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Q Tