5. The Fourier method

observation plane, where a CCD camera sensor is placed to register an image of the resulting

In Figure 7, a set of four phase-shifted interferograms with phase shifts of π/2 is shown. A first interferogram with θ = 0 is seen in Figure 7a, and subsequent interferograms with θ = π/2, θ = π and θ = 3π/2 are observed in Figure 7b–d, respectively. The wrapped phase is showed in Figure 7e, and the unwrapped phase related with the shape of the optical element being tested is shown in Figure 7f. A three-dimensional representation of the unwrapped phase seen in

In applied phase-shifting interferometry, there are concerns about the presence of errors that may affect the accurate phase extraction from phase-shifted interferograms. A typical systematical source of error introduced PZT arises when there is miscalibration of the phaseshifting actuator, causing detuning in the phase extraction process. The inclusion of phaseshifting algorithms with more than three or four interferograms can be implemented in order

Figure 7. Four-step phase-shifted interferograms from (a) to (d), the wrapped phase (e) and the unwrapped phase (f).

interferogram that is then stored in a PC for subsequent processing.

Figure 7f is presented in Figure 8.

10 Optical Interferometry

to reduce this systematic error [5].

The method of Takeda or the Fourier method was developed in 1982 [6]. Unlike phase-shifting methods, see Ref. [7], a single interferogram with open fringes can be analyzed. This is useful when the object under study changes dynamically or environmental disturbances (vibrations or air turbulence) do not allow the use of phase-shifting methods unless special, and often, very expensive hardware is used to acquire several images simultaneously. However, in general, the accuracy and the dynamical range of the phase that can be measured are reduced. The Fourier method makes use of the fast Fourier transform technique to separate, in the frequency domain, the background and phase terms of the interferogram. Employing complex notation, an interferogram can be written as follows:

$$I(\mathbf{x}, y) = a(\mathbf{x}, y) + \frac{1}{2}b(\mathbf{x}, y)e^{i[\phi(\mathbf{x}, y) + \phi(\mathbf{x}, y)]} + \frac{1}{2}b(\mathbf{x}, y)e^{-i[\phi(\mathbf{x}, y) + \phi(\mathbf{x}, y)]},\tag{20}$$

where a(x, y) and b(x, y) are the background intensity and the modulation term, respectively. The phase term is denoted by φ(x, y) and finally, the symbol ϕ(x, y) denotes the lineal carrier function or tilt that is introduced usually by tilting the reference mirror in a two arm interferometer. The Fourier transform of the above expression can be written as:

$$\mathcal{F}\{I(\mathbf{x},y)\} = \hat{I}(\boldsymbol{\mu},\boldsymbol{\upsilon}) = \delta(\boldsymbol{\mu},\boldsymbol{\upsilon}) + E(\boldsymbol{\mu}+\boldsymbol{a},\boldsymbol{\upsilon}+\boldsymbol{\beta}) + E^\*(\boldsymbol{\mu}-\boldsymbol{a},\boldsymbol{\upsilon}-\boldsymbol{\beta}),\tag{21}$$

where (u, v) is the coordinates in the frequency domain, δ(u, v) is a delta function and E(u + α, v + β) and E\*(u-α, v-β) are complex conjugate functions that correspond to the transforms of the second and third terms of Eq. (20), respectively. The introduction of the linear carrier function, ϕ(x, y) = αx + βy, shifts the terms E(u, v) and E\*(u, v) in opposite directions in the frequency spectrum as can be seen noting that:

$$E(u,v) = \mathcal{F}\left\{\frac{1}{2}b(x,y)e^{i\phi(x,y)}\right\},\tag{22}$$

and

$$E(\mu+\alpha, \upsilon+\beta) = \mathcal{F}\left\{\frac{1}{2}b(\mathbf{x}, y)e^{i\phi(\mathbf{x}, y)}e^{i\wp(\mathbf{x}, y)}\right\},\tag{23}$$

similarly

$$E^\*(\mu, \upsilon) = \mathcal{F}\left\{\frac{1}{2}b(\mathbf{x}, y)e^{-i\phi(\mathbf{x}, y)}\right\},\tag{24}$$

and

$$\mathcal{E}^\*(\mu-\alpha, \upsilon-\beta) = \mathcal{F}\left\{\frac{1}{2}b(\mathbf{x}, y)e^{-i\phi(\mathbf{x}, y)}e^{-i\psi(\mathbf{x}, y)}\right\}.\tag{25}$$

Figure 9. Interferogram with open fringes (a) and its Fourier spectrum (logarithm of the module) (b).

The separation of the terms in the Fourier spectrum due to the introduction of a linear carrier can be observed in Figure 9. An open-fringe interferogram and its Fourier spectrum can be seen in Figure 9a and b. It can be noted that the three terms of Eq. (21) are clearly separated. The central peak corresponds to the delta function δ(u, v) while the adjacent lobules correspond to E(u + α, v + β) and E\*(u - α, v - β). The interferogram was constructed as follows:

$$a(x, y) = 120e^{-(x^2 + y^2)} \tag{26}$$

$$b(x, y) = 124e^{-(x^2 + y^2)},\tag{27}$$

$$\phi(\mathbf{x}, y) = 2\pi \left\{ 4e^{-4\left[\left(\mathbf{x} - 0.2\right)^2 + \left(y - 0.3\right)^2\right]} \mathbf{-} \mathbf{S} e^{-6\left[\left(\mathbf{x} + 0.2\right)^2 + \left(y + 0.3\right)^2\right]} \right\},\tag{28}$$

and

function or tilt that is introduced usually by tilting the reference mirror in a two arm interfer-

where (u, v) is the coordinates in the frequency domain, δ(u, v) is a delta function and E(u + α, v + β) and E\*(u-α, v-β) are complex conjugate functions that correspond to the transforms of the second and third terms of Eq. (20), respectively. The introduction of the linear carrier function, ϕ(x, y) = αx + βy, shifts the terms E(u, v) and E\*(u, v) in opposite directions in the frequency

> 2 bðx, yÞe

> > 2 bðx, yÞe

2 bðx, yÞe

2 bðx, yÞe

iφðx, yÞ

> iφðx,yÞ e iϕðx,yÞ

−iφðx,yÞ

> −iφðx, yÞ e −iϕðx,yÞ

ðu−α, v−βÞ, (21)

, (22)

, (23)

, (24)

: (25)

ometer. The Fourier transform of the above expression can be written as:

spectrum as can be seen noting that:

and

12 Optical Interferometry

similarly

and

FfIðx, yÞg ¼ ~Iðu, vÞ ¼ δðu, vÞ þ Eðu þ α, v þ βÞ þ E�

Eðu, vÞ ¼ F 1

ðu, vÞ ¼ F 1

ðu−α, v−βÞ ¼ F 1

Figure 9. Interferogram with open fringes (a) and its Fourier spectrum (logarithm of the module) (b).

Eðu þ α, v þ βÞ ¼ F 1

E�

E�

$$
\varphi(\mathbf{x}, y) = 2\pi (16\mathbf{x} + 20y) \tag{29}
$$

where x and y vary from 1 to −1 along the vertical and horizontal directions and the interferogram was multiplied by a circular function of radius one.

In order to recover the phase, we need to isolate one of the lateral lobules of the Fourier domain. To this end, we employ a band-pass filter. The filtered spectrum is then transformed back to the spatial domain to obtain If(x, y) and the wrapped phase is found with the arctangent function of the ratio of the imaginary and real parts of If(x, y) as shown in Figure 10. Figure 10a shows the band-pass filter H(u, v), the filtered spectrum can be observed in Figure 10b and finally the wrapped phase is shown in Figure 10c.

The band-pass filter has the following form:

$$H(\mu, \upsilon) = e^{-3000 \left[ \left( \mu - 0.2 \right)^2 + \left( \upsilon - 0.2 \right)^2 \right]^2},\tag{30}$$

Figure 10. Band-pass filtering process of the Fourier spectrum seen in Figure 9. Pass-band filter (a), filtered spectrum (b) and recovered wrapped phase (c).

where u and v vary from 1 to −1 in vertical and horizontal directions, respectively. The filtered spectrum becomes,

$$
\tilde{I}\_f(\mu, \upsilon) = \tilde{I}(\mu, \upsilon) H(\mu, \upsilon). \tag{31}
$$

Transforming back to spatial domain we obtain:

$$I\_f(\mathbf{x}, y) = \mathcal{F}^{-1}\left\{\tilde{I}\_f(\mu, v)\right\}.\tag{32}$$

The wrapped phase is found by:

$$\psi\_w(\mathbf{x}, y) = \operatorname{atan2}\left\{ \operatorname{imag}[I\_f(\mathbf{x}, y)], \operatorname{real}[I\_f(\mathbf{x}, y)] \right\} \tag{33}$$

where the atan2() function accepts two arguments corresponding to the sine and cosine and returns the results modulo 2π. This wrapped phase, however, is not the desired one because it contains the introduced tilt that is not part of the object information. The wrapped desired phase is found with:

$$\phi\_w(\mathbf{x}, y) = \operatorname{atan2}\left\{ \sin[\psi\_w(\mathbf{x}, y) - \varphi(\mathbf{x}, y)], \cos[\psi\_w(\mathbf{x}, y) - \varphi(\mathbf{x}, y)] \right\}.\tag{34}$$

The final step is to apply an unwrapping method to obtain the continuous phase related with the object under study. This last procedures are shown in Figure 11a and b where the wrapped phase ϕw(x, y) and the unwrapped reconstructed phase ϕr(x, y) can be appreciated, respectively.

The Fourier method is not the unique procedure for phase retrieval from one interferogram with open fringes. Besides the Fourier approach there are other procedures in the spatial domain including phase locked loop [8] and spatial carrier phase-shifting methods [9, 10].

Figure 11. Phase reconstruction results. Wrapped phase (a) and unwrapped continuous phase (b).

## 6. Phase unwrapping

where u and v vary from 1 to −1 in vertical and horizontal directions, respectively. The filtered

n ~Ifðu, vÞ o

where the atan2() function accepts two arguments corresponding to the sine and cosine and returns the results modulo 2π. This wrapped phase, however, is not the desired one because it contains the introduced tilt that is not part of the object information. The wrapped desired

The final step is to apply an unwrapping method to obtain the continuous phase related with the object under study. This last procedures are shown in Figure 11a and b where the wrapped phase ϕw(x, y) and the unwrapped reconstructed phase ϕr(x, y) can be appreciated, respectively. The Fourier method is not the unique procedure for phase retrieval from one interferogram with open fringes. Besides the Fourier approach there are other procedures in the spatial domain including phase locked loop [8] and spatial carrier phase-shifting methods [9, 10].

imag½Ifðx, yÞ�,real½Ifðx, yÞ�

sin½ψwðx, yÞ−ϕðx, yÞ�, cos½ψwðx, yÞ−ϕðx, yÞ�

Ifðx, yÞ ¼ F−1

n

ψwðx, yÞ ¼ atan2

n

Figure 11. Phase reconstruction results. Wrapped phase (a) and unwrapped continuous phase (b).

