vacuum, the relative permittivity of water, respectively. 47 H, D, and ψ P, respectively, are the surface-surface separation between the colloidal particles, Debye length, surface potential of the colloidal particle. The pair potential between the substrate and the colloidal particle (VWP ) is given by

O[P Q =

RS* ST

F 2EUP + U[

@

\

2UP U[ 1 + exp−Q/Y 2Q ln = ? − ln =1 − exp ]− ^?_. 9 1 − exp−Q/Y Y UP + U[

In Eq. (9), H and ψW represent the surface-surface separation between the wall and the colloidal particle and surface potential of the wall, respectively. When the pair potential between the substrate and the colloidal probe is calculated, d P and ψP in Eq. (9) are replaced by the diameter of the colloidal probe (2rCP ) and surface potential of the colloidal probe (ψ B ), respectively. The reason why we do not use DLVO with constant surface potential is that its potential shape between the colloidal particle and the substrate is unrealistic when the signs of their surface potentials are identical (see Fig. S3). 24 To simplify and unify the model potentials, we used the DLVO potential with the constant surface charge density for both cases. In addition, we disregarded the van der Waals potential and used rigid potential as the steric repulsive potential in order to simplify the analysis, which is helpful for extracting the essential physical characteristics. We calculated the radial distribution function between the colloid particles in the bulk using ‘a radial symmetric OZ-HNC’. Substituting the radial distribution function into ‘OZ-HNC for between a flat wall and spherical particles’, we calculated g WP. In the calculation, the number of meshes is 16384, the grid spacing is d P/100 nm, and T is set at 298 K.

11



As a supplement, we conducted verification test of the transform theory within computationally closed cycle. Results of the verification test are shown in Fig. S4, in which the benchmark structure of g WP and the input force curve are prepared by using the OZ-HNC. We compared the benchmark structure and the g WP transformed from the input force curve and discussed the conversion accuracy in Fig. S4. In the comparison, we found that the phase coherent between the benchmark and the transformed structures is not so good. However, the benchmark and the transformed structures were similar in shape. Thus, we concluded from Fig. S4 that the transform theory can predict g WP from f P qualitatively or semi-quantitatively.

■ RESULTS AND DISCUSSION Particle-induced force curve. Before obtaining g WP , the particle-induced force curve being f P is prepared by using the raw force curves measured by Piech and Walz. 24 They measured the raw force curves using the silica colloidal probe and the silica substrate in several colloidal dispersions. In their experiment, T = 298 K, rCP = 1.75 µm, ψB and ψW are both ~−20 mV (we approximate that ζ potentials 24 measured by an electrophoresis apparatus or another analyzer as the surface potentials). The obtained f P curves are shown in Fig. 1. In Figs. 1(a) and 1(b), the colloidal particles are the polystyrene nanoparticles. In Fig. 1(c) of the red and green curves, the colloidal particles are polystyrene and silica nanoparticles, respectively. In Fig. 1(a), the volume fraction of the colloidal particles (φ) is varied but the bulk number density of the colloidal particles (ρ 0 ) is fixed. In Fig. 1(b), ρ 0 is varied but φ is fixed. In Fig. 1(c), ψP is varied but ρ 0 and φ are fixed.

12


Page 12 of 32

Page 13 of 32 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Number density distribution predicted from the combined method. The combined method of CP-AFM and the transform theory requires only f P, T, ρ 0 , rCP , and d P . Figures 2(a), 2(b), and 2(c) show the g WP transformed from the data in Figs. 1(a), 1(b), and 1(c), respectively. As seen in Fig. 2(a), the amplitude of g WP is large when φ is high. Similarly, Fig. 2(b) shows that the amplitude of g WP is large when ρ 0 is high. The interfacial stratification is enhanced by increases in both φ and ρ 0. The result of Fig. 2(c) is notable. In the condition of Fig. 2(c), surfaces of the substrate and colloidal particles are negatively charged. The colloidal particles with large negative surface potentials are expected to be repelled from the wall and the stratification is expected to be decayed. However, Fig. 2(c) shows that such colloidal particles with large negative surface potential enhance the stratification. The result is peculiar from viewpoint of the electrostatic interactions between the substrate and the colloidal particles. However, mechanism of the stratification can be understood as follows: Repulsions among the colloidal particles are strong when the absolute value of the surface potential of the colloidal particle is high. It makes the degree of the crowding in the bulk high. In this case, some of the colloidal particles are pushed out to the wall surface in order to relax the crowding.

Behavior of the number density distribution on the wall. In this section, a purely theoretical method is used to qualitatively compare behavior of g WPs from the combined method and the purely theoretical method. As the theoretical method, an integral equation theory (OZ-HNC) is used. 45,46 In Figs. 3(a),

13



3(b), 3(c), and 3(d), the effects of φ, ρ 0, ψP , and ψ W on gWP are shown, respectively. The parameters for these figures are decided with reference to the experiment condition conducted by Piech and Walz. 24 In all of these calculations, T = 298 K, and the relative permittivity is calculated using an equation proposed by Malmberg and Maryott. 47 The parameters used in Fig. 3 are listed in Table. 1. The effect of φ on g WP is shown in Fig. 3(a), which reveals that the stratification is enhanced by the increase in φ. This behavior is consistent with the result shown in Fig. 2(a). In Fig. 3(a), d P increases with increasing φ, however, the oscillation length is almost the same. A possible mechanism is as follows: (I) The increase in φ involves the increase in d P ; (II) If we consider only a pair of the colloidal particles here, it is assumed that the increase in d P increases the correlation length between the pair particles; (III) However, the increases in d P and φ causes crowding of the system (here, many colloidal particles around the pair particles are considered); (IV) Then, due to the crowding, the correlation length between the pair particles is shortened; (V) Eventually, the correlation length is almost not changed, and therefore the oscillation length shown in Fig. 3(a) is also almost not changed by the increase in φ. The effect of ρ 0 on g WP is shown in Fig. 3(b), which reveals that the height of the first peak in g WP increases as ρ 0 increases. This behavior is consistent with the result shown in Fig. 2(b). By the way, the oscillation length in Fig. 3(b) is increased as ρ 0 is reduced. In the situation, dP increases with reducing ρ 0, and thus the behavior of the oscillation length is likely. However, as a supplemental, we write a possible mechanism as follows: (i) The decrease in ρ 0 involves the increase in d P; (ii) If we consider only a pair of the colloidal particles here, it is assumed that the increase in d P increases the correlation length between the pair particles; (iii) It seems that the increase in dP causes crowding of the system, but the volume fraction remains constant and low (φ = 2.4 %); (iv) The system is not so crowded even when d P is relatively large,

14


Page 14 of 32

Page 15 of 32 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


and hence it can increase the correlation length with increasing d P ; (v) Therefore, the oscillation length shown in Fig. 3(b) is also increased by the increase in d P (by the decrease in ρ 0 ). The effect of ψP on g WP (see Fig. 3(c)) is not so straightforward in comparison with the effects of φ and ρ 0 . As |ψP | increased (from 0 mV to −200 mV), the repulsion between the substrate and colloidal particle is increased, which may cause decay of the stratification. However, the behavior shown in Fig. 3(c) is different from the above prediction. The heights of the peaks in g WP are increased by an increase in |ψ P |. Notably, this behavior is the same with that in Fig. 2(c). Mechanism of the stratification is explained as follows (see Fig. 4): When |ψ P | is high, the colloidal particles push each other in the bulk and the space is relatively crowded. In this case, some of the colloidal particles are driven to the wall surface despite of the repulsion between the substrate and colloidal particle. Then, the crowding in the bulk is somewhat relaxed. The mechanism will be discussed in terms of energy and entropy in the next section. Finally, in Fig. 3(d), effect of ψ W on g WP is checked. As |ψW | is increased (from 0 mV to −200 mV), the position of the first peak in g WP moves away from the wall. The behavior is consistent with results obtained from a Monte Carlo simulation and an integral equation theory. 48 The height of the first peak in g WP gradually decreases as |ψ W | increases, and the height is almost reaching a constant value when |ψ W | is sufficiently large. It is an interesting behavior, because it seems that the increase in |ψ W | makes the pair of the substrate and the colloidal particle more repulsive. Moreover, as shown in the figure, the position of the first peak in g WP is moving away from the wall with increasing |ψ W |. This behavior can be explained as follows: When the value of |ψW | is relatively high, a position (H1 ) near the first peak in g WP is sufficiently far away from the wall. Then, a following inequality, exp(–H 1 /D) 0), it is rewritten as, O[P Q@ = 2RS* ST UP U[ exp−Q@ − Ylna/Y . 11

Consequently, from Eq. (11), it can be understood that the change from ψW to θψW corresponds to sliding of the distance Dlnθ. 48 Therefore, the position of the first peak in g WP was moving away from the wall and the peak height was almost not changed by the increase in |ψ W |.

Roles of energetic and entropic components. In this section, we see energetic and entropic contributions in the stratification. It is known that g WP is directly related to the potential of mean force between the substrate and the colloidal particle in the particle suspension (Φ WP ) through an exact relationship: Φ WP (d WP ;T) = −k B Tln(gWP (d WP;T)), where T is the absolute temperature. In the calculation of OZ-HNC, isothermal-isochoric system is considered, and hence the potential of mean force corresponds to the change in Helmholtz free energy. Φ WP is expressed in terms of energetic (EWP ) and entropic (S WP ) components: Φ WP = E WP − TS WP. Total differentiation of Φ WP is given by a following equation: ∆Φ WP = ∆EWP − T∆S WP − S WP ∆T. In a reversible process, a following equation is realized, ∆E WP = T∆SWP − P∆V, and hence ∆Φ WP = − P∆V − S WP ∆T. (P and V are the osmotic pressure originating from the colloidal particles48 and the system volume, respectively.)

16


Page 16 of 32

Page 17 of 32 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Therefore, S WP can be calculated by differentiating Φ WP with respect to T (S WP = −∂Φ WP /∂T). In the numerical differentiation, we set ∆T as ±1 K. In addition, relative permittivity of water47 and the Debye length were changed according to ∆T. The energetic component is obtained from E WP = Φ WP + TS WP . In Fig. 5, Φ WP, EWP, −TS WP, and V WP are shown, where ψP = −50 mV. The thermodynamic quantities Φ WP , E WP, and −TS WP are calculated from the data on the dotted line in Fig. 3(c). As shown in Fig. 5, the shapes of Φ WP and V WP are not similar due to existences of the colloidal particles. There is a local minimum of Φ WP near d WP ~ 40 nm. The local minimum is appeared as a consequence of relaxation of the bulk crowding (see Fig. 4). An interesting point is that the local minimum exists at repulsive region for VWP. The local minimum is mainly formed by the entropic component. That is, the entropic component stabilizes the colloidal particles in the vicinity of the wall. This entropic effect is related to the excluded volume effect of Asakura-Osawa theory. 49,50 In the theory, contact of the colloidal particles on the wall increases the volume for translational motion of the colloidal particles in the bulk, giving rise to the entropic gain. However, in the very vicinity of the wall, Φ WP rapidly increases due to the rapid increase in E WP (VWP ).

■ CONCLUSIONS We have demonstrated the combined method of CP-AFM and the transform theory to obtain gWP from the measured force curve. We have obtained qualitatively valid g WP through the combined method. The effects of φ, ρ 0, ψP, and ψW on g WP have also been found. The roles of the energetic and entropic components in the stratification have been discussed. 17



The present transform theory can obtain the qualitatively valid result from the force curve, but there are the modified Kirkwood superposition approximation and the rigidization of the colloidal probe surface in the theory. To transform it more accurately, we recently derived a new transform theory51,52 without using these approximations. Although the new transform theory is more complicated and requires more input data (a particle-particle structure factor in a bulk), we believe that we can obtain more accurate results from the new transform theory in the future. The present study revealed the stratification mechanism of the colloidal particles, which is important knowledge for developments of colloidal technologies. Thus, we believe that combination of CP-AFM and the transform theory can support the developments of the colloidal technologies. To find other (unknown) mechanisms of the stratification at the colloidal particle-wall interface, we will study effects of pH, ionic strength, polyelectrolyte solutions, and surface modifications on g WP in the future.

■ ASSOCIATED CONTENT S Supporting Information ○

The Supporting Information is available free of charge on the ACS Publications website at DOI: *****************. Figures S1-S4 (PDF)

■ AUTHOR INFORMATION 18


Page 18 of 32

Page 19 of 32 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Corresponding Author *E-mail: [email protected] ORCID Ken-ichi Amano: 0000-0002-4055-8813 Notes The authors declare no competing financial interest.

■ ACKNOWLEDGEMENTS We thank Ryosuke Sawazumi for assisting the force curve fitting. We thank Satoshi Furukawa for supporting preparation of Fig. S4 and simplifying Eq. (7). We appreciate advice and comments from Atsushi Ikeda and Hiroshi Onishi. This work was supported by Grant-in-Aid for Young Scientists (B) from Japan Society for the Promotion of Science (15K21100), and partly supported by Grant-in-Aid for Scientific Research (B) from Japan Society for the Promotion of Science (15H03877) and JSPS Bilateral Open Partnership Joint Research Projects.

■ REFERENCES (1)

Matsuoka, H.; Ise, N.; Okubo, T.; Kunugi, S.; Tomiyama, H.; Yoshikawa, Y. Ordered Structure in Dilute Solutions of Biopolymers as Studied by Small-Angle X-Ray Scattering. J. Chem. Phys. 1985, 83 (1), 378–387.

(2)

Zhang, F.; Skoda, M. W. A.; Jacobs, R. M. J.; Zorn, S.; Martin, R. A.; Martin, C. M.; Clark, G. F.; Goerigk, G.; Schreiber, F. Gold Nanoparticles Decorated with 19



Oligo(ethylene Glycol) Thiols: Protein Resistance and Colloidal Stability. J. Phys. Chem. A 2007, 111 (49), 12229–12237. (3)

Fukasawa, T.; Sato, T. Versatile Application of Indirect Fourier Transformation to Structure Factor Analysis: From X-Ray Diffraction of Molecular Liquids to Small Angle Scattering of Protein Solutions. Phys. Chem. Chem. Phys. 2011, 13 (8), 3187–3196.

(4)

Sumi, T.; Imamura, H.; Morita, T.; Isogai, Y.; Nishikawa, K. Model-Potential-Free Analysis of Small Angle Scattering of Proteins in Solution: Insights into Solvent Effects on Protein–protein Interaction. Phys. Chem. Chem. Phys. 2014, 16 (46), 25492–25497.

(5)

Ye, X.; Narayanan, T.; Tong, P.; Huang, J. Neutron Scattering Study of Depletion Interactions in a Colloid-Polymer Mixture. Phys. Rev. Lett. 1996, 76 (24), 4640–4643.

(6)

Lutterbach, N.; Versmold, H.; Reus, V.; Belloni, L.; Zemb, T. Charge-Stabilized Liquidlike Ordered Binary Colloidal Suspensions. 2. Partial Structure Factors Determined by Small-Angle Neutron Scattering. Langmuir 1999, 15 (6), 345– 352.

(7)

Nygård, K.; Konovalov, O. Decay of Interfacial Fluid Ordering Probed by X-Ray Reflectivity. Soft Matter 2012, 8, 5180–5186.

(8)

Gerstenberg, M. C.; Pedersen, J. S. Surface Induced Ordering of Triblock Copolymer Micelles at the Solid-Liquid Interface. 1. Experimental Results. Langmuir 2002, 18 (iv), 4933–4943.

(9)

Skoda, M.; Schreiber, F.; Jacobs, R. Protein Density Profile at the Interface of Water with Oligo(ethyleneglycol) Self-Assembled Monolayers. Langmuir 2009, 25 (9), 4056–4064.

(10) Kim, M. H.; Im, S. H.; Park, O. O. Fabrication and Structural Analysis of Binary

20


Page 20 of 32

Page 21 of 32 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Colloidal Crystals with Two-Dimensional Superlattices. Adv. Mater. 2005, 17 (20), 2501–2505. (11) Kaplan, P. D.; Rouke, J. L.; Yodh, A. G.; Pine, D. J. Entropically-Driven Surface Phase Separation in Binary Colloidal Mixtures. Phys. Rev. Lett. 1994, 72 (4), 582–585. (12) Richert, R.; Yang, M. Surface Induced Glass Transition in a Confined Molecular Liquid. J. Phys. Chem. B 2003, 107 (4), 895–898. (13) Stenberg, M.; Nygren, H. Computer Simulation of Surface-Induced Aggregation of Ferritin. Biophys. Chem. 1991, 41 (2), 131–141. (14) Miura, A.; Hikono, T.; Matsumura, T.; Yano, H.; Hatayama, T.; Uraoka, Y.; Fuyuki, T.; Yoshii, S.; Yamashita, I. Floating Nanodot Gate Memory Devices Based on Biomineralized Inorganic Nanodot Array as a Storage Node. Jpn. J. Appl. Phys. 2006, 45 (1–3), L1–L3. (15) Kruglova, O.; Demeyer, P.-J.; Zhong, K.; Zhou, Y.; Clays, K. Wonders of Colloidal Assembly. Soft Matter 2013, 9 (38), 9072–9087. (16) Mezger, M.; Shiröder, H.; Reichert, H.; Schramm, S.; Okasinski, J. S.; Schöder, S.; Honkimäki, V.; Deutsch, M.; Ocko, B. M.; Ralston, J.; et al. Molecular Layering of Fluorinated. Sicence 2008, 322 (17), 424–428. (17) Odiachi, P. C.; Prieve, D. C. Effect of Added Salt on the Depletion Attraction Caused by Non-Adsorbing Clay Particles. Colloids Surfaces A Physicochem. Eng. Asp. 1999, 146 (1–3), 315–328. (18) Wei, X.; Gong, X.; Ngai, T. Interactions between Solid Surfaces Mediated by Polyethylene Oxide Polymers: Effect of Polymer Concentration. Langmuir 2013, 29 (35), 11038–11045. (19) Gong, X.; Hua, L.; Wei, J.; Ngai, T. Tuning the Particle-Surface Interactions in Aqueous Solutions by Soft Microgel Particles. Langmuir 2014, 30 (44), 13182–

21



13190. (20) Toikka, G.; Hayes, R. a.; Ralston, J. Surface Forces between Spherical ZnS Particles in Aqueous Electrolyte. Langmuir 1996, 12 (16), 3783–3788. (21) Borkovec, M.; Szilagyi, I.; Popa, I.; Finessi, M.; Sinha, P.; Maroni, P.; Papastavrou, G. Investigating Forces between Charged Particles in the Presence of Oppositely Charged Polyelectrolytes with the Multi-Particle Colloidal Probe Technique. Adv. Colloid Interface Sci. 2012, 179–182, 85–98. (22) Sennerfors, T.; Fröberg, J.; Tiberg, F. Adsorption of Polyelectrolyte-Nanoparticle Systems on Silica: Influence on Interaction Forces. J. Colloid Interface Sci. 2000, 228 (1), 127–134. (23) Zeng, Y.; Von Klitzing, R. Oscillatory Forces of Nanoparticle Suspensions Confined between Rough Surfaces Modified with Polyelectrolytes via the Layer-by-Layer Technique. Langmuir 2012, 28 (15), 6313–6321. (24) Piech, M.; Walz, J. Y. Direct Measurement of Depletion and Structural Forces in Polydisperse, Charged Systems. J. Colloid Interface Sci. 2002, 253 (1), 117– 129. (25) Thormann, E.; Claesson, P. M.; Mouritsen, O. G. Tuning Structural Forces between Silica Surfaces by Temperature-Induced Micellization of Responsive Block Copolymers. Phys. Chem. Chem. Phys. 2010, 12 (36), 10730. (26) Richetti, P.; Kékicheff, P. Direct Measurement of Depletion and Structural Forces in a Micellar System. Phys. Rev. Lett. 1992, 68, 1951–1954. (27) Amano, K.; Liang, Y.; Miyazawa, K.; Kobayashi, K.; Hashimoto, K.; Fukami, K.; Nishi, N.; Sakka, T.; Onishi, H.; Fukuma, T. Number Density Distribution of Solvent Molecules on a Substrate: A Transform Theory for Atomic Force Microscopy. Phys. Chem. Chem. Phys. 2016, 18 (23), 15534–15544. (28) Grandner, S.; Zeng, Y.; Klitzing, R. V.; Klapp, S. H. L. Impact of Surface

22


Page 22 of 32

Page 23 of 32 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Charges on the Solvation Forces in Confined Colloidal Solutions. J. Chem. Phys. 2009, 131 (15). (29) Arai, S.; Tanaka, H. Surface-Assisted Single-Crystal Formation of Charged Colloids. Nat. Phys. 2017, 13 (5), 503–509. (30) E. J. W. Verwey; J. TH. G. Overbeek. Theory of the Stability of Lyophobic Colloids; Elsevier: Amsterdam, 1948. (31) Usui, S. Interaction of Electrical Double Layers at Constant Surface Charge. J. Colloid Interface Sci. 1973, 44 (1), 107–113. (32) Ohtsuka, Y.; Seki, T.; Takeoka, Y. Thermally Tunable Hydrogels Displaying Angle-Independent Structural Colors. Angew. Chemie - Int. Ed. 2015, 54 (51), 15368–15373. (33) Gottardo, S.; Sapienza, R.; García, P. D.; Blanco, A.; Wiersma, D. S.; López, C. Resonance-Driven Random Lasing. Nat. Photonics 2008, 2 (7), 429–432. (34) Akaike, H. A New Look at the Statistical Model Identification. IEEE Trans. Automat. Contr. 1974, 19 (6), 716–723. (35) Hurvich, C. M.; Tsai, C.-L. Regression and Time Series Model Selection in Small Samples. Biometrica 1989, 76, 297–307. (36) Derjaguin, B. V.; Rabinovich, Y. I.; Churaev, N. V. Measurement of Forces of Molecular Attraction of Crossed Fibres as a Function of Width of Air Gap. Nature 1977, 265 (5594), 520–521. (37) Oettel, M. Depletion Force between Two Large Spheres Suspended in a Bath of Small Spheres: Onset of the Derjaguin Limit. Phys. Rev. E 2004, 69, 41404. (38) Rentsch, S.; Pericet-camara, R.; Papastavrou, G.; Borkovec, M. Probing the Validity of the Derjaguin Approximation for Heterogeneous Colloidal Particles. Phys. Chem. Chem. Phys. 2006, 8, 2531–2538. (39) Kirkwood, J. G.; Boggs, E. M. The Radial Distribution Function in Liquids. J.

23



Chem. Phys. 1942, 10, 394–402. (40) Kubota, Y.; Akiyama, R. Assessment of Three-Dimensional Distribution Functions and Triplet Distribution Functions for Hard Spheres Calculated Using a Three-Dimensional Ornstein-Zernike Equation with Hypernetted-Chain Closure. J. Phys. Soc. Japan 2012, 81, SA017. (41) Chialvo, A. A.; Vlcek, L. Can We Describe Graphene Confined Water Structures as Overlapping of Approaching Graphene–Water Interfacial Structures? J. Phys. Chem. C 2016, 120, 7553−7561. (42) Amano, K.; Takahashi, O. Simple Transform Methods of a Force Curve Obtained by Surface Force Apparatus to the Density Distribution of a Liquid near a Surface. Physica A 2015, 425, 79–89. (43) Amano, K.; Tanaka, E.; Kobayashi, K.; Onishi, H.; Nishi, N.; Sakka, T. Force Measurement Reveals Structure of a Confined Liquid: Observation of the Impenetrable Space. Surf. Sci. 2015, 641, 242–246. (44) Amano, K.; Iwaki, M.; Hashimoto, K.; Fukami, K.; Nishi, N.; Takahashi, O.; Sakka, T. Number Density Distribution of Small Particles around a Large Particle: Structural Analysis of a Colloidal Suspension. Langmuir 2016, 32, 11063−11070. (45) Hansen, J.-P.; McDonald, I. R. Theory of Simple Liquids; Academic Press: London, 2006. (46) Karanikas, S.; Dzubiella, J.; Moncho-Jordá, A.; Louis, A. A. Density Profiles and Solvation Forces for a Yukawa Fluid in a Slit Pore. J. Chem. Phys. 2008, 128, 204704. (47) Malmberg, C. G.; Maryott, A. A. Dielectric Constant of Water from 0° to 100°C. J. Res. Natl. Bur. Stand. 1956, 56, 1. (48) González-Mozuelos, P.; Alejandre, J. Rogers-Young Approximation for the

