The Boundary Element Method for Fluctuating Active Colloids DOI: http://dx.doi.org/10.5772/intechopen.86738

with u∞ð Þ¼� xs vsð Þ xs on S? Constructing an effective flow field would allow us to obtain Fdrag, τdrag, and S by Faxén laws, without having to solve the complete hydrodynamic problem posed in Section II. Moreover, an understanding of how vsð Þ x determines Fdrag, τdrag, and S would pave the way towards development of a hybrid BEM-Stokesian Dynamics scheme, since these quantities are central to SD.

As our starting point, we write the Taylor expansion of u∞ð Þ x :

$$u\_i^{\infty}(\mathbf{x}) = u\_i^{\infty}(\mathbf{0}) + \left. \frac{\partial u\_i^{\infty}}{\partial \mathbf{x}\_j} \right|\_{\mathbf{x} = \mathbf{0}} \propto\_j. \tag{80}$$

To obtain u∞ i ð Þ 0 , we integrate both sides of Eq. 80 over the surface of the sphere:

$$\int u\_i^{\infty}(\mathbf{x})d\mathbf{S} = \int u\_i^{\infty}(\mathbf{0})d\mathbf{S} + \left. \int \frac{\partial u\_i^{\infty}}{\partial \mathbf{x}\_j} \right|\_{\mathbf{x}=\mathbf{0}} \mathbf{x}\_j d\mathbf{S}.\tag{81}$$

We identify u∞ i ð Þ x on the surface of the sphere as �vsð Þ xs . The second integral on the right hand side of Eq. 81 vanishes, giving

$$\mathbf{u}^{\infty}(\mathbf{0}) = -\frac{1}{4\pi R^2} \left[ \mathbf{v}'(\mathbf{x}\_i) \, \text{d}\mathbf{S}.\tag{82}$$

Using Eq. 75, we obtain

u∞

Non-Equilibrium Particle Dynamics

i ð Þ¼ x u∞

and the (anti-symmetric) vorticity tensor

force on the sphere (see Ref. [59] for details):

where the angular velocity of the fluid ω � 1

2 xjσiknk þ xiσjknk � � � 1

S ¼ 20

where S is defined as an integral over the particle surface

the stresslet [59, 83, 84]

Sij ¼

flow field

68

ð 1

i ð Þþ 0 ∂u∞ i ∂xj � � � � x¼0 xj þ 1 2 ∂2 u∞ i ∂xj∂xk

Now we recall the definitions of the (symmetric) rate of strain tensor eij,

∂ui ∂xj þ ∂uj ∂xi � �

> ∂ui ∂xj

Using the Lorentz reciprocal theorem, one can obtain Faxén's law for the drag

R2 6 ∇2 � �

(In our shorthand notation, the Laplacian is first applied to u∞ð Þ x and then

R2 10 ∇2 � �

� �

u∞ð Þ� x0 U

τdrag ¼ 8πμR3 ω∞ ð Þ ð Þ� x0 Ω , (76)

ð Þ xkσklnl δij � μ uinj þ ujni

u∞ð Þ¼ x u∞ð Þþ 0 e∞ � x þ ω∞ � x (79)

� ∂uj ∂xi � �

eij ¼ 1 2

Wij ¼ 1 2

The vorticity tensor can be related to the vorticity vector w ¼ ∇ � u by

W ¼ 1 2

Here, ε is the Levi-Civita tensor. The first derivative in Eq. 70 can be

∂u∞ i ∂xj ¼ e ∞ ij þ W∞

decomposed into symmetric and anti-symmetric contributions:

Fdrag ¼ 6πμR 1 þ

evaluated at x0.) One can also obtain Faxén law for the drag torque:

3 πμR3 1 þ

3

question. Consider an active sphere with a slip velocity vsð Þ x . Comparing the boundary conditions in Eq. 3 and Eq. 69, can we construct an ambient linear

� � � �

So far we have only presented standard results, but now we raise the following

� � � � x¼0

xjxk þ … (70)

, (71)

, (72)

ε � w: (73)

ij : (74)

2 w. Finally, there is a Faxén law for

e∞ð Þ x0 , (77)

dS: (78)