~Ifðu, vÞ ¼ ~Iðu, vÞHðu, vÞ: (31)

o

: (32)

o

: (34)

(33)

spectrum becomes,

14 Optical Interferometry

phase is found with:

Transforming back to spatial domain we obtain:

φwðx, yÞ ¼ atan2

The wrapped phase is found by:

Phase unwrapping is a common step to finally find a continuous phase for several fringe analysis techniques such as phase shifting, Fourier, the phase synchronous and others methods that use the arctangent function of the ratio of the sine and cosine of the phase to obtain a wrapped phase. In its simplest form, phase unwrapping consists in adding or subtracting 2π terms to the pixel being unwrapped if a difference greater that π is found with a previous pixel already unwrapped [11]. The phase unwrapping problem in one dimension can be observed in Figure 12. The wrapped phase found with the arctangent function is seen in Figure 12a and the unwrapped continuous phase is showed in Figure 12b.

The described procedure works well only for wrapped phases with no inconsistencies and low noise levels, however, delivers wrong results when dealing with noisy wrapped phases or those obtained from interferograms with broken or unconnected fringes. A more consistent approach is achieved with the least squares phase unwrapping method [12]. The least square technique integrates the discretized laplacian of the phase. To this end, the laplacian of the phase is calculated as follows:

$$L\_{i,j} = \phi\_{i+1,j}^x - \phi\_{i,j}^x + \phi\_{i,j+1}^y - \phi\_{i,j}^y,\tag{35}$$

where

$$\phi\_{i,j}^{x} = \operatorname{atan2}\left[\sin\left(\phi\_{i,j}^{w} - \phi\_{i-1,j}^{w}\right), \cos(\phi\_{i,j}^{w} - \phi\_{i-1,j}^{w})\right] p\_{i,j} p\_{i-1,j},\tag{36}$$

and

$$\phi\_{i,j}^y = \operatorname{atan2}\left[\sin\left(\phi\_{i,j}^w - \phi\_{i,j-1}^w\right), \cos(\phi\_{i,j}^w - \phi\_{i,j-1}^w)\right] p\_{i,j} p\_{i,j-1}.\tag{37}$$

In the last equations, we have used pixel subscript notation in order to limit the extension of the equations. The pupil function pi, j is defined as equal to one where we have valid data and zero otherwise. One may note that if pi, j = pi+1, j = pi-1, j = pi, j+1 = pi, j-1 = 1, then

$$L\_{i,j} = -4\phi\_{i,j} + \phi\_{i+1,j} + \phi\_{i-1,j} + \phi\_{i,j+1} + \phi\_{i,j-1}.\tag{38}$$

Solving for the phase in the above equation, we obtain that the unwrapping problem under the least square approach consist in the resolution of a linear system of equations, as:

Figure 12. Wrapped (a) and unwrapped phase (b).

An iterative technique that solves the above system of linear equations is the overrelaxation method in which the following equation is iterated until convergence:

$$
\phi\_{i,j}^{k+1} = \phi\_{i,j}^k - \frac{(d\phi\_{i,j}^k - \phi\_{i-1,j}^{k+1} - \phi\_{i+1,j}^k - \phi\_{i,j-1}^{k+1} - \phi\_{i,j+1}^k + L\_{i,j})w}{d},\tag{40}
$$

where

$$d = p\_{i-1,j} + p\_{i+1,j} + p\_{i,j-1} + p\_{i,j+1}.\tag{41}$$

In the last equations, k is the iteration number and w is a parameter of the overrelaxation method that must be set between 1 and 2. Results of the least square technique are showed in Figure 13. A two-dimensional wrapped phase is seen Figure 13a, the phases differences are shown in Figure 13b and c, respectively. The laplacian of the phase is showed in Figure 13d,

Figure 13. Least square results to unwrap the phase. Wrapped phase (a), phase differences in the x (b) and y (c) directions, laplacian of the phase (d), unwrapped phase (e) and rewrapped phase (f).

the reconstructed phase in a two-dimensional view is observed in Figure 13e and finally, for comparison purposes the reconstructed rewrapped phase is also showed in Figure 13f. One may observe that the original wrapped phase and the reconstructed wrapped phase are slightly different, this is because the least square method recovers the phase with an arbitrary constant term, however this term is usually neglected since doesn't carry any information of the object being measured.

If desired, the constant term in the retrieved phase may be corrected easily in the following form:

$$\phi\_c = \phi\_r \text{--attan2}[\sin\left(\phi\_r \neg \phi\_w\right), \cos(\phi\_r \neg \phi\_w)].\tag{42}$$

The wrapped phase seen in Figure 13a was constructed as follows:

$$\phi\_w = \operatorname{atan2}[\sin(\phi), \cos(\phi)],\tag{43}$$

where

An iterative technique that solves the above system of linear equations is the overrelaxation

In the last equations, k is the iteration number and w is a parameter of the overrelaxation method that must be set between 1 and 2. Results of the least square technique are showed in Figure 13. A two-dimensional wrapped phase is seen Figure 13a, the phases differences are shown in Figure 13b and c, respectively. The laplacian of the phase is showed in Figure 13d,

Figure 13. Least square results to unwrap the phase. Wrapped phase (a), phase differences in the x (b) and y (c) directions,

laplacian of the phase (d), unwrapped phase (e) and rewrapped phase (f).

− φkþ1 i,j−1− φk

i,jþ1 þ Li,jÞw

d ¼ pi−1,j þ piþ1,j þ pi,j−1 þ pi,jþ1: (41)

d , (40)

method in which the following equation is iterated until convergence:

φkþ1 i,j ¼ φk i,j − ðdφk i,j − φkþ1 i−1,j − φk iþ1,j

where

16 Optical Interferometry

$$\phi\_{i,j} = 2\pi \left\{ \begin{array}{c} 16\left[ \left( \mathbf{x}\_i \right)^2 + \left( \mathbf{y}\_j \right)^2 \right]^2 - 18\left[ \left( \mathbf{x}\_i \right)^2 + \left( \mathbf{y}\_j \right)^2 \right] + \\ 5y\_j(\mathbf{x}\_i)^2 - 4\mathbf{x}\_i(y\_j)^2 + 3y\_j\mathbf{x}\_i + 2\mathbf{x}\_i - 2y\_j + 0.56 \end{array} \right\}. \tag{44}$$

In the above equations xi and yj are range variables that vary from −1 to 1, the wrapped phase was multiplied by an annular pupil function with an exterior radius of 198 pixels, while the interior radius was 60 pixels for an image size of 400 + 400. The convergence of the reconstructed phase seen in Figure 2e was reached after 700 iterations; the overrelaxation parameter w was set to 1.99, which is usual for large images. The reconstructed phase with the constant term corrected and the phase error, ε = φ = φc, can be appreciated in Figure 14a and b. The maximum error was about 0.00005 radians.

Finally, results on a noisy wrapped phase are presented. Random noise with uniform distribution in the range of −π/4 to π/4 radians were added to the phase to obtain the wrapped phase

Figure 14. Unwrapped phase and phase error. Three-dimensional view of the reconstructed and corrected phase (a) and phase error (b).

Figure 15. Results on noisy measurements. Wrapped phase with noise (a), retrieved phase (b) and wrapped retrieved phase (c).

seen in Figure 15a, the unwrapped phase is shown in Figure 15b, and the rewrapped reconstructed phase is observed in Figure 15c. As can be noticed the least square method is a very reliable technique that works with any pupil configuration and stands noisy measurements.

### 7. Phase recovery from lateral shearing interferograms

Lateral shearing interferometry is a very important field in experimental optical measurements, in which, the test beam interferes with a laterally displaced version of itself instead of a reference beam. The resulting fringe patterns are thus related with the object wavefront derivative in a given direction. This is very useful when the object information of interest is related with the derivative as in strain analysis or when the dynamical range of the object wave front is too high that cannot be measured with direct interferometry. Let us consider a laterally shear interferogram with a beam displacement in the x direction a quantity Δx, we obtain:

$$I^{\mathbf{x}}(\mathbf{x},y) = B(\mathbf{x},y) + \mathbb{C}(\mathbf{x},y)\cos[\psi^{\mathbf{x}}(\mathbf{x},y)],\tag{45}$$

where

$$
\psi^x(\mathbf{x}, y) = \phi(\mathbf{x}, y) \neg \phi(\mathbf{x} - \Delta \mathbf{x}, y). \tag{46}
$$

In Eq. (45) B(x, y) is the background intensity and C(x, y) is the modulation term. The objective is to retrieve the undisplaced phase ϕ(x, y). To this end, we need, at least, another laterally shear interferogram with a beam displacement in the y direction another quantity Δy, obtaining:

$$I^y(\mathbf{x}, y) = B(\mathbf{x}, y) + \mathbb{C}(\mathbf{x}, y) \cos[\psi^y(\mathbf{x}, y)],\tag{47}$$

where

$$
\psi^y(\mathbf{x}, y) = \phi(\mathbf{x}, y) \neg \phi(\mathbf{x}, y - \Delta y). \tag{48}
$$

Let us considered that we have retrieved the phase differences ψx (x, y) and ψy (x, y) by means of a phase-shifting technique as is depicted in Figure 16 and Figure 17 for the x and y directions, respectively. A set of four phase-shifted interferograms can be seen in Figure 16a. The wrapped phase differences ψwx(x, y) and the unwrapped phase ψx (x, y) are observed in Figure 16e and f, respectively. The modulation term and a sheared pupil function px (x, y) are shown in Figure 16g and h, respectively. The sheared pupil px (x, y) is found after normalization from zero to one and thresholding the normalized modulation term. A second set of phase-shifted interferograms are seen Figure 17a and d. The wrapped phase differences ψwy(x, y) and the unwrapped phase ψy (x, y) are observed in Figure 17e and f, respectively. The modulation term and a sheared pupil function py (x, y) are shown in Figure 4g and h, respectively. In a similar way to the first set of phase-shifted interferograms, the pupil py (x, y) is found after normalization from zero to one and thresholding the normalized modulation term. The sheared pupils px (x, y) and py (x, y) are useful to find the undisplaced pupil p(x, y), where the original wave front ϕ(x, y) have valid data, since px (x, y) = p(x, y)p(x − Δx, y) and py (x, y) = p(x, y)p((x, y) − Δy)

Once the phase differences θx (x, y) and θy (x, y) are known, we can use the next procedure to find the searched phase ϕ(x, y) observing that:

seen in Figure 15a, the unwrapped phase is shown in Figure 15b, and the rewrapped reconstructed phase is observed in Figure 15c. As can be noticed the least square method is a very reliable technique that works with any pupil configuration and stands noisy measurements.

Figure 15. Results on noisy measurements. Wrapped phase with noise (a), retrieved phase (b) and wrapped retrieved

Lateral shearing interferometry is a very important field in experimental optical measurements, in which, the test beam interferes with a laterally displaced version of itself instead of a reference beam. The resulting fringe patterns are thus related with the object wavefront derivative in a given direction. This is very useful when the object information of interest is related with the derivative as in strain analysis or when the dynamical range of the object wave front is too high that cannot be measured with direct interferometry. Let us consider a laterally shear interferogram with a beam displacement in the x direction a quantity Δx, we obtain:

ðx, yÞ ¼ Bðx, yÞ þ Cðx, yÞcos½ψx

In Eq. (45) B(x, y) is the background intensity and C(x, y) is the modulation term. The objective is to retrieve the undisplaced phase ϕ(x, y). To this end, we need, at least, another laterally shear interferogram with a beam displacement in the y direction another quantity Δy,

ðx, yÞ ¼ Bðx, yÞ þ Cðx, yÞcos½ψy

ðx, yÞ�, (45)

ðx, yÞ�, (47)

ðx, yÞ ¼ φðx, yÞ−φðx−Δx, yÞ: (46)

7. Phase recovery from lateral shearing interferograms

I x

I y ψx

where

phase (c).

18 Optical Interferometry

obtaining:

where

Figure 16. Recovery of the phase differences in the x direction. Set of four sheared interferograms acquired under a phaseshifting technique (a) to (d), wrapped phase differences (e), unwrapped phase differences (f), modulation (g) and recovered sheared pupil (h).

Figure 17. Recovery of the phase differences in the y direction. Set of four sheared interferograms acquired under a phase-shifting technique (a) to (d), wrapped phase differences (e), unwrapped phase differences (f), modulation (g) and recovered sheared pupil (h).

$$
\psi\_{i,j}^{\mathbf{x}} p\_{i-a,j} - \psi\_{i+a,j}^{\mathbf{x}} p\_{i+a,j} = \phi\_{i,j} (p\_{i-a,j} - p\_{i+a,j}) - \phi\_{i-a,j} p\_{i-a,j} - \phi\_{i+a,j} p\_{i+a,j}, \tag{49}
$$

and

$$
\psi\_{i,j}^y p\_{i,j-b} - \psi\_{i,j+b}^y p\_{i,j+b} = \phi\_{i,j} (p\_{i,j-b} - p\_{i,j+b}) - \phi\_{i,j-b} p\_{i,j-b} - \phi\_{i,j+b} p\_{i,j+b}.\tag{50}
$$

In the above equations we have changed the (x, y) dependence by pixel subscript notation. As described before, pi, j is the undisplaced pupil and the displacement quantities Δx and Δy are rounded to the nearest integer in pixel dimensions obtaining a and b, respectively. Adding Eq. (49) and Eq. (50) and solving for the phase, we obtain:

$$
\phi\_{i,j} = \frac{F + G}{H},
\tag{51}
$$

where

$$H = p\_{i+a,j} + p\_{i-a,j} + p\_{i,j+b} + p\_{i,j-b},\tag{52}$$

$$F = \psi\_{i,j}^x p\_{i-a,j} - \psi\_{i+a,j}^x p\_{i+a,j} + \psi\_{i,j}^y p\_{i,j-b} - \psi\_{i,j+b}^y p\_{i,j+b},\tag{53}$$

and

Digital Processing Techniques for Fringe Analysis http://dx.doi.org/10.5772/66474 21

$$G = \phi\_{i-a,j} p\_{i-a,j} + \phi\_{i+a,j} p\_{i+a,j} + \phi\_{i,j-b} p\_{i,j-b} + \phi\_{i,j+b} p\_{i,j+b}.\tag{54}$$

Eq. (51) represents a linear system of equations; however, it is ill posed because there are more unknowns than equations due to the effects of the sheared pupils. Nevertheless, a regularization term may be aggregated to overcome this problem [13–15]. The regularization term is in the form of discrete Laplacians of the phase among adjacent pixels. The following equation that incorporates the regularization term is iterated until convergence:

$$\phi\_{i,j}^{k+1} = \phi\_{i,j}^k - \frac{H\phi\_{i,j}^k - (F+G) + a(L\_{i+1,j}^x - 2L\_{i,j}^x + L\_{i-1,j}^x + L\_{i,j+1}^y - 2L\_{i,j}^y + L\_{i,j-1}^y)}{H},\tag{55}$$

where

$$L\_{i,j}^x = (\phi\_{i+1,j} - 2\phi\_{i,j} + \phi\_{i-1,j})p\_{i+1,j}p\_{i-1,j},\tag{56}$$

and

ψx i,j pi−a,j −ψx iþa,j

recovered sheared pupil (h).

20 Optical Interferometry

ψy i,j pi,j−b−ψy

Eq. (49) and Eq. (50) and solving for the phase, we obtain:

F ¼ ψx i,j pi−a,j −ψx iþa,j

and

where

and

piþa,j ¼ φi,j

i,jþbpi,jþb ¼ φi,j

ðpi−a,j

Figure 17. Recovery of the phase differences in the y direction. Set of four sheared interferograms acquired under a phase-shifting technique (a) to (d), wrapped phase differences (e), unwrapped phase differences (f), modulation (g) and

In the above equations we have changed the (x, y) dependence by pixel subscript notation. As described before, pi, j is the undisplaced pupil and the displacement quantities Δx and Δy are rounded to the nearest integer in pixel dimensions obtaining a and b, respectively. Adding

φi,j ¼ F þ G

piþa,j þ ψy

i,j

pi,j−b−ψy

−piþa,j

Þ−φi−a,j

pi−a,j

H ¼ piþa,j þ pi−a,j þ pi,jþb þ pi,j−b, (52)

−φiþa,j

ðpi,j−b−pi,jþbÞ−φi,j−bpi,j−b−φi,jþbpi,jþb: (50)

H ; (51)

i,jþbpi,jþb, (53)

piþa,j

, (49)

$$L\_{i,j}^y = (\phi\_{i,j+1} - 2\phi\_{i,j} + \phi\_{i,j-1})p\_{i,j+1}p\_{i,j-1}.\tag{57}$$

Here, α is a parameter that controls the effects of the regularization term. The phase reconstruction is seen in Figure 18. A two-dimensional view of the retrieved phase is observed in Figure 18a, the same phase but in a three-dimensional perspective is shown in Figure 18b and the phase error (the actual phase minus the reconstructed one) can be appreciated in Figure 18c. This reconstruction was achieved after 800 iterations with a parameter α = 0.1 obtaining a maximum error of about 0.0004 radians.

The actual phase was constructed as follows:

$$\phi\_{i,j} = 2\pi \left\{ \begin{aligned} 20\left[\left(\mathbf{x}\_i\right)^2 + \left(\mathbf{y}\_i\right)^2\right]^2 - 24\left[\left(\mathbf{x}\_i\right)^2 + \left(\mathbf{y}\_i\right)^2\right] \neg 6\left(\mathbf{x}\_i\right)^2 y\_j \\ + 8\mathbf{x}\_i \left(\mathbf{y}\_j\right)^2 + 5\mathbf{x}\_i y\_j + 4\mathbf{x}\_i \neg 5y\_j + 0.45 \end{aligned} \right\}, \tag{58}$$

Figure 18. Phase reconstruction from lateral sheared interferograms. Reconstructed phase (a), three-dimensional view of the reconstructed phase (b) and the phase error (c).

where xi and yj are range variables that vary from −1 to 1 in both vertical and horizontal directions. The pupil function is a circular one with a radius of 170 pixels. The shear distances were set to Δx = a = 12 pixels and Δy = b = 12 pixels.


**Decoupling the Contributions of ZnO and Silica in the Characterization of Industrially-Mixed Filled Rubbers by Combining Small Angle Neutron and X-Ray Scattering**

**Mariapaola Staropoli 1,\*, Dominik Gerstner 2, Aurel Radulescu 3, Michael Sztucki 4, Benoit Duez 2, Stephan Westermann 1, Damien Lenoble 1 and Wim Pyckhout-Hintzen 5**

Received: 28 January 2020; Accepted: 22 February 2020; Published: 25 February 2020

**Abstract:** Scattering techniques with neutrons and X-rays are powerful methods for the investigation of the hierarchical structure of reinforcing fillers in rubber matrices. However, when using only X-ray scattering, the independent determination of the filler response itself sometimes remains an issue because of a strong parasitic contribution of the ZnO catalyst and activator in the vulcanization process. Microscopic characterization of filler-rubber mixtures even with only catalytic amounts of ZnO is, therefore, inevitably complex. Here, we present a study of silica aggregates dispersed in an SBR rubber in the presence of the catalyst and show that accurate partial structure factors of both components can be determined separately from the combination of the two scattering probes, neutrons, and X-rays. A unique separation of the silica filler scattering function devoid of parasitic catalyst scattering becomes possible. From the combined analysis, the catalyst contribution is determined as well and results to be prominent in the correction scheme. The experimental nano-structure of the ZnO after the mixing process as the by-product of the scattering decomposition was found also to be affected by the presence or absence of silica in the rubber mixture, correlated with the shear forces in the mixing and milling processes during sample preparation. The presented method is well suited for studies of novel dual filler systems.

**Keywords:** SANS; SAXS; silica; zinc oxide; partial structure factors; small angle scattering

#### **1. Introduction**

Nanofiller particles added to elastomers are known to form hierarchical structures varying from clusters of the order of a few nm up to agglomerates in the range of several μm. As a consequence of this, the resulting composite materials show strongly improved mechanical properties, which lead to important applications as, e.g., most prominently in tires [1–4]. The complex arrangement of the nanoparticles in the mixtures is thereby at the basis of the associated reinforcing mechanisms. The understanding of the correlation between the filler structure and the ultimate macroscopic properties of the filled rubbers widely employs Small Angle Scattering (SAS) methods [5–11].

These techniques exploit the different sensitivity of the respective radiation probe for the components in the mixture, allowing the identification of the structural distribution of the different

species. Whereas for Small Angle X-ray Scattering (SAXS) the difference in the electron density between typical soft and hard constituents determines the contrast, in the case of Small Angle Neutron Scattering (SANS) different contrasts depending on nuclear properties rather than on electronic ones may exist. In the ideal case, these contrasts can be tuned in order to elaborate the scattering of specific components only [12–14]. For multicomponent systems, the combination of both SAXS and SANS can be therefore used to highlight the contribution of different components to the total scattering function, due to the different but complementary scattering contrasts obtained on the same sample. The underlying manuscript will deal with an application of these two probes to identify the ZnO structure for the first time in such industrial silica-containing composites, allowing its subtraction from the structural contribution of the reinforcing fillers.

ZnO is typically used as an activator in the vulcanization process and is known to improve the tire abrasion resistance and heat transfer at braking as well as to reduce the rubber shrinking during the molding process [15–17].

For conventional ZnO nanoparticles with a radius between 20 and 90 nm [18], however, the scattering contribution will occur in the same scattering vector range as the silica filler aggregates. This interference impairs, therefore, a direct evaluation of the scattering response of the reinforcing fillers, which is the only component determining the performance of the rubber. The problem of the interference and also additional inhomogeneities have been discussed in the literature [5,6,14,19–22]. Due to the difficulty in disentangling the contribution from the catalyst and the filler particles, the ZnO contribution was sometimes neglected or a simple weighted subtraction of a ZnO-containing background was applied as data correction. Recently, a more accurate correction was reported [22], based on different scattering techniques. The weighted subtraction assumed a priori an identical contribution of ZnO in filled and non-filled compounds. In addition, more recently, the use of contrast variation in such mixtures was discussed. With anomalous small angle X-ray scattering (ASAXS) at the Zn edge, which can be realized only at synchrotron sources varying the energy of the X-ray beam, a certain variation of the scattering length of the ZnO might be obtained, making it only partially contributing to the total intensity [19]. Alternatively, but for the case of neutron scattering another contrast variation procedure, in which the nuclear polarization of hydrogen atoms was modified, was applied for such compounds. However, the latter is technically difficult and is not compatible with standard neutron instrumentation [23]. In the underlying work using a combination of scattering techniques, we aim to extract the form and structure factors belonging to the silica filler complex in the presence of the full cure package and simultaneously of the necessary presence of ZnO.

Dealing with these problems is an important aspect for an enhanced understanding and the development of structure-property relationships. For the extraction of the single contributions to the total scattering function, we make use of a polymer-based approach in analogy to the Singular Value Decomposition (SVD) method widely applied to multi-component 'green' nanocomposites in the presence of solvents [14,24] but which relies on several samples with different labeling and contrasts. For our purpose, the unfilled rubber will be treated as a three-component mixture formed by the processing oil, matrix polymer, and ZnO [25], all leading to different scattering contrasts. The same approach is then extended to the case of silica-filled compounds in order to separate the contribution of silica and ZnO. This approximation is reasonable as these components have the strongest scattering contribution due to not only their concentration in the mixture. As we show, the approach relies on the combination of X-ray and neutron scattering intensities on identical samples and on the different contrast factors involved in the two methods to decouple the partial contributions to the total intensity. The ZnO correction is required, and its contribution appears in all samples analyzed.

#### **2. Materials and Methods**

#### *2.1. Samples and Preparation*

The SBR used in this work has high styrene (40%) and medium vinyl content (24%). The random copolymer is characterized by a glass transition temperature of −34.5 ◦C and is pre-mixed with 37.5 phr (parts in weight per 100 parts of rubber) of an aromatic distillate oil (TDAE). The rubber was mixed with amorphous precipitated silica (1165 MP, Rhodia/Solvay Group, Aubervilliers, France) of a specific surface of 165.8 m2/g (N2 BET). Several samples with different silica volume fractions were prepared in a standard way [21] using an internal mixer of the Banbury type, followed by a two-roll milling process. Rubber and fillers were mixed in the non-productive stage using bis-3-triethoxysilylpropyldisulfide (Si266) as a coupling agent. In this stage n-(1,3)-dimethylbutyl-n-phenyl-p-phenylenediamine was used as an antioxidant while N -cyclohexyl-2-benzothiazolesulfeneamide and diphenylguanidine, respectively, as first and secondary accelerators. During the productive stage, the curing package was introduced, including ZnO with a specific surface of 5 m2/g used as the catalyst. Samples with different silica mass fractions: 0 (A), 30 (B), 60 (C) phr, and an unfilled sample free of ZnO (E) were studied in this work. The sample list and the silica filling degree are specified, respectively, in Tables 1 and 2. After the milling process, the slabs were cured by compression-molding vulcanization at 170 ◦C for 10 min in a hydraulic press, and square sheet samples of thickness ~0.7mm were obtained.

**Table 1.** Composition of investigated samples. From left to right column the complexity is decreased. The compositions are given in parts per 100 g rubber (phr).

**Table 2.** Silica content in the different samples. phr is parts per 100 g of rubber.

#### *2.2. Small Angle X-Ray Scattering (SAXS)*

SAXS experiments were performed at ID02 [26], ESRF, (Grenoble, France) at detector distances 31, 8, and 2 m and at a wavelength 1 Å. The usable scattering vector *q*, defined as *q* = 4π λ sin θ 2 , extended from 5 × 10−4 < *q* < 0.4 Å−1. θ is the scattering angle. The *q*-range differs along *x* and *y* direction due to the asymmetric beam stop. 2D data were binned to 1920 × 1920 channels with 0.09 × 0.09 mm2 width. They were corrected for empty beam, dark current, and radially averaged (where possible) using standard procedures of the beamline. Absolute scattering intensities in [cm−1] units were obtained by calibration with the scattering of water at room temperature. A background, which was assigned to the left wing of the amorphous halo, was subtracted linearly as only data up to roughly *q* = 0.1 Å−1 were used. For reference sake, sample E was re-measured and calibrated independently vs. a glassy carbon standard at the University of Erlangen. Intensities were found to match in excellent agreement.

#### *2.3. Small Angle Neutron Scattering (SANS)*

SANS experiments were conducted at the KWS-2 diffractometer of MLZ, (Garching, Germany) using three detector distances of respectively 20, 8, and 2 m at a wavelength of 10 Å. The scattering vector range spanned between 1 × 10−3 < *q* < 0.4 Å−1. 2D-detector data were obtained in 142 × 142 channels of 8 × 8 mm2 width. The wavelength distribution Δλ λ was 10%. For the absolute scaling to [cm−1] the incoherent scattering level of a secondary standard Plexiglass was used. The two-dimensional data in 142 × 142 channels were corrected pixelwise for empty beam scattering, detector sensitivity, and background noise using B4C as beam blocker and subsequently radially averaged. In the case of anisotropic scattering patterns, sectors with total opening angles of 10◦ along the main axes of the anisotropic patterns were applied to the scattering patterns. The incoherent background arising from the hydrogenous polymer fraction and the other additives was determined from the high *q*-range and was found to be in good agreement with the estimated incoherent scattering. Likewise, as for SAXS, sample E was re-measured and put to an absolute scale at the KWS-3 diffractometer of MLZ using the direct beam method. Moreover, here, the calibrated intensities of KWS-2 and KWS-3 matched ideally.

#### **3. Results**

The scattering model approach used in this work is developed for the non-filled samples with and without ZnO and further extended to the case of filled systems. Due to the strong scattering of the ZnO catalyst, this component cannot be neglected in the analysis despite its irrelevance in the reinforcing mechanism and mechanical performance of the rubber. It leads, therefore, to treatments that necessarily involve its structure. Although the matrix is ideally expected to contribute as a simple background to the scattering of filled rubbers, it is known that inhomogeneities due to the crosslinks possibly lead to an additional contribution [13,14]. This contribution is normally observed in the lowest *q* region as a characteristic power law. The pure vulcanized SBR/oil sample has often been modeled then by means of a classical Ornstein-Zernike approach in analogy to the case of a semi-dilute polymer solution or swollen gel assuming that the processing oil acts as a contrast agent for the meshes of the network [25]. In the presence of the curing activator, on the other hand, at low *q* ZnO nanocrystals or domains could become visible due to their strong and different contrast with the embedding matrix and at high *q* they might exhibit Bragg peaks. The overall dimensions of the ZnO crystals could be estimated in principle with an additional contribution taken into account by a Debye-Bueche law. In the literature, the simple incoherent superposition of Ornstein-Zernike and Debye-Bueche laws was applied on a regular basis with limited success [25,27–29]. However, the estimation of the ZnO contribution to the total scattering could be more complex due to its interaction with other components of the mixture as sulfur and accelerators, which can be absorbed on the ZnO particles surface. These interactions could lead to a different average contrast between the ZnO and the matrix and possibly to a non-negligible deviation from the incoherent sum of two contrast-weighted partial structure factors.

The coherent scattering intensity of a general three-component mixture *I*(*q*) is theoretically written as

$$I(q) = \sum\_{i,j=1}^{n=3} \rho\_i \rho\_j \mathbf{S}\_{ij}(q) \tag{1}$$

where the pre-factors of the partial structure factors *Sij*(*q*) are their scattering length densities ρ*i*, ρ*j* (here abbreviated as SLD). The total scattering function is then defined as a linear combination of all partial structure factors resulting from all the intra (*Sii*) and inter-component (*Sij*) correlation functions. The approach is first applied to the determination of different contributions in non-filled samples, which can be approximated to two- and three-components systems. In the following section, we discuss first the case of the unfilled rubber and generalize the same concept then to the reinforced case.

#### *3.1. The Unfilled Case*

In the following section, the polymer-based approach will be applied to the description of the scattering function of an unfilled rubber. Sample E consists of the rubber mixture but was vulcanized without ZnO. The SBR and oil constitute then a classical two-component system. On the contrary, Sample A is already a three-component but two-phase mixture containing SBR polymer, processing extender oil, and ZnO. The other curing agents can be neglected in scattering evaluations. Taking sample A as the working example in further treatment and making use of the incompressibility condition, one obtains for the intensity [25]:

$$I(q) = \left(\rho\_{\rm pol} - \rho\_{\rm oil}\right)^2 \mathcal{S}\_{\rm pol} + \left(\rho\_{\rm ZnO} - \rho\_{\rm oil}\right)^2 \mathcal{S}\_{\rm ZnO} + 2\left(\rho\_{\rm pol} - \rho\_{\rm oil}\right)\left(\rho\_{\rm ZnO} - \rho\_{\rm oil}\right)\mathcal{S}\_{\rm pol-ZnO} \tag{2}$$

Where ρ*pol*, ρ*oil* and ρ*ZnO* are the SLD for the matrix, oil, and ZnO, respectively. Here, the oil is considered as the background component (or solvent in the polymer-based approach). *Spol*, *SZnO* are the partial structure factors respectively for the polymer and the ZnO while *Spol*−*ZnO* takes into account the interaction between these two components. The q-dependence of the structure factors was omitted for readability. The intensity from a background of oil does not appear explicitly therefore in the q-dependent scattering functions. It can be easily seen that this general three-component treatment for sample A can be simplified to obtain the case of sample E. If no ZnO is present in the compound, then the second and third term of the equation vanish, and the intensity for the resulting two-component system is written in the usual well-known form as *I*(*q*) = ρ*pol* − ρ*oil*2 *Spol* For sample E, the incompressibility condition leads to ∅*pol* + ∅*oil* = 1. ∅ corresponds to the volume fraction of the components. The scattering contrast remains, therefore, a delicate fitting parameter. For sample A, within the approximation that only the main components contribute to the scattering function, the following condition is valid: ∅*pol* + ∅*oil* + ∅*zn* = 1. The small amount of activator allows neglecting interactions between neighboring ZnO nanocrystals. Therefore, the scattering contribution of the activator could ideally be described in terms of a particle form factor. Furthermore, the low concentration, as well as the non-specific interaction of the inorganic ZnO particles with the oil or SBR, are very good reasons to neglect also the cross-term *Spol*−*ZnO* in the scattering equation of the three-component system [12,25,29,30] which is then reduced to:

$$I(q) = \left(\rho\_{pol} - \rho\_{oil}\right)^2 \mathcal{S}\_{pol} + \left(\rho\_{ZnO} - \rho\_{oil}\right)^2 \mathcal{S}\_{ZnO} \tag{3}$$