24


Page 24 of 32

Page 25 of 32 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Concentration Profile of a Colloidal Suspension in Front of a Highly Repulsive Wall. J. Chem. Phys. 1996, 105 (14), 5949–5955. (49) Asakura, S.; Oosawa, F. Interaction between Particles Suspended in Solutions of Macromolecules. J. Polym. Sci. 1958, 33, 183–192. (50) Kinoshita, M. Roles of Translational Motion of Water Molecules in Sustaining Life. Front. Biosci. 2009, 14, 3419–3454. (51) Amano, K.; Hashimoto, K.; Sawazumi, R. Conversion of a Force Curve between Chemically the Same Surfaces into the Number Density Distribution of the Particles on the Surface Using a Structure Factor. arXiv 2016, 1612.07114. (52) Hashimoto, K.; Amano, K. A Theory for Calculating the Number Density Distribution of Small Particles on a Flat Wall from Pressure between the Two Walls. arXiv 2017, 1702.02720.

25



Page 26 of 32

Table Table 1. Parameters used in Fig. 3. In (a), (b), (c), and (d), φ, ρ0 , ψ P, and ψ W are variable parameters, respectively (underlined). In (a), d P is also varied according to the changes in φ to fix ρ0 , while in (b), it is varied according to the changes in ρ0 to fix φ. Fig. 3

φ (%)

ρ 0 (#/m 3 )

ψ P (mV)

ψW (mV)

D (nm)

d P (nm)

(a)

1 to 20

2.0×1021

−92

−20

8.8

(6φ/(πρ 0 ))1/3

(b)

2.4

10 21 to 10 23

−92

−20

10.3

(6φ/(πρ 0 ))1/3

(c)

2.3

4.7×1021

−200 to 0

−20

8.8

21

(d)

2.3

4.7×1021

−92

−200 to 0

8.8

21

26


Page 27 of 32 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Figures

Figure 1. Particle-induced force curves calculated from the raw force curves measured by Piech and Walz. 24 (a) ρ 0 is fixed at 2.0 × 10 21 #/m 3 . (b) φ is fixed at 2.4%. (c) ρ 0 is fixed at 4.7 × 10 21 #/m 3 .

27



Figure 2. g WP obtained from the particle-induce force curve. (a) gWP from Fig. 1(a). (b) g WP from Fig. 1(b). (c) g WP from Fig. 1(c). g WP is set to zero in the range from 0 to the (intrinsic) radius of the colloidal particle.

28


Page 28 of 32

Page 29 of 32 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Figure 3. Contour map of g WP as a function of a parameter calculated by OZ-HNC. (a) Effect of φ on gWP . (b) Effect of ρ 0 on g WP. (c) Effect of ψP on g WP. (d) Effect of ψW on g WP. d WP represents distance between the wall surface and the center of the colloidal particle.

29



Figure 4. Illustration of mechanism of the stratification of the colloidal particles near the wall. Here, the surface potentials of the colloidal particle and the wall are assumed to be both negative (positive). When the absolute value of the surface potential of the colloidal particle is high, the repulsion between the colloidal particle and the wall is strong. However, the colloidal particles tend to be densely structured on the wall as shown in Figs. 2(c) and 3(c). In the situation, the bulk space is highly crowded due to the strong repulsions among the colloidal particles. Hence, the interfacial stratification occurs in order to relax the crowding in the bulk.

30


Page 30 of 32

Page 31 of 32 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Figure 5. Decomposition of Φ WP into E WP and −TS WP . The thermodynamic quantities Φ WP , EWP , and −TS WP are calculated from the data on the dotted line in Fig. 3(c), where ψP = −50 mV. For a comparison, VWP is also shown.

31



TOC graphic

32


Page 32 of 32

Stratification of Colloidal Particles on a Surface: Study by a Colloidal

Recommend Documents