: (75)

$$\mathbf{F}^{\text{drag}} = 6\pi\mu R \left[ -\frac{1}{4\pi R^2} \int \mathbf{v}'(\mathbf{x}\_i) \, dS - \mathbf{U} \right]. \tag{83}$$

If we consider a force-free swimmer, Fdrag ¼ 0, giving the result:

$$\mathbf{U} = -\frac{1}{4\pi R^2} \int \mathbf{v}^\prime(\mathbf{x}\_\prime) \, dS. \tag{84}$$

This equation is one of the major results obtained in Ref. 37 by use of the Lorentz reciprocal theorem. However, our rederivation and interpretation in terms of an effective ambient flow field u∞ is (to our knowledge) novel. To obtain the vorticity associated with u∞, we multiply Eq. 80 by εlmixm and integrate over the sphere surface:

$$\int \varepsilon\_{lmi} \boldsymbol{\mathcal{x}}\_{m} \boldsymbol{u}\_{i}^{\infty}(\mathbf{x}) \, d\mathbf{S} = \int \boldsymbol{u}\_{i}^{\infty}(\mathbf{0}) \boldsymbol{e}\_{lmi} \boldsymbol{\mathcal{x}}\_{m} \, d\mathbf{S} + \left. \int \frac{\partial \boldsymbol{u}\_{i}^{\infty}}{\partial \boldsymbol{\mathcal{x}}\_{j}} \right|\_{\mathbf{x}=\mathbf{0}} \varepsilon\_{lmi} \boldsymbol{\mathcal{x}}\_{m} \, d\mathbf{S}. \tag{85}$$

The first integral on the right hand side of Eq. 85 vanishes. For the second integral on the right hand side, we use the identity

$$
\int \mathbf{x}\_m \mathbf{x}\_j \, d\mathbf{S} = \frac{4\pi R^4}{3} \delta\_{jm} \,. \tag{86}
$$

We obtain:

$$\int \varepsilon\_{lmi} \mathbf{x}\_m \, u\_i^{\infty}(\mathbf{x}) \, d\mathbf{S} = \frac{4\pi R^4}{3} \varepsilon\_{lji} \frac{\partial u\_i^{\infty}}{\partial \mathbf{x}\_j} \bigg|\_{\mathbf{x} = \mathbf{0}} \tag{87}$$

$$-\int \mathbf{x} \times \mathbf{v}'(\mathbf{x}\_t) \, d\mathbf{S} = \frac{4\pi R^4}{3} \nabla \times \mathbf{u}^\infty \Big|\_{\mathbf{x}=0} = \frac{8\pi R^4}{3} \boldsymbol{\alpha}^\infty(\mathbf{0}),\tag{88}$$

so that

$$\boldsymbol{\phi}^{\infty}(\mathbf{0}) = -\frac{3}{8\pi R^4} \left[ \mathbf{x} \times \mathbf{v}'(\mathbf{x}\_t) \, d\mathbf{S}.\tag{89}$$

Using Eq. 76, we obtain:

$$\boldsymbol{\sigma}^{\text{drag}} = 8\pi\mu R^3 \left( -\frac{3}{8\pi R^4} \int \mathbf{x} \times \mathbf{v}^\epsilon(\mathbf{x}\_s) d\mathbf{S} - \boldsymbol{\mathfrak{Q}} \right). \tag{90}$$

For a torque-free swimmer, τdrag ¼ 0, and we obtain a second major result from Ref. [37]:

$$\mathbf{\dot{\Omega}} = -\frac{3}{8\pi R^4} \left[ \mathbf{x} \times \mathbf{v}'(\mathbf{x}\_t) \, d\mathbf{S}.\tag{91}$$

Finally, we consider how to obtain the stresslet S. We multiply Eq. 80 by xm and integrate over the surface of the sphere:

$$\int \boldsymbol{u}\_{i}^{\infty}(\mathbf{x}) \boldsymbol{\varkappa}\_{m} d\mathbf{S} = \int \boldsymbol{u}\_{i}^{\infty}(\mathbf{0}) \boldsymbol{\varkappa}\_{m} d\mathbf{S} + \left. \int \frac{\partial \boldsymbol{u}\_{i}^{\infty}}{\partial \boldsymbol{\varkappa}\_{j}} \right|\_{\mathbf{x}=\mathbf{0}} \boldsymbol{\varkappa}\_{m} \boldsymbol{\varkappa}\_{j} d\mathbf{S}.\tag{92}$$

The first integral on the right hand vanishes, giving

$$\int u\_i^{\infty}(\mathbf{x}) \varkappa\_m \, dS = \frac{4\pi R^4}{3} \frac{\partial u\_i^{\infty}}{\partial \varkappa\_m} \Big|\_{\mathbf{x}=0} \,. \tag{93}$$

Swapping the indices i and m, we can also write:

$$\int u\_m^{\infty}(\mathbf{x}) \propto\_i d\mathbf{S} = \frac{4\pi R^4}{\mathfrak{Z}} \frac{\partial u\_m^{\infty}}{\partial \mathbf{x}\_i} \bigg|\_{\mathbf{x}=\mathbf{0}}.\tag{94}$$

Author details

William E. Uspal

HI, United States

71

\*Address all correspondence to<EMAIL_ADDRESS>
The Boundary Element Method for Fluctuating Active Colloids

DOI: http://dx.doi.org/10.5772/intechopen.86738

provided the original work is properly cited.

Department of Mechanical Engineering, University of Hawai'i at Manoa, Honolulu,

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

Adding these two equations and dividing by two, we obtain

$$\frac{1}{2}\int \left[u\_i^{\infty}(\mathbf{x})\mathbf{x}\_m + u\_m^{\infty}(\mathbf{x})\mathbf{x}\_i\right]d\mathbf{S} = \frac{4\pi R^4}{3}e\_{im}^{\infty}(\mathbf{0}).\tag{95}$$

Accordingly,

$$e\_{im}^{\infty}(\mathbf{0}) = -\frac{3}{8\pi R^4} \int [v\_{\boldsymbol{\epsilon},i}(\mathbf{x})\boldsymbol{\epsilon}\_m + v\_{\boldsymbol{\epsilon},m}(\mathbf{x})\boldsymbol{\epsilon}\_i] d\mathbf{S}.\tag{96}$$

Using the Faxén Law in Eq. 77, we obtain:

$$\mathbf{S} = -\frac{5\mu}{2R} \int [\mathbf{v}\_{\boldsymbol{\epsilon}}(\mathbf{x}\_{\boldsymbol{\epsilon}})\,\mathbf{x} + \mathbf{x}\,\mathbf{v}\_{\boldsymbol{\epsilon}}(\mathbf{x}\_{\boldsymbol{\epsilon}})]d\mathbf{S}.\tag{97}$$

This is the major result obtained in Ref. 84 via the Lorentz reciprocal theorem. As before, this manuscript provides a novel alternative derivation and interpretation of Eq. 97 in terms of an effective ambient flow field. (Note that, due to the linearity of the Stokes equation, our approach is easily extended to model active particles in a real ambient flow field.)

The Boundary Element Method for Fluctuating Active Colloids DOI: http://dx.doi.org/10.5772/intechopen.86738

so that

Ref. [37]:

Using Eq. 76, we obtain:

Non-Equilibrium Particle Dynamics

integrate over the surface of the sphere:

i ð Þ x xm dS ¼

The first integral on the right hand vanishes, giving

ð u∞

Swapping the indices i and m, we can also write:

ð u∞

imð Þ¼� 0 3

S ¼ � 5μ 2R ð

Using the Faxén Law in Eq. 77, we obtain:

1 2 ð u∞

e ∞

particles in a real ambient flow field.)