This approximation applies to sample A containing less than 1% of ZnO (2.5phr). However, the contrast factors in this solid cross-linked or vulcanized mixture differ from those of the system in which ZnO would be considered as one component in an effective matrix of SBR and oil due to interaction with the curing package. The subtraction of *SZnO* from the scattering function is, therefore, not straightforward and requires an estimation of the contrast factors involved. Moreover, the size of the particular ZnO component is a pre-requisite for adequate correction. Different scenarios could

determine the effect of the ZnO on the scattering function of the filled rubber. In the case of large ZnO particles, the Guinier regime of this component is expected at too small scattering vectors q. The only contribution would, therefore, be its surface scattering with its approximately observed characteristic q−4 slope in addition to the Ornstein-Zernike behavior expected for the rubbery matrix. This particular q−4 at a low scattering vector occurs in the case of smooth particles that have a well-defined flat and strong interface with the surrounding matrix. In the case of rough or fractal surfaces, however, the detected power law would differ from −4. The power law exponent X depends on the surface fractal dimension *Ds* by X = (6 − *Ds*). On the other hand, if the mixing and milling process results in better dispersed, relatively small ZnO particles, their form factor would interfere with the scattering contribution resulting from the fillers and corrections that require the estimation of contrast factors become necessary. Unlike a SVD method based on a number of contrast parameters that exceeds the number of unknown partial structure factors [12–14,23,27,29,30], in our particular case, it is impossible to induce additional contrast factors into a cross-linked rubber without a previous labeling of components by, e.g., deuteration and inevitably leading to intrinsically different samples [25]. For this reason, the decomposition of the measured signal intensity has been performed using a linear combination of experimental data obtained by X-ray and neutron scattering measurements applied to the same sample. Due to the different contrast factors in the two experiments, the partial structure factors for fillers and ZnO can be extracted for each q in analogy to the SVD approach. The resulting set of two linear equations with two unknowns, i.e., the partial structure factors can be solved exactly. On the other hand, the merging of SANS and SAXS data into one evaluation may be affected by the different q-resolution of the two methods. Especially in the case of SANS, resolution and polydispersity effects are fully correlated. In the lowest q range, this does not affect the data. Resolution- and non-resolution corrected data differ only in the order of 2%–3% and do not influence the results. As such, the determination of the scattering functions strongly depends on the quality of the absolute calibrations as well as on the correct evaluation of the SLD parameters for all the components taken into account within this approximation. Therefore, absolute care in the absolute normalization of the intensities is important.

In our presented approach, the partial form factors result from the exact solution of a system of two linear equations. For sample A the expression of *Spol* and *SZnO* is given by:

$$S\_{pol} = \frac{\left(\rho\_{\rm ZnO} - \rho\_{\rm oil}\right)\_X^2 I\_{\rm AN} - \left(\rho\_{\rm ZnO} - \rho\_{\rm oil}\right)\_N^2 I\_{\rm AX}}{\left(\rho\_{\rm pol} - \rho\_{\rm oil}\right)\_N^2 \left(\rho\_{\rm ZnO} - \rho\_{\rm oil}\right)\_X^2 - \left(\rho\_{\rm pol} - \rho\_{\rm oil}\right)\_X^2 \left(\rho\_{\rm ZnO} - \rho\_{\rm oil}\right)\_N^2} \tag{4}$$

$$S\_{\rm ZnO} = \frac{\left(\rho\_{\rm pol} - \rho\_{\rm oil}\right)\_N^2 I\_{\rm AX} - \left(\rho\_{\rm pol} - \rho\_{\rm oil}\right)\_X^2 I\_{\rm AN}}{\left(\rho\_{\rm pol} - \rho\_{\rm oil}\right)\_N^2 \left(\rho\_{\rm ZnO} - \rho\_{\rm oil}\right)\_X^2 - \left(\rho\_{\rm pol} - \rho\_{\rm oil}\right)\_X^2 \left(\rho\_{\rm ZnO} - \rho\_{\rm oil}\right)\_N^2} \tag{5}$$

The indices N and X stand for neutron and X-ray, respectively. IAN and IAX correspond to the experimentally measured absolute intensities of sample A in both N and X case. They are corrected for instrumental backgrounds of any source (see Experimental). It is obvious that the contrasts as weighting factors are of extreme importance in further evaluation. Therefore, SLDs of the components, as well as the consistency of absolute calibration of intensities, have to be verified with the greatest care. We remind that the structure factor is defined as S(q) = φ V P(q) with V the volume of the scattering entity and P(q) its q-dependent form factor [25]. The weighed subtractions in Equations (7) and (8) were obtained in the common SANS *q-* range after a cubic spline was fitted to the SAXS data.

#### *3.2. Contrast Considerations*

As discussed above, deriving accurate partial structure factors of single components that give rise to the experimental scattering intensity requires accurate estimations of the contrasts of the particular components in the mixtures. Contrast is generally defined as the square of the difference in scattering

length densities (SLD) between two scattering entities. Whereas for neutrons the scattering length *b* is a nuclear property, for X-rays the equivalent parameter is a function of the electronic number of the component. The SLD is then the sum of the scattering lengths per volume and is defined for neutrons and X-ray respectively as: *SLDN* = *n i*=1 *bi Vmolecule* , *SLDX* = *n i*=1 *ei Vmolecule re* where *n* is the number of atoms of type *i* and *e* is the number of electrons. *re* is the radius of the electron, 2.8 × 10−13 cm.

The calculated parameters for the relevant components using the available chemical information are summarized in Table 3.

**Table 3.** Mass densities and scattering length densities for the most significant components of the mixture (amount > 2.5 phr). Only for SBR, the experimental value for the SLD in the case of X-rays was optimized against experimental data (see text).

The contrast factors between the significant components were then estimated on the basis of the calculated SLD and are reported in Table 4. Sample E, considered as an ideal two-component system was used to verify the estimated contrast factors with the experimental scattering intensities. Whereas the SLD of SBR and the corresponding contrast of SBR-to-oil in sample E is in good agreement for the SANS case, a clear discrepancy was observed in SAXS, which led to higher-than-expected intensities. Doubts about the absolute calibrated intensity levels can be excluded, however, in view of the elaborate cross-references at different large-scale facilities. The reason for the reversed intensity order is then to be sought in the assumptions about the components.

Deviations can also be partly due to uncertainties in the thickness of each sample, the statistical and systematic errors of the data themselves, and the quality and intrinsic accuracy of the absolute calibrations. Inaccurate estimation of the SLD of SBR could be also due to the neglect of the components present in phr amounts <2.5 phr. On the other hand, the SLD of the oil in both neutron and X-ray data, calculated from the element analysis can be accurately determined.

To comply with the aforementioned higher SAXS intensity, the SLD of the SBR in the X-ray case should then be 8.46× 1010 cm−4, i.e., lower by 4%–5% lower than the theoretical value. It is worth noting that the SANS method is insensitive, in terms of contrast, to the alkane-like compounds of the matrix. The coherent SLD contribution of a CH2 section is about 0 for SANS (bC = 6.65 fm, bH = −3.74 fm). This is not the case for SAXS where each atom always adds positively to the electron density. Thus, the optimized value for SBR indicated in Table 4 is assumed to affect only SAXS intensities. The silane component, used as a coupling agent in the filled samples, is in the same way involved in the uncertainty of the SLD determination due to its reaction with both the silica and the polymer matrix. In our estimation, we assume that it is distributed evenly in the matrix.

#### *3.3. The Filled Case*

The previous relations and considerations applied to the case of unfilled silica rubbers. In the more interesting case of silica-filled systems, the former three-component system now evolves to at least a four-component mixture. A further complication, besides the addition of silica, in the decomposition of the intensities of samples B and C (30 and 60 phr silica, respectively) stems from the presence of the silane component used as a coupling agent. For this reason, the estimation of the contrast factors is based on the estimation of an effective SLD of the matrix including the silane weighted volume fraction. We then obtain again workable equations like before. As in the case of the non-filled rubbers, we neglected all other ingredients with volume fraction below 2.5 phr as they contribute mainly to the incoherent background. The scattering intensity for samples B and C can then be expressed in terms of the partial contributions as:

$$\mathbf{I}(\boldsymbol{\varrho}) = \left(\rho\_{\mathrm{sil}} - \rho\_{\boldsymbol{\varepsilon}ff}\right)^2 \mathbf{S}\_{\mathrm{sil}} + \left(\rho\_{\mathrm{ZnO}} - \rho\_{\boldsymbol{\varepsilon}ff}\right)^2 \mathbf{S}\_{\mathrm{ZnO}} + 2\left(\rho\_{\mathrm{sil}} - \rho\_{\boldsymbol{\varepsilon}ff}\right)\left(\rho\_{\mathrm{ZnO}} - \rho\_{\boldsymbol{\varepsilon}ff}\right) \mathbf{S}\_{\mathrm{sil}-\mathrm{ZnO}} \tag{6}$$

Note the similarity with Sample A, whereas now the effective matrix replaces the oil. Again, the cross term appearing in Equation (6) can be neglected due to the low fraction of ZnO, leading to negligible correlations with silica. The partial form factors can then be expressed using the same approach as before:

$$S\_{\rm sil} = \frac{\left(\rho\_{\rm ZuO} - \rho\_{\rm eff}\right)\_{\rm X}^{2} I\_{\rm N} - \left(\rho\_{\rm ZuO} - \rho\_{\rm eff}\right)\_{\rm N}^{2} I\_{\rm X}}{\left(\rho\_{\rm sil} - \rho\_{\rm eff}\right)\_{\rm N}^{2} \left(\rho\_{\rm ZuO} - \rho\_{\rm eff}\right)\_{\rm X}^{2} - \left(\rho\_{\rm sil} - \rho\_{\rm eff}\right)\_{\rm X}^{2} \left(\rho\_{\rm ZuO} - \rho\_{\rm eff}\right)\_{\rm N}^{2}} \tag{7}$$

$$\mathcal{S}\_{\rm ZnO} = \frac{\left(\rho\_{\rm sil} - \rho\_{\rm eff}\right)\_N^2 I\_\text{X} - \left(\rho\_{\rm sil} - \rho\_{\rm eff}\right)\_\text{X}^2 I\_\text{N}}{\left(\rho\_{\rm sil} - \rho\_{\rm eff}\right)\_\text{N}^2 \left(\rho\_{\rm ZnO} - \rho\_{\rm eff}\right)\_\text{X}^2 - \left(\rho\_{\rm sil} - \rho\_{\rm eff}\right)\_\text{X}^2 \left(\rho\_{\rm ZnO} - \rho\_{\rm eff}\right)\_\text{N}^2} \tag{8}$$

where ρ*sil* and ρ*eff* are respectively the SLD for silica and effective matrix including the silane component. In this way, the relevant structure factors of both contributing components can be extracted. A similar approach is lacking up-to-now in the literature. For the first time, microscopic details of the mixing process and/or vulcanization process in a reactive mixture could be accessible.

