Number Density Distribution of Small Particles around a Large Particle

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Number Density Distribution of Small Particles around a Large Particle: Structural Analysis of a Colloidal Suspension Ken-ichi Amano, Mitsuhiro Iwaki, Kota Hashimoto, Kazuhiro Fukami, Naoya Nishi, Ohgi Takahashi, and Tetsuo Sakka Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.6b02628 • Publication Date (Web): 29 Sep 2016 Downloaded from http://pubs.acs.org on October 4, 2016

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Number Density Distribution of Small Particles around a Large Particle: Structural Analysis of a Colloidal Suspension

Ken-ichi Amano, *a Mitsuhiro Iwaki, bc Kota Hashimoto,a Kazuhiro Fukami,d Naoya Nishi, a Ohgi Takahashi,e and Tetsuo Sakkaa

a

Department of Energy and Hydrocarbon Chemistry, Graduate School of Engineering, Kyoto University, Kyoto 615-8510, Japan. E-mail: [email protected] b Quantitative Biology Center, RIKEN, Suita, Osaka 565-0874, Japan. c Graduate School of Frontier Biosciences, Osaka University, Suita, Osaka, 565-0874, Japan. d Department of Materials Science and Engineering, Graduate School of Engineering, Kyoto University, Kyoto 606-8501, Japan. e Faculty of Pharmaceutical Sciences, Tohoku Medical and Pharmaceutical University, Sendai 981-8558, Japan.

ABSTRACT Some colloidal suspensions contain two types of particles—small and large particles—to improve the lubricating ability, light absorptivity, etc. Structural and chemical analyses of such colloidal suspensions are often performed to understand their properties. In a structural analysis study, the observation of the number density distribution of small particles around a large particle (g LS ) is difficult because these particles are randomly moving within the colloidal suspension by Brownian motion. We obtained g LS using the data from a line optical tweezer (LOT) which can measure the potential of mean force between two large colloidal particles (Φ LL). We propose a theory that transforms Φ LL into g LS . The transform theory is explained in detail and tested. We demonstrate for the first time that LOT can be used for the structural analysis of a colloidal suspension. LOT combined with the transform theory will facilitate structural analyses of the colloidal suspensions, which is important for both understanding colloidal properties and developing colloidal products.

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1 Introduction Colloidal suspensions are used in various industrial products, such as coating materials, cosmetics, pharmaceuticals, and food products. Optimization of the physical and chemical properties of colloidal suspensions is indispensable to improve the quality of the products, and information about the microscopic structure of the colloidal particles is important to understand the various properties of the colloidal suspensions. The structural information for mixtures of two or more different colloidal particles is also important, and such mixtures are often used to improve the lubricating ability, light absorptivity, heat capacity, durability, etc., of the colloidal suspension. In the present paper, we focus on the structure of a binary colloidal suspension where small and large colloidal particles exist, and we experimentally obtain the number density distribution of small colloidal particles around a large colloidal particle (g LS ). However, the observation of g LS is difficult because the particles are randomly moving by Brownian motion. This difficulty is overcome by using a line optical tweezer (LOT)1 and proposing a transform theory. LOT can measure the potential of mean force between two large colloidal particles in a sea of small colloidal particles, and the transform theory can provide g LS from the potential of mean force. The combination of LOT and the transform theory is useful for the observation of g LS . The neutron contrast variation method of small-angle neutron scattering (SANS)2 is known as an experimental method for measuring g LS . SANS is helpful for measuring the structural properties of colloidal suspensions; however, it cannot be performed in a general lab space. Hence, LOT could provide a convenient method for the measurement of g LS. LOT can measure the potential of mean force between two large colloidal particles (Φ LL), but until now, g LS has not been obtained from Φ LL. For that reason, we propose a theory that can transform Φ LL into g LS. The transform theory is derived from a statistical mechanics a simple liquid. 3 The small and large colloidal particles are modeled as small and large spheres, respectively. In the transform theory, the dispersing media (liquid) is treated as a continuous fluid. The combination of LOT and the transform theory enables us to measure g LS in a general lab space without deuteration. This method will facilitate structural analyses of the colloidal suspensions, which is important for both understanding and developing colloidal products. Several apparatuses can measure subtle interactions between two surfaces. For example, LOT, one of the aforementioned optical tweezers, can measure the potential of mean force between two large colloidal particles. 1 Stepwise movements of kinesins4 and myosins5,6 have been measured by optical tweezers (kinesin and myosin are linear motor proteins). The forces during the stretching of a DNA chain 7 and a protein8,9 have also been measured by optical tweezers. Another technique for measuring this type of subtle interaction is atomic force microscopy (AFM). AFM can measure the oscillatory force 10–14 originating from the solvation layers on the surface of the substrate, which is 2


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the so-called solvation force. Furthermore, with a colloidal probe, AFM can measure the force between two colloidal particles. 15 Surface force apparatus (SFA) can also measure the oscillatory forces in liquids 16–19 and colloidal suspensions. 20 LOT, AFM, and SFA can measure the subtle interactions between two surfaces; however, the observation of the number density distribution of small spheres on a sample surface from the measured subtle interaction is one of the difficult tasks in this area of study. The theories that transform from the subtle interactions to the number density distributions in the research fields of AFM 21–24 and SFA 25,26 have already been studied. However, such a study has not been performed for LOT. The layer structures of colloidal particles on some surfaces have been studied from theoretical and experimental points of view. For example, fundamental measure theory (FMT)27–29 and integral equation theory (IET)30–32 are well known within the field. The simplest and most constitutive model of a colloidal suspension is an ensemble of small hard spheres, and FMT and IET can very precisely calculate the layer structure using the model. From these theories, it has been found that the granularity (i.e., the volumes of the colloidal particles themselves) is essential for the formation of the layer structure. Experimentally, the layer structure is studied by observing the oscillatory force curve between a colloidal prove and a sample surface using colloidal probe AFM. 33–35 The oscillatory force curve represents the existence of the layer structure. The influences of the volume fraction, surface roughness, etc., on the oscillatory force curve are also studied using colloidal probe AFM. The observation of the oscillatory force is a remarkable achievement; however, it would be valuable if we could also evaluate the number density distribution of the colloidal particles on the sample surface from the force data. We believe that the proposition of a theory that can transform from the measured force curve into the number density distribution would enable us to analyze the layer structure in more detail. Moreover, the transform theory would bridge the gap between theory and experiment. In this paper, we propose a transform theory to calculate g LS (the number density distribution of small colloidal particles around a large colloidal particle) from the input data of Φ LL (the potential of mean force between the two large colloidal particles). This paper is composed of five chapters. In Chapter 2, the transform theory is explained in detail. We propose two processes for the calculation of g LS . In one process, we focus on the contact number density of the small colloidal particles around the large colloidal particle. Thus, it is called the contact density (CD) process. In the other process, we focus on the pressure on a surface element of the large colloidal particle’s excluded volume. Hence, it is called the surface element (SE) process. Eventually, both processes are proven to be the same. Hence, we collectively call them particle pair (PP) transform. In our opinion, both processes are important because the CD process justifies the unclear Derjaguin’s assumption 36–38 used in the SE process. In Chapter 3, the computational details for the verification tests of the PP transform are described. In 3


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Chapter 4, the results of the theoretical (computational) tests of the PP transform are shown. In addition, the experimentally obtained Φ LL1 is transformed into g LS. Finally, we present the conclusions in Chapter 5.

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2 Theory 2.1 Calculation of the density distribution using the CD process We illustrate the model system for the CD (contact density) process in Fig. 1, where large spheres 1 and 2 with radii, r L, are immersed in an ensemble of small spheres. The radius of the small sphere is rS . The bulk number density of the small spheres is ρ 0 . The separation between the centers of the two large spheres is expressed as s. The distance between the center of large sphere 1 and the center of the small sphere in contact with large sphere 2 is expressed as l. The meaning of θ is illustrated in Fig. 1.

Figure 1. System configuration of the CD process.

The large sphere has a radial density distribution of small spheres around itself, and the radial density distributions of the two large spheres overlap when they approach each other. The overlap creates a complex distribution of small spheres within the confined space. In a simple form, the overlap can be expressed by the Kirkwood superposition approximation: 39–41 ; , 1

where g W, g LS1 , and g LS2 represent the normalized number density distribution of the small spheres in the whole system and the normalized number density distributions around large spheres 1 and 2, respectively. We notice that g LS1 and g LS2 are the same (g LS1 = g LS2 = g LS ), and they are the distributions in the bulk (in the isolated condition). Hereafter, we simply mention the normalized number density distribution as the density distribution. The three-dimensional (3D) vector, ɤ, starts from the center of large sphere 1, and it indicates an arbitrary position from the center. The 3D vector, s, 5


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represents the separation between the centers of large spheres 1 and 2 and the direction. The Kirkwood superposition approximation is exact when large spheres 1 and 2 are sufficiently separated and/or when there are only three bodies (i.e., one small sphere and two large spheres) in the system. However, as the bulk number density of the small spheres increases, the approximation becomes invalid. Next, we employ an approximation that the force (f) acting on large sphere 2 depends only on the contact density of the small spheres. 25,29,42 That is, the two-body potential between the large and small spheres is approximated as a rigid potential. In this case, f is expressed as follows: 20,25,29,42

= 2

(/

, !sin cos ' , 2

where f simply means:

=

' * +

. 3

'

The functions Φ LL and u LL represent the potential of mean force and the (bare) two-body potential between the two large spheres, respectively. The subtracted product (Φ LL − u LL) represents the potential induced by the small spheres. Φ LL is exactly related to the normalized number density between the two spheres (g LL) as follows:

= exp1* /2 3, 4

where k B and T are the Boltzmann constant and absolute temperature, respectively. The symbol g LSC represents the number density at the contact point; π is the circular constant and r ≡ r L + rS . The symbol, f B (constant), is the force acting on the back (right side) of large sphere 2, which is expressed as:

= 2

(/

sin cos ' = . 5

As shown in Fig. 1, l can be expressed as:

= 6 + 2 sin . 6

Differentiation of l with respect to θ is written as:

6


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64 + ' = . 7

' 2 Furthermore, sinθ and cosθ are written as:

+ , 8

2

sin

=

cos

=;

4 + . 9

4

Consequently, Eq. (2) can be rewritten as: √> + = >BA

? @A ?

+

' . 10

Eq. (10) can be expressed in the form of a matrix operation, as follows:

D∗ = FG , 11

where F * corresponds to the left-hand side of Eq. (10). G1 and H correspond to g LS1 (l) and all the remaining parts, respectively. We use sufficiently large matrices of F *, G 1 , and H . The boundary conditions of F * are (f(2r) + f B )/(ρ 0 k B Tπg LSC ) at the upper side and f B /(ρ 0 k B Tπg LSC ) at the lower side. The latter boundary condition originates from f(s∞ ) = 0, where s ∞ represents a sufficiently long separation. The boundary conditions of G 1 are g LS1 (r) at the upper side and 1 at the lower side. g LS1 (r) is the contact value of the density distribution that is the same as g LSC , and it is an unknown value at this stage. To solve Eq. (11), we set H as a square matrix whose variables are l and s. A cell of matrix H is composed of l(s2 + r 2 + l 2 )dl/s 2 inside the integral range and 0 outside the integral range. At the lower right area of H , where correspondent g LS1 (l) converges to 1, the integral range protrudes from the matrix area. For such an area, the block matrix is a square unit matrix multiplied by f B /(ρ 0 k B Tπg LSC ). To obtain G1 , an arbitrary initial value is substituted into g LSC , and a trial F * is prepared. G1 is numerically predicted by multiplying the inverse matrix of H by F *. Because large spheres 1 and 2 are equivalent, when the output g LS1 (r) at the contact is equal to g LSC of the input value, the numerical calculation converges. The convergence is, for instance, operated by a bisection method.

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2.2 Calculation of the density distribution using the SE process The outline of the SE (surface element) process is as follows. First, the force curve between the two large spheres is converted to pressure on a projected surface element of the large sphere’s excluded volume (see Fig. 2). The excluded volume of the large sphere is defined as the volume that the center of the small sphere cannot enter. We call this conversion the FPSE conversion, where FPSE means force to pressure on a surface element. Second, the pressure on the surface element is transformed into a density distribution of small spheres around the large sphere by applying the transform theory recently proposed by Amano. 25

Figure 2 . System configuration of the SE process.

The FPSE conversion is an exact version of the Derjaguin approximation. 36–38 The Derjaguin approximation has been applied in many studies due to its universality and validity. However, its applicability is limited to large particles, and it is restricted to very short surface–surface separations. Bhattacharjee and Elimelech also proposed an exact version of the Derjaguin approximation called the surface element integration (SEI). 43,44 The difference between the SEI and FPSE conversion is that SEI can convert the interaction between two parallel flat substrates into the interaction between two spheres, while the FPSE conversion can change its inverse transformation. The applicability of the FPSE conversion is not limited to large particles and is not restricted to very short surface–surface separations because the FPSE conversion is an exact version of the Derjaguin approximation. (We note that the three methods, the Derjaguin approximation, SEI conversion, and FPSE conversion, each contain Derjaguin’s assumption, which is explained in the next paragraph and Eq. (12).) The system configuration for the SE process is shown in Fig. 2. There are many small spheres with a bulk density, ρ 0 ; large spheres 1 and 2 are immersed in the ensemble of small spheres. The space where the center of the small sphere cannot enter 8


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is the excluded volume of the large sphere, and r is the radius of the excluded volume. rS and r L are the radii of the small and large spheres, respectively. The meaning of θ is illustrated in Fig. 2. The separation between the centers of large spheres 1 and 2 is represented as s. The length of the horizontal line between a pair of surface elements of the excluded volumes is represented as lʹ. If the force between the two large spheres, f, is expressed by a summation of pressures on the surface elements. Thus, the force can be written as:

= H I J KL ′; , 12

NO

where P is the pressure along the x axis and A 2 x is the projected flat area of the surface element of the excluded volume of large sphere 2. We note that Eq. (12) represents Derjaguin’s assumption. 36–38 In the present case, Eq. (12) can be rewritten as: P

= 2 I′ sin cos ' . 13

Here, P is equal to the pressure from the left side plus that from the right side of large sphere 2. In accordance with Fig. 2, l′ can be written as:

J = 2sin . 14

Hence, the following two expressions are obtained:

cos ' = 1/2 '′, 15

sin = ′ /2 . 16

Then, Eq. (13) is rewritten as:

2 / =

>

>BA

I J J '′. 17

If we differentiate both sides of Eq. (17) with respect to s, we obtain: > 2 Q

= 2I 2 + I J ' J . 18 Q >BA

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Moreover, if we eliminate the integral term of Eq. (18), the Derjaguin approximation for the granular system 37 can be obtained. The Derjaguin approximation for the granular system indicates that (i) the medium around the large sphere is not a continuum fluid but an ensemble of small spheres and (ii) the surface element of the excluded volume is considered as the surface element. However, in the SE process, we would like to obtain P from f without eliminating the integral term. Then, we make use of the matrix operation below:

D# = ST, 19

where F # corresponds to the left-hand side of Eq. (17). P and J correspond to P(lʹ) and all the remaining parts, respectively. We use sufficiently large matrixes of F #, P, and J . The boundary conditions of F # are 2f(2r)/π at the upper side and 0 at the lower side because f(s) = 0 at a sufficiently large separation. The boundary conditions of P are P(0) at the upper side and 0 at the lower side. To obtain P from Eq. (19), we set J as a square matrix whose variables are lʹ and s. A cell of matrix J is composed of (s – lʹ)dlʹ inside the integral range and 0 outside the integral range. At the lower right of J , the integral range protrudes from the matrix area. For such an area, the block matrix is a square unit matrix multiplied by 2r 2. P is numerically obtained by multiplying the inverse matrix of J by F # . This is an example of the solving process of Eq. (19), and the FPSE conversion is finished here. We found that the result of the FPSE conversion is similar to the result of the Derjaguin approximation for the granular system; when r L >> rS (r ≈ r L), both results are almost the same. Next, we transform P (i.e., P(lʹ)) into g LS1 . In this step, we approximate that the force acting on large sphere 2 depends only on the contact density of the small spheres. 25,29,42 By applying this approximation, g LS1 is readily calculated as follows:25

|V + WJ | =

I ′

+ 1, 20

where

=

1 + 61 + 4I0 /

. 21

2

The bold characters r and l′ represent the 3D vectors of r and l′, respectively. The start point of r is the center of large sphere 1 and its end point is connected to the start point of l′.