Accordingly,

70

ð u∞ ω∞ð Þ¼� 0 3

Ω ¼ � 3

ð u∞

8πR4

τdrag ¼ 8πμR3 � 3

8πR4

8πR4

ð x � vs

ð x � vs

For a torque-free swimmer, τdrag ¼ 0, and we obtain a second major result from

Finally, we consider how to obtain the stresslet S. We multiply Eq. 80 by xm and

ð ∂u∞ i ∂xj � � � � x¼0

∂u∞ i ∂xm � � � � x¼0

∂u∞ m ∂xi � � � � x¼0

> 3 e ∞

vs,ið Þ x xm þ vs,mð Þ x xi ½ �dS: (96)

vsð Þ xs x þ xvsð Þ xs ½ �dS: (97)

i ð Þ 0 xm dS þ

3

3

i ð Þ x xm dS ¼ 4πR4

m ð Þ x xi dS ¼ 4πR4

m ð Þ x xi � �dS ¼ 4πR4

This is the major result obtained in Ref. 84 via the Lorentz reciprocal theorem. As before, this manuscript provides a novel alternative derivation and interpretation of Eq. 97 in terms of an effective ambient flow field. (Note that, due to the linearity of the Stokes equation, our approach is easily extended to model active

Adding these two equations and dividing by two, we obtain

i ð Þ x xm þ u∞

8πR4

ð

ð x � vs

� �

ð Þ xs dS: (89)

ð Þ xs dS: (91)

: (90)

xm xj dS: (92)

: (93)

: (94)

imð Þ 0 : (95)

ð Þ xs dS � Ω


**New Properties and Applications of Polyvinylidene-Based Ferroelectric Polymer**

J.L. Wang, X.J. Meng and J.H. Chu

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/60946

### **Abstract**

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[9] Tay, S.; Blanche, P.A.; Voorakaranam, R.; et al. *Nature*, 2008, *451*, 694-698.

[10] Blanche, P.A.; Bablumian, A.; Voorakaranam, R.et al. *Nature*, 2010, *468*, 80-83.

[11] Khoo, I.C. *Liquid Crystal Photorefractive Optics.* In: Yu, F.T.S (ed) *Photorefractive Optics*;

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*Applications II*; Springer: 2002, 87-156.

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150 Ferroelectric Materials – Synthesis and Characterization

Academic Press: 2000.

8678-8686.

There are different kinds of novel properties and applications of polyvinylidene difluoride (PVDF)-basedferroelectricpolymerfilms.Severalissues associatedwiththe structure, properties, and applications of PVDF-based ferroelectric polymer films are discussed. The main achievements of the research include high electric tunability of relaxorferroelectricLangmuir–Blodgett(LB)terpolymerfilms,the creepprocess ofthe domain switching in poly(vinylidene fluoride-trifluoroethylene) ferroelectric thin films, transition from relaxor to ferroelectric-like phase in poly(vinylidene fluoridetrifluoroethylene -chlorofluoroethylene) terpolymer ultrathin films, abnormal polarization switching of relaxor terpolymer films at low temperatures, huge electrocaloric effectin LB ferroelectric polymerthin films, self-polarization in ultrathin LB polymerfilms, enhanced dielectric and ferroelectric properties in artificial polymer multilayers, and transition of polarization switching from extrinsic to intrinsic in ultrathin PVDF homopolymer films.

In addition, the recent progresses of PVDF-based ferroelectric polymer films are connected with potential applications.

**Keywords:** Poly(vinylidene fluoride), Langmuir–Blodgett technique, Ferroelectric polymer, Ferroelectric relaxor


*Edited by Sadia Ameen, M. Shaheer Akhtar and Hyung-Shik Shin*

This book provides the fundamental understanding of the functioning of solar cellsand the materials for the effective utilization of energy resources. The main objective of writing this book is to create a comprehensive and easy-to-understand source of information on the advances in the rapidly growing research on solar cells. *Emerging Solar Energy Materials* comprises 12 chapters written by the experts in the solar cell field and is organized with the intention to provide a big picture of the latest progress in the solar cell field and at the same time give an in-depth discussion on fundamentals of solar cells for interested audiences. In this book, each part opens with a new author's essay highlighting their work for contribution toward solar energy. Critical, cutting-edge subjects are addressed, including: Photovoltaic device technology and energy applications Functional solar energy materials New concept in solar energy Perovskite solar cells Dye-sensitized solar cells Organic solar cells Thin-film solar cells The book is written for a large and broad readership including researchers and university graduate students from diverse backgrounds such as chemistry, physics, materials science, and photovoltaic device technology. The book includes enough information on the basics to be used as a textbook undergraduate coursework in engineering and the sciences.

ISBN 978-1-78923-582-1

Emerging Solar Energy Materials

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