While interparticle interactions can be excluded for ZnO as well as interactions between the ZnO activator and silica filler particles, the same assumption cannot necessarily be applied to the silica particles. *Ssil* as the cluster form factor generally consists of the product of a particle form factor with an intra-cluster and even an inter-cluster structure factor taking into consideration the interaction between them as we recently showed [31]. However, the full description of the hierarchical structure of the fillers is out of the scope of this work. Here, the decomposition of the different contributions in a mixed-filler system and the evaluation of the ZnO effect is in the foreground.

In order to obtain a common *q-* range and identical step size between the data points for X-ray and neutron results and allow the weighed subtractions in Equations (7) and (8), a cubic spline is applied to the SAXS data.

#### **4. Results and Discussion**

First, the unfilled samples are analyzed. Figure 1 shows the comparison between absolutely calibrated total SANS and SAXS results obtained for samples E and A, before the subtraction of the incoherent background. For sample A, the contribution of 2.5 phr ZnO is immediately evidenced by a ~ 10-fold increased intensity between A and E and an associated different *q*-dependence in the q-range 10−3 < q < 10−2 A−1 and a crystalline Bragg peak around 0.15 A−1.

**Figure 1.** Absolutely calibrated SAXS (black) and SANS (green) data for samples E (left) and A (right). The data are not background-corrected.

In Figure 2, we compare extracted *Spo*l and *SZn0* with the SAXS intensities of sample A and E. The contribution of the ZnO form factor to the SANS curves results to be irrelevant as we show later. The partial form factors *Spol* and *SZnO*, respectively in red and orange, were shifted by an arbitrary factor along the *y* axis for the sake of comparison with the experimental data.

**Figure 2.** Partial structure factors *Spol* (red) and *SZnO* (orange) as derived from sample A in comparison with absolutely calibrated SAXS data for samples A (black) and E (green). *Spol* was arbitrarily shifted to be compared with sample E.

The figure shows that the scaled *Spol* as derived from sample A is perfectly congruent with the intensity of sample E, leading to the conclusion that—maybe unexpectedly—the vulcanization with and without ZnO would not lead to differences in the matrix structure detectable by scattering in length scales smaller than ~3000 Å. On the other hand, *SZnO* shows a visible power law contribution for *q* < 10−1*,* indicative for additional scattering to *Spol*.

The same approach defined for the extraction of the single contributions from the total scattering function can be transferred to the case of the filled samples.

Going to filled rubbers, Figure 3 shows a comparison between the SAXS and SANS averaged scattering intensities obtained for samples B and C (respectively, 30 and 60 phr silica). Besides the different absolute intensity and *q*-range due to instrumental factors, SANS and SAXS curves show different low-to-intermediate *q* < 0.01 Å−1 behavior. In addition, it can be seen that for *q* > 0.01 Å−1 the silica particle structure dominates in both SAXS and SANS.

**Figure 3.** Experimental SANS and SAXS data in absolute intensity units cm−1 for the ZnO-containing filled rubbers, sample B (left) and sample C(right). The number of data points of the ultrahigh-resolution SAXS data was about 4500 and was reduced to a SANS-compatible lower resolution of about 350 data points by means of a cubic spline fitting procedure.

The power law exponent in this scale range differs from −4 and stands for a deviation from the ideal spherical particle behavior. The value of approximately −3.7 observed here was also found in the literature for the same type of silica particles. Since this slope is not affected by particle polydispersity it indicates a rough silica surface. For the SANS data of sample C a closer inspection of the lower *q* < 0.01 Å−1 reveals a power law region which is the signature of silica aggregates with a mass fractal dimension of ~2.2 and an onset of a Guinier plateau is seen. In sample B a Guinier region cannot be detected due to the obvious formation of larger clusters. This finding corroborates the finding that the cluster size decreases when the filling degree is increased. In addition, some orientation seems to be visible in the higher filling degree sample (C), and therefore, the data must be analyzed along two perpendicular directions. We have selected only the vertical direction as it allows obtaining lower *q*-values due to the horizontal alignment of the beam stop stick in SAXS.

Besides the different extension in the *q*-range due to instrumental factors, the SAXS data show in both cases a shallow peak or shoulder around *q* ~ 4 10−3 Å−1, which cannot be seen in the SANS results. This is attributed to ZnO, which has a much stronger contribution in the X-ray intensity than in the neutron data, as already observed for the unfilled rubbers. Interestingly, sample C presents a more pronounced peak with respect to sample B, probably due to the lower contribution of the smaller clusters to the scattering function. In Figure 4, the partial contributions *SZnO* and *SSil* are multiplied by the contrast factors, as shown in Equation (6) and the experimental SAXS intensities are re-constructed. ZnO contributes to 10% to the total intensity for sample B (30 phr) but amounts to ~50% for C. The scattering at high q is fully determined by silica and consolidates a rather particle-like morphology for ZnO, its form factor dropping with *q*−*4*.

**Figure 4.** Experimental SAXS data in absolute intensity units cm−1 for samples B (30 phr) (left) and C (60 phr) (right). Partial form factors for silica (*Spol*, red) and ZnO (*SZnO*, blue) were multiplied by the contrast factors to re-construct the intensity.

In addition and as anticipated, the power law exponent of approximately −2, found in the ultra-low *q* region of the SAXS data could be attributed to the connectivity of clusters into a network when the volume fraction is increased above their percolation threshold [2–4].

Except for these observations, however, the main result and highlight of this first decomposition of SAXS intensities in partial structure factors is the important extraction of the scattering contribution of minute amounts of ZnO and some microscopic details from a clear *q*-dependence. A comparison with Figure 5 showing the same decomposition for SANS data, highlights that due the different contrast, the contribution of ZnO to the total scattering intensity is about two orders of magnitude lower than the one of the silica. However, even the catalytic content of ZnO impedes the analysis in the full *q* range since the intensity of the extracted partial structure factor drops very fast to below the sensitivity limit of the SAXS or SANS machine. Both SANS curves highlight that the main contribution comes almost entirely from silica. The catalyst has an almost negligible effect on the scattering profile. In SAXS, however, both partial intensities contribute to almost the same amount.

**Figure 5.** Experimental SANS data in absolute intensity units cm−1 for samples B (30 phr) (left) and C (60 phr) (right). Partial form factors for silica (*Spol*, red) and ZnO (*SZnO*, blue).

Thus, this analysis allowed us to evaluate the total scattering intensities as the sum of contributions of 2 components, similar to the case of a dual filler system. The structure of both in the mixture, as well as their mutual influence, can thereby be captured.

Let's first discuss ZnO. The stronger effect of the ZnO contribution in sample C over B is tentatively correlated to a simultaneous decrease of the silica contribution of smaller clusters.

The structure of ZnO, i.e., their *q*-dependence in the filled rubbers is, however, very different in comparison to that, extracted from the unfilled sample, as shown in Figure 6. In the case of sample A, the full scattering range could be accessed, and a typical power-law-like behavior as for silica is found. The slope at high *q* is close to −3.8 and indicates some deviation from smooth spherical behavior. Interestingly, the ZnO partial structure factor extracted from sample A does not exhibit any Guinier-like behavior at the lowest *q*, as shown by samples B and C. This indicates that either large ZnO clusters are formed in the case of the unfilled rubber in comparison to the silica-filled ones, which prevent the appearance of the Guinier region in the accessible *q*-range, or huge micron-sized strongly polydisperse and rough particles are still present, the size of which is larger than *R* > 1/qmin =1000 Å =0.1 μ.

**Figure 6.** Partial structure factors for ZnO extracted from sample A (black), B (cyan), and C (magenta), and theoretical calculation of a spherical particle form factor (blue line).

The situation is entirely different if ZnO is mixed into the rubber together with a much larger amount of silica. Figure 6 clearly shows that after the mixing and vulcanization rather monodisperse single ZnO particles are dispersed, although the high *q* region cannot be detected precisely anymore. We confirm this by comparing with a calculation of a monodisperse particle form factor (blue line in Figure 6). The computed intensity of a single ZnO particle follows in reasonable agreement the calculated spherical form factor, i.e., the Guinier plateau and the characteristic steep drop of the order of about 100 at q ~ 0.01 Å−1, i.e., *qR* ~ *4.5*. The silica nanofillers, therefore, seem to affect the size of the ZnO nanocrystals and/or their aggregation. This could be explained considering the different shear forces imposed on the sample during the milling process. Such forces typically increase due to the presence of silica fillers in the rubber, which increase the melt viscosity by orders of magnitude and lead to a break-down of ZnO aggregates or crystals.

In order to compare the size of silica aggregates and ZnO species in the filled samples, the low *q* scattering for both silica and ZnO partial structure factors is presented in a Guinier-type representation in Figures 7 and 8. The slopes are then related to –*R*2 *g*/3.

Next, silica is discussed. For the silica-extracted form factor, shown in Figure 7, an average radius of gyration *Rg* can be determined only for sample C because sample B does not exhibit a Guinier-like behavior or is more polydisperse. A cluster size of about 280 Å is obtained from the linear fit of the Guinier plot of *Ssil*(*q*) and an increase is expected with a lower volume fraction. However, it has to be taken into account that the extraction of the silica cluster size from the Guinier-like description consists also of an approximation, as sample C does show a deviation from the simple Guinier behavior at low *q.* This extrapolation neither takes into account the polydispersity nor the interactions between the clusters. A more precise estimation of the silica cluster sizes is out of the scope of this work. A fit to a hierarchical scattering model is treated in a recently published work [31].

For samples B and C the extrapolation of the ZnO size, reported in Figure 8 yields almost comparable sizes of about 450–480 Å. It could not be applied to the ZnO form factor extracted from sample A as its low *q*-dependence clearly deviates from the Guinier behavior. Note that the maximum detectable size in the available q-range would approximately be ~1000 Å. Thus, the size of ZnO does not differ considerably from the silica aggregates in the case of the filled samples. Therefore, we infer that in the presence of silica, the processing conditions probably have a comparable effect on the de-agglomeration or destruction of the ZnO granulates. Besides this, the values found for the ZnO radius are comparable with the ones reported in the literature [22]. In other works, the size determination by means of other techniques for ZnO dispersed particles ranged from 300 to 500 Å as the result of the balance of fracture of aggregates and material resistance, i.e., intrinsic hardness in milling processes [32,33].

**Figure 7.** Guinier extrapolation for the silica cluster radius for sample C (black). The fit cannot be uniquely applied to sample B (green) due to the strong deviation from linearity.