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2.3 Relation between the CD and SE processes Here, we demonstrate that the CD and SE processes are identical, where we suppose that the force acting on the large sphere depends only on the contact density of the small spheres. We show that Eq. (17) in the SE process leads to Eq. (10) in the CD process. First, the pressure on the surface element of large sphere 2 is given by: 25

I′ = X ′ /KL ′ = |V + WJ | . 22

Note that Eq. (22) indicates that the force acting on the surface element (f SE) is the sum of the forces from the right and left sides. Substituting Eq. (22) into Eq. (17), we obtain:

=

> |V + WJ | 1 J '′. 23

2 >BA

Next, the relation between l and lʹ can be obtained by comparing Figs. 1 and 2:

J = /. 24

Thus, Eq. (23) is rewritten as:

=

√>? @A ?

>BA

|V + WJ | 1 +

'. 25

Since the value of g LS1(|r + l ʹ|) is equal to that of g LS1 (l), Eq. (25) is rewritten as:

=

√>? @A ?

>BA

+

' . 26

Eq. (26) is the same as Eq. (10). Therefore, the values of g LS1 calculated by the CD and SE processes are the same. In our opinion, the CD process is important for corroborating the validity of Derjaguin’s assumption in the SE process.

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3 Computational details Because the CD (contact density) and SE (surface element) processes produce the same results, we collectively call these processes the PP (particle pair) transform. In this chapter, we explain the computational details of the tests for the PP transform. In the tests, we employ rigid systems and a one-dimensional Ornstein–Zernike equation coupled with a hypernetted-chain closure (1D-OZ-HNC). 3,45–47 A bridge function proposed by Verlet, Choudhury, and Ghosh (VCG)30 is incorporated in 1D-OZ-HNC to precisely perform the verification test. The incorporated VCG bridge function is useful because the function becomes a bridge function of the Verlet type 31,32 (a highly sophisticated bridge function for the rigid system) when the system is rigid. The 1D-OZ-HNC with the VCG bridge function generates reliable results for the tests. The verification test is performed in a computationally closed cycle. The outline of the verification test is as follows. In step (A), an ensemble of small spheres is prepared in a radially symmetric space, and a pair correlation function between the two small spheres is computed. In step (B), a single large sphere is immersed in the ensemble of small spheres, and the density distribution of the small spheres around the large sphere (g O ) is calculated. We mention g O as the original density distribution. In step (C), one additional large sphere is immersed in the system, and the potential of mean force between the two large spheres (Φ LL) is computed. In step (D), Φ LL is changed to the mean force through Eq. (3). Then, in step (E), the density distribution of the small spheres around the large sphere (g LS ) is calculated from the force curve using the PP transform. The reproducibility of the PP transform is validated when the g LS calculated using the PP transform agrees with the original one (g O). The model systems for the verification tests are as follows. The diameters of the large and small spheres are d L and d S, respectively. Here, d S is an arbitrary short length, which is a unit length for the calculation. Several values of d L were tested, but in this paper, only the results obtained in the case of d L = 10d S are shown. (Almost the same reproducibilities were also obtained when d L = d S, 2d S , 3d S, 4d S , 5d S , 15d S , 20dS , etc.) The grid spacing is set at dS /50, and the number of grid points is 2048. The number of grid points (2048) is sufficiently large because the center of the large sphere is placed on the boundary of the grid line. The bulk number density of the small spheres is ρ 0 . The tested volume fractions (φ S) are 0.1, 0.2, 0.3, and 0.4 (φ S = ρ 0 d S 3 π/6). The tested results obtained for the rigid systems are shown in Section 4.1, and those for the non-rigid systems are shown in the Supporting Information. In the derivation process for the PP transform (CD and SE processes), the two-body potential between the large and small spheres is supposed to be rigid, but the other two-body potentials were arbitrary. Hence, the PP transform can be used in an ensemble of soft small spheres. We found that the PP transform can also work well in the ensemble of soft small spheres. Moreover, the PP transform can be used even when the two-body 12


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potential between the large and small spheres is soft. However, we found that when the depth of the potential minimum is much lower than –k B T, the reproducibility of the PP transform is not very high. As a result, we conclude that the PP transform works well when the two-body potential between the large and small spheres is close to the rigid potential.

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4 Results and discussion 4.1 Tests for the PP transform We show the results of the tests conducted for the four volume fractions. Fig. 3 shows the potential of mean force between the large spheres (Φ LL) as a function of the distance between the centers of the large spheres calculated by 1D-OZ-HNC-VCG. As observed in Fig. 3, the amplitude of the curve increases as the volume fraction (φ S ) increases. By using the PP transform, the curves are transformed into the number density distributions of the small spheres around the large sphere (see Fig. 4). The bold solid line represents the original number density, gO, which is the benchmark structure. The thin solid line represents g LS obtained from Φ LL. As shown in Fig. 4, the reproducibility of the number density is high when the volume fraction is low. This trend originates from the Kirkwood superposition approximation because the approximation becomes precise as the volume fraction decreases.

Figure 3 . Potential of mean force between the two large spheres calculated by 1D-OZ-HNC-VCG.

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Figure 4 . Comparison among g O, g LS , and ğ LS . φS = 0.1, 0.2, 0.3, and 0.4 in (A), (B), (C), and (D), respectively.

The thin solid line in Fig. 4(D) shows the wrongly underestimated minimum around r LS /d S = 5.8, the shape of which is unusual compared to the benchmark structure. To improve the g LS , it is approximated as follows:

= exp1* /2 3 1 * /2 , 27

where Φ LS is the potential of mean force between the large and small spheres. Then, the modified density distribution, ğ LS , is empirically expressed as:

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ğ = when ≥ 1, 28a ğ = exp1 13 when < 1. 28b The introduction of the method corrects the wrongly underestimated g LS. In all of the tested cases, we found that ğ LS is better than g LS. The layer spacings of g LS (ğ LS ) are shorter than that of gO. The reason for the decrease is as follows. When the small spheres are confined between the two large spheres, the layer structure is compressed. However, the PP transform cannot correct for the compression (shortening) of the layer spacings because it uses the Kirkwood superposition approximation to treat the density distribution of the small spheres in the confined space. To obtain more precise results, this problem should be solved in the future.

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4.2 Practical use of the PP transform In this section, the PP transform is applied to LOT (line optical tweezer) to obtain the density distribution of small colloidal particles around a large colloidal particle from the experimental data. Crocker et al. 1 measured the potential of mean force between poly(methyl methacrylate) (PMMA) particles in a colloidal suspension of polystyrene (PS) particles. The diameters of the PS and PMMA particles are 83 nm (Seradyn, Inc.) and 1100 ± 15 nm (Bangs Labs, Inc.) in products, respectively. The colloidal suspension contains 5 mM NaCl and 5 mM sodium dodecyl sulfate, which prevents aggregation. 1 The screening lengths of the repulsive interactions between the colloid particles are about 3 nm. 1 Since the screening length is so small compared to the diameters of the colloid particles, they are treated as hard spheres. 1 However, the repulsive interaction between the PS particles causes its effective diameter to be slightly larger than the actual diameters. Crocker et al. applied the Asakura-Oosawa theory, 48,49 and estimated the effective diameter of a PS particle as 97 ± 6 nm. 1 However, it is likely that the effective diameter of a PS particle is overestimated because the diameter was determined in the low volume fractions. Therefore, we assume the effective diameter of a PS particle as 91 nm (97 nm – 6 nm, taken from “± 6”). However, we do not need to determine the effective diameter of a PMMA particle here because the PS-induced potential between the PMMA particles (Fig. 5) almost automatically yields the value. The PS-induced potential between the PMMA particles in φ PS = 0.34 (φPS : volume fraction of PS particles) is calculated by subtracting the (bare) two-body potential between the PMMA particles from the potential of mean force between the PMMA particles. The (bare) two-body potential is obtained from the potential of mean force measured as φ PS = 0.00. The volume fraction of PMMA particles is less than 10 –7 in both cases (φ PS = 0.00 and 0.34). 1 The fitting is performed using a polynomial function of sexp , where sexp is the separation between the centers of the PMMA particles.

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Figure 5 . Potential between PMMA particles in the colloidal suspension. The thin and bold solid lines are PS-induced potentials (the same thing). 1 The bold solid line is used for the input of the PP transform. The dotted line is an arbitrarily drawn potential curve that contains PS-induced and steric repulsive potentials.

The PP transform is performed using the bold solid line in Fig. 5. At first, the PS-induced potential (bold solid line) is transformed into the pressure (P(l′)) on the projected surface element of PMMA’s excluded volume using the FPSE conversion. It corresponds to the pressure between the two flat surfaces. Secondly, the point of the first maximum peak of P(l′) is determined, which corresponds to P(0). When the separation between two flat surfaces is just equal to the diameter of a small colloidal particle, the force between the two flat surfaces takes a maximum value. 50,51 Thirdly, the contact density of PS particles around a PMMA particle is calculated using Eq. (21). Finally, the density distribution of PS particles around a PMMA particle is obtained using Eq. (20). Fig. 6 is the density distribution of PS particles around a PMMA particle (ğ exp ), where the modification (Eq. (28)) is used. The ratios of the amplitudes of the potential and the density are not the same (by comparing Figs. 5 and 6). There are three observable layers in ğexp , which is very similar to the liquid structure around a solute. 52–58 Since three PS layers are semi-adsorbed around a PMMA particle, the 18


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layers may move with the PMMA particle. That is, when the dynamics of the PMMA particle are studied in the PS colloidal suspension, the layer structure around PMMA should be considered.

Figure 6. Normalized number density of PS particles around a PMMA particle.

Next, we use Eq. (1) to calculate the density distribution of the PS particles around a pair of PMMA particles. Fig. 7(A) shows the density distribution at sexp = 1.3 µm. The repulsive force arises between the PMMA particles when s exp = 1.3 µm (from the slope of Fig. 5). In this situation, the contact density of PS particles on the inside of PMMA particles is expected to be high compared to that on the outside of the PMMA particles. Actually, as shown in Fig. 7(B), the inside contact density is relatively high compared to the outside contact density. This density distribution generates the repulsive force between the PMMA particles.

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Figure 7. Normalized number density of PS particles around a pair of PMMA particles (A), and its magnified view (B). The separation between the centers of PMMA particles (s exp) is 1.3 µm. The white areas represent the excluded volumes of the PMMA particles, in which the centers of the PS particles cannot enter. (C) Color scale of the normalized number density.

Crocker et al. predicted “the formation of shells around the large sphere” 1 from their experimental results. Consequently, we were able to visualize their prediction. Furthermore, they discovered a dramatic slowing of Brownian motion for the trapped PMMA particles for a high volume fraction of PS particles. In our opinion, the slowing is closely related to the layer structure of PS particles around a PMMA particle.

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5 Conclusions We have proposed a theory (PP transform) that transforms from the potential of mean force between two large colloidal particles to the density distribution of small colloidal particles around the large colloidal particles. The PP transform was derived through two processes: the CD (contact density) and SE (surface element) processes. We demonstrated that the CD and SE processes are mathematically identical. Verification tests of the PP transform were conducted in a computationally closed cycle. The PP transform reproduced the density distribution of small spheres around a large sphere. The potential of mean force between the PMMA particles measured by LOT (line optical tweezer) was transformed into the density distribution of PS particles around a PMMA particle for the first attempt using the PP transform for practical use. We demonstrated that LOT combined with the PP transform can be used for the structural analysis of colloidal suspensions. We believe that LOT combined with the PP transform will help studies of shear force, lubricating ability, etc., because the density distribution of the colloidal particles is closely related to the mechanisms. In addition, we expect that the combination of fluorescence microscopy and the PP transform can also be used for the structural analyses of colloidal suspensions because fluorescence microscopy can also detect the potential of mean force between two labeled colloidal particles. We believe that this study makes the structural analyses of the colloidal suspensions easier and facilitates developments of colloidal products (e.g., coating materials, cosmetics, pharmaceuticals, and food products) because LOT combined with the PP transform can measure the density distribution without SANS and deuteration.

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Associated Content Supporting Information The Supporting Information is available free of charge on the ACS Publications website at DOI: . Results of the computational test of PP transform in non-rigid systems (PDF)

Author Information Corresponding author *E-mail: [email protected] Notes The authors declare no competing financial interest

Acknowledgements We appreciate support from Masahiro Kinoshita, Kozue Maruta, and Ryosuke Sawazumi. This work was supported by a Grant-in-Aid for Young Scientists (B) from Japan Society for the Promotion of Science (15K21100) and partly supported by a Grant-in-Aid for Scientific Research (B) from Japan Society for the Promotion of Science (15H03877).

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Graphical abstract (TOC graphic)

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Number Density Distribution of Small Particles around a Large Particle

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