**Figure 8.** Guinier extrapolation of the ZnO radius for sample C (black) and B (green). Sample A (orange) clearly shows a deviation from the Guinier dependence.

#### **5. Conclusions**

We have shown that in industrially mixed rubber compounds a decomposition of the SAXS scattering intensity into its single contributing components, i.e., silica and ZnO is possible. We showed that SANS is hardly affected as it was postulated. The structure factor, the size and morphology of the silica aggregates and of the ZnO nanocrystals as an additional element of analysis can now be obtained. Latter strongly depends on the presence of silica during the mixing process. We observe that ZnO in silica-free samples forms aggregates or is of the micron-size and is characterized by a fractal surface. It appears to be structured on the same length scale as the silica filler. Therefore, a dependence of the size on the shear force exerted on the samples during the mixing process is suspected. Furthermore, the size of the ZnO catalyst is nearly constant for filled samples containing 30 and 60 phr of silica. As a consequence of this study, corrections for ZnO scattering based on computation with well-estimated contrast factors could be formulated. With estimates for the size of the resulting particle and assumptions for the average SLD of the effective matrix for the contrast, the contribution of *SZnO* can be calculated theoretically for standard SAXS analyses. The proposed analysis is general and can be applied in the future to novel dual filler systems, which scatter both but differently strong in both SANS and SAXS.

**Author Contributions:** Conceptualization, M.S. (Mariapaola Staropoli) and W.P.; data curation, M.S. (Mariapaola Staropoli), A.R., M.S. (Michael Sztucki) and W.P.; formal analysis, M.S. (Mariapaola Staropoli) and W.P.; funding acquisition, S.W., D.L., B.D. and D.G.; investigation, D.G., M.S. (Michael Sztucki), A.R., M.S. (Mariapaola Staropoli) and W.P.; methodology, M.S. (Mariapaola Staropoli), W.P., D.G.; project administration, D.L., S.W., D.G. and B.D.; software, M.S. (Mariapaola Staropoli) and W.P.; supervision, D.L. and W.P.; visualization, M.S. (Mariapaola Staropoli); writing—original draft, M.S. (Mariapaola Staropoli) and W.P.; writing—review & editing, M.S. (Mariapaola Staropoli), D.G. and W.P. All authors have read and agreed to the published version of the manuscript.

**Funding:** The present project was supported by the National Research Fund Luxembourg (No (IPBG16/11514551/TireMat-Tech) whose financial support is acknowledged.

**Acknowledgments:** The authors thank T. Unruh and W. Gruber of the Friedrich-Alexander-Universität Erlangen-Nürnberg for the control measurement of the SAXS intensity of sample E and E. Scolan for scientific support. They thank V. Pipich of the MLZ for the corresponding service at KWS-3. The authors are grateful to the MLZ for the provision of neutron beam time at KWS-2 and KWS-3 in the framework of internal JCNS-1 access time. The authors thank Goodyear S.A. for the permission to publish the paper.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## *Article* **E**ffi**cient Chain Formation of Magnetic Particles in Elastomers with Limited Space**

**Shota Akama 1,2, Yusuke Kobayashi 1,2, Mika Kawai 1,2 and Tetsu Mitsumata 1,2,\***

Received: 27 December 2019; Accepted: 26 January 2020; Published: 1 February 2020

**Abstract:** The magnetic response of the storage modulus for bimodal magnetic elastomers containing magnetic particles with a diameter of 7.0 μm and plastic beads with a diameter of 200 μm were investigated by varying the volume fraction of plastic beads up to 0.60 while keeping the volume fraction of the magnetic particles at 0.10. The storage modulus at 0 mT for monomodal magnetic elastomers was 1.4 × 104 Pa, and it slightly increased with the volume fraction of plastic beads up to 0.6. The storage modulus at 500 mT for bimodal magnetic elastomers at volume fractions below 0.25 was constant, which was equal to that for the monomodal one (=7.9 × 104 Pa). At volume fractions of 0.25–0.40, the storage modulus significantly increased with the volume fraction, showing a percolation behavior. At volume fractions of 0.40-0.60, the storage modulus was constant at 2.0 × 105 Pa, independently of the volume fraction. These results indicate that the enhanced increase in the storage modulus was caused by the chain formation of the magnetic particles in vacancies made of plastic beads.

**Keywords:** soft material; stimuli-responsive gel; magnetic elastomer; percolation

#### **1. Introduction**

Magnetic elastomers are a type of stimuli-responsive soft material [1–5] and its physical properties alter in response to magnetic fields. The magnetic response for a magnetic elastomer is in general drastic; therefore, the material attracts considerable attention as actuators in the next generation of materials [6–8]. Magnetic elastomers consist of polymeric matrices, such as polyurethane, and magnetic particles with nano or micron sizes in diameter. When a magnetic field is applied to a magnetic elastomer, the elasticity increases due to the chain structure formation (restructuring) of magnetic particles, which is called the magnetorheological (MR) effect. So far, we have investigated the MR effect for polyurethane-based magnetic elastomers and found that bimodal magnetic elastomers with magnetic and nonmagnetic particles exhibit a significant MR effect compared with monomodal magnetic elastomers [9–13].

In our past studies, zinc oxide with a diameter of 10.6 μm or aluminum hydroxide with a diameter of 1.4 μm was used as nonmagnetic particles [9,11]. The increase in the storage modulus for bimodal magnetic elastomers containing zinc oxide particles of 12 vol.% was 500 kPa, which was achieved by applying a magnetic field of 500 mT, which is 4.2 times that used for the monomodal one (120 kPa) [9]. The increase in the storage modulus for bimodal magnetic elastomers containing aluminum hydroxide particles of 6.6 vol.% was 3.27 MPa, which is 4.3 times that used for the monomodal one (754 kPa) [11]. Figure 1 shows the schematic illustrations representing the mechanism for the MR effect for bimodal magnetic elastomers in our previous study and the scenario for that in the present study. In the previous study, the enhanced magnetorheology mentioned above is caused by bridging the discontinuous chains

of magnetic particles with nonmagnetic particles. However, this mechanism is not efficient since a considerable amount of nonmagnetic particles is needed to raise the elasticity of magnetic elastomers.

**Figure 1.** Schematic illustrations representing the mechanism for the magnetorheological (MR) effect for bimodal magnetic elastomers in our previous study and the scenario in the present study.

In our present study, we proposed a new concept for the enhanced magnetorheology of bimodal magnetic elastomers. A large bead of poly(methyl-2-methylpropenoate) (PMMA) with a diameter of 200 μm was used as a non-magnetic particle. The diameter of the bead was large, and the interaction with the matrix of polyurethane was weak compared with nonmagnetic particles in past studies. Beads with a large diameter made vacancies in the magnetic elastomer, and magnetic particles were localized in the vacancies made of the plastic beads. The density of the plastic was low compared with inorganic compounds; therefore, the beads were not precipitated during the synthesis; this contributed to the random dispersion of vacancies in the elastomer. Bimodal magnetic elastomers with plastic beads also have advantages of transparency and weight saving. The density of zinc oxide and aluminum hydroxide is 5.8 and 2.4 g/cm3, respectively, meanwhile, the density of the plastic bead was 1.2 g/cm3. Magnetic elastomers with both a lightweight and huge magnetic response could be obtained. Bimodal magnetic elastomers with plastic beads have a possibility to transmit visible light, although the light may reflect at the interface between the matrix and plastic beads. This should be helpful for biologists or medical scientists who investigate the effect of substrate elasticity on the cell behavior using magnetic elastomers [14].

Here, we prepared bimodal magnetic elastomers with various volume fractions of plastic beads while keeping the volume fraction of magnetic particles constant, and discuss the addition effect of plastic beads on the magnetic response, the transmissibility of visible light, and weight saving.

#### **2. Experimental Procedures**

#### *2.1. Synthesis of Magnetic Elastomers*

Polypropylene glycols (P2000, G3000B, Adeka Co., Tokyo, Japan) with molecular weights of *M*w = 2000 and 3000 were used for the matrix of magnetic elastomers. Tolyrene diisocyanate (Wako Pure Chemical Industries. Ltd., Osaka, Japan) and dioctyl phthalate (DOP, Wako Pure Chemical Industries. Ltd., Osaka, Japan) were used as a crosslinker and plasticizer, respectively. Carbonyl iron with a median diameter of 7.0 μm (CS Grade BASF SE., Ludwigshafen am Rhein, Germany) was used for magnetic particles. The saturation magnetization of carbonyl iron particles was 190 emu/g measured using a SQUID magnetometer (MPMS, Quantum Design Inc., San Diego, CA, USA). Plastic beads made of polymethylmethacrylate (PMMA) (Techpolymer, MBX-200, Sekisui Plastics Co., Ltd., Tokyo, Japan) with a mean diameter of 200 μm were used as nonmagnetic particles. Magnetic elastomers were synthesized using a prepolymer method. Polypropylene glycols are crosslinked with tolyrene diisocyanate. The concentration of the crosslinker remarkably affects not only the off-field modulus, but also the magnetorheological response. The molar ratio of –NCO to –OH group for the prepolymer

was constant at 2.01 (=[NCO]/[OH]). Magnetic particles and plastic beads were mixed with prepolymer, linear polymer, plasticizer, and catalysis. The mixed liquid was poured into a silicon mold and cured for 30 min at 100 ◦C. DOP has a good chemical affinity with polypropylene glycols, which is a good solvent for crosslinked PPG elastomers. Similar to the crosslinker concentration, the concentration of the plasticizer strongly affects both the off-field modulus and the magnetorheological response. The weight concentration of DOP to the matrix without magnetic particles was fixed at 60 wt.%. The densities of the magnetic particles and plastic beads were 7.57 and 1.20 g/cm3, respectively. The volume fraction of magnetic particles was kept at 0.1; meanwhile, for plastic beads, it was varied from 0.10 to 0.60.

#### *2.2. Dynamic Viscoelastic Measurement*

Dynamic viscoelastic measurements were carried out for magnetic elastomers using a rheometer (MCR301, Anton Paar Pty. Ltd., Graz, Austria) at 20 ◦C. The strain was varied from 10−5 to 1 and the frequency was kept at 1 Hz. The sample was a disk that was 20 mm in diameter and 1.5 mm in thickness. The normal force initially applied to the magnetic elastomer was approximately 0.3 N.

#### *2.3. SEM Observation*

The shape of the magnetic and nonmagnetic particles in the powder state and the particle morphology for monomodal and bimodal elastomers were observed using scanning electron microscopy (SEM, JCM-6000 Neoscope JEOL Ltd. Tokyo, Japan) with an accelerating voltage of 15 kV without a Au coating. SEM photographs for magnetic particles and plastic beads (nonmagnetic particles) are presented in Figure 2.

**Figure 2.** SEM photographs for (**a**) magnetic particles and (**b**) plastic beads (nonmagnetic particles).

#### **3. Results and Discussion**

Figure 3a exhibits the magnetic field response of the storage modulus at a strain of 10−4 (in the linear viscoelastic regime) for monomodal and bimodal magnetic elastomers containing plastic beads. A magnetic field of 500 mT was applied to the magnetic elastomers every 60 s. It was observed for all magnetic elastomers that the storage modulus was altered synchronously with the magnetic field, and completely recovered to the original modulus after removing the field. The storage modulus for bimodal magnetic elastomers at φPB < 0.3 rapidly increased due to the application of the magnetic field. This result coincided with our previous study showing that the alignment time improved by adding aluminum hydroxide nonmagnetic particles [11]. Meanwhile, at φPB > 0.35, the storage modulus gradually increased with time and it was not saturated within 60 s. It is considered that only magnetic particles move and form a chain structure at φPB < 0.3. At φPB > 0.35, large plastic beads moved, accompanying the movement of the magnetic particles, resulting in long relaxation time.

**Figure 3.** Magnetic field response of the storage modulus at strains of (**a**) 10−4 and (**b**) 1 for monomodal and bimodal magnetic elastomers containing plastic beads with various volume fractions.

Figure 3b demonstrates the magnetic-field response of the storage modulus at a strain of 1 (in the non-linear viscoelastic regime) for monomodal and bimodal magnetic elastomers containing plastic beads. Similar to the results at low strain, all magnetic elastomers responded to the magnetic field at high strain. However, the time development of the storage modulus was opposite at high strain. A gradual increase of the storage modulus with time was observed for monomodal and bimodal magnetic elastomers with φPB < 0.3, and a rapid increase of the storage modulus with time was observed for bimodal magnetic elastomers with φPB > 0.35.

Figure 4a depicts the storage modulus at 10−4 for monomodal and bimodal magnetic elastomers as a function of the volume fraction of plastic beads. The storage modulus at 0 mT was almost constant, although the volume fraction of the plastic beads increased. In general, when the volume fraction of fillers was high, the storage modulus for the composite elastomers was higher than that for the matrix. The low storage modulus seen at φPB = 0.60 indicated that the plastic beads were randomly dispersed in the elastomer without direct contact between the beads. Of course, the interfacial effect between the matrix and plastic beads with a large diameter should be negligibly small compared with the small particles being several microns in size. Actually, when a plastic bead with a diameter of 8 μm was used, the storage modulus at 0 mT for the bimodal magnetic elastomer at φPB = 0.60 was 3.6 × 105 Pa, which was far higher than that for the elastomer with large particles. In addition, the change in the storage modulus was very small and negative. This strongly indicated that the total interfacial area was dominant for the storage modulus of magnetic elastomers, i.e., the storage modulus could be raised by increasing the interfacial area, even though the interfacial interaction between the matrix and plastic beads was weak. The storage modulus at 500 mT for bimodal magnetic elastomers was equal to that of the monomodal one, and it was almost independent of the volume fraction at φPB < 0.25. This behavior was unusual, and it indicated that bridging did not occur between discontinuous chains of magnetic particles via plastic beads, even though the volume fraction of plastic beads was considerably high. At 0.25 < φPB < 0.40, the storage modulus significantly increased with the volume fraction of the plastic beads. It was considered that the magnetic particles bridged the gap between the plastic beads. An interesting thing was that the significant increase in the storage modulus was only seen at 500 mT and did not appear at 0 mT. In addition, the influence of the matrix elasticity on the magnetorheological effect was investigated. The storage modulus for polyurethane elastomers with (φPB=0.40) and without plastic beads was (3.8 ± 0.35) × 104 Pa and (3.3 ± 0.34) × 104 Pa, respectively.

**Figure 4.** (**a**) Storage modulus at a strain of 10−4 at 0 and 500 mT as a function of the volume fraction of plastic beads and (**b**) storage modulus as a function of φPB − φc PB for monomodal and bimodal magnetic elastomers.

Therefore, an important factor for the stress transfer between plastic beads is the chain structure of magnetic particles, not the elasticity of the matrix. The onset volume fraction for the percolation is considered to be 0.25 < φPB < 0.30, which is close to the value of the cite percolation for a body-centered cubic structure (=0.248) [15,16]. Figure 4b shows the relation between the storage modulus and the volume fraction of plastic beads (φPB − φc PB) for bimodal magnetic elastomers. The storage modulus at φPB > 0.25 obeyed a power low of *G*500~(φPB − φc PB) 0.64, although the critical exponent was relatively high compared with that of a 3D lattice (≈0.4) [15,16]. Accordingly, the plastic beads were packed in the structure of the body-centered cubic structure, and magnetic particles were filled in the vacancies between the beads. At φPB > 0.4, the storage modulus was not raised, although the volume fraction of plastic beads was increased. This indicated that the percolation possibility reached a maximum at φPB ≈ 0.5, which was 74% of the maximum packing ratio for the lattice of a body-centered cubic structure. It might be that the effective paths contributing to the storage modulus were not increased at φPB > 0.4 due to the increase in the number of branches of the percolated paths. Figure 5 exhibits the schematic illustrations representing the mechanism of the elasticity increase by the magnetic field for bimodal magnetic elastomers with plastic beads with a large diameter. The distance between the nearest neighbor beads was calculated to be 31 μm. Magnetic particles are forced to be localized in the vacancy and form a chain structure by the magnetic field, for example, a qualitative representation of the microstructure [17] and computed tomography images [18]. In our previous study, the storage modulus for bimodal magnetic elastomers containing small nonmagnetic particles (1.4 or 10.6 μm in diameter) gradually increased with the volume fraction of nonmagnetic particles, and a clear percolation threshold was not observed. This strongly indicated that the chains of magnetic particles were gradually connected via nonmagnetic particles with increases in the volume fraction of nonmagnetic particles [9–11]. The mechanism presented here was completely different from that observed in the past. Figure 5 also shows a realization of percolation behavior for bimodal magnetic elastomers with plastic beads. Each white circle represents a plastic bead, and the black part represents the magnetic elastomer, i.e., polyurethane and magnetic particles. The percolation of the stress between the plastic beads occurred via chains of magnetic particles at φPB = 0.25. The percolation path increased with the volume fraction of plastic beads at 0.25 < φPB < 0.40. At 0.40 < φPB < 0.60, both the percolation path and the branch of the paths increased with the volume fraction.

**Figure 5.** Schematic illustrations representing (**left**) the mechanism of the MR effect and (**right**) a realization of percolation behavior for bimodal magnetic elastomers with large plastic beads.

Figure 6a,e indicates the SEM photographs for monomodal magnetic elastomers taken at different magnifications. Figure 6e demonstrates the fact that no clear aggregations of magnetic particles were found in the polyurethane matrix. However, the photo showed that there were two regions with different concentrations of magnetic particles, i.e., densely and sparsely dispersed parts. This result coincided with the previous results showing that carbonyl iron particles form secondary particles within the polyurethane matrix when ultrasonication is not carried out [19]. Figure 6c,g displays the SEM photographs for bimodal magnetic elastomers with plastic beads above the percolation threshold (φPB = 0.4). It was clear that magnetic particles were localized in the vacancy made by plastic beads, meaning the success of our scenario given that magnetic particles were gathered in a limited space. The hole made by removing the plastic bead showed a very smooth surface, suggesting that the interaction between the plastic beads and the matrix was weak (see Figure 6f). Figure 6d,h shows the SEM photographs for bimodal magnetic elastomers containing plastic beads with the maximum volume fraction (φPB = 0.60). It was clearly observed that magnetic particles were distributed in the vacancy.

**Figure 6.** *Cont*.

**Figure 6.** SEM photographs for (**a**,**e**) monomodal magnetic elastomers, (**b**–**d**,**f**–**h**) bimodal magnetic elastomers with various volume fractions of plastic beads (**b**,**f**) 0.25, (**c**,**g**) 0.40, and (**d**,**h**) 0.60.

Figure 7 demonstrates the photographs representing the transparency and the density for bimodal magnetic elastomers with plastic beads obtained in this study. The transparency for bimodal magnetic elastomers with plastic beads (φPB~0.60) was compared with that for monomodal ones. The increase in the storage modulus for the bimodal magnetic elastomer (=(8.00 ± 0.53) × 104 Pa) was equal to that for monomodal one (=(7.1 ± 0.26) × 104 Pa). In this experiment, the volume fraction of the magnetic particles was kept at 0.07 because the difference in the transparency for these elastomers was not clear to see. The thickness of the monomodal and bimodal magnetic elastomers was approximately 180 and 290 μm, respectively. It was found that the paint, the mountain with snow, could only be seen through the bimodal magnetic elastomer. The density for the bimodal magnetic elastomers with plastic beads (φPB ≈ 0.4, 0.6) was also compared with that for a monomodal one. The increase in the storage modulus was (2.7 ± 0.99) × 105 Pa for monomodal, (1.8 ± 0.01) × 105 Pa for bimodal with φPB ≈ 0.4 and (1.9 ± 4.2) × 105 Pa for bimodal with φPB ≈ 0.6. The density was 2.11 g/cm3 for monomodal, 1.37 g/cm3 for bimodal with φPB ≈ 0.4 and 1.15 g/cm3 for bimodal with φPB ≈ 0.6. Weight saving of −59% was achieved by creating the limited space in the magnetic elastomer. It was the case that the particle distribution or the local rearrangement of magnetic particles [20,21] was considered and designed for an efficient MR effect in the vacancy.

**Figure 7.** *Cont*.

**Figure 7.** Photographs representing the transparency for monomodal and bimodal magnetic elastomers: (**a**) original paint, and (**b**) monomodal, and (**c**) bimodal magnetic elastomers were put on the original paint. (**d**) The density for monomodal and bimodal magnetic elastomers with plastic beads of φPB ≈ 0.4 and 0.6.

#### **4. Conclusions**

The magnetorheological response for bimodal magnetic elastomers with large plastic beads was investigated. It was found that the increment in the storage modulus for the bimodal magnetic elastomers was higher than twice that of the monomodal one. It could be considered that the enhanced magnetorheology was caused by an efficient chain formation of magnetic particles in a vacancy made by the plastic beads that packed into the body-centered cubic structure. In addition, it was found that the bimodal magnetic elastomer could partially transmit natural light, although the clarity was very low. At approximately half the weight of conventional elastomers, the bimodal magnetic elastomers showed the same level of elasticity changes. We believe that the magnetorheological features presented here are useful not only for applications but also for designing materials for new magnetorheological elastomers with a high efficiency and multiple functions.

**Author Contributions:** The idea for this work came from T.M. M.K. conceived and designed all the experiments; S.A. performed the sample synthesis, rheological experiment, and SEM observations for magnetic elastomers and wrote the paper; and Y.K. carried out the rheological experiment and drew the graphs and graphic abstract. Presentation of the results and the draft structure were discussed between all authors. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was partially supported by a grant from the Cooperative Research Program of Cooperative Research Program of the Network Joint Research Center for Materials and Devices (No. 20191257).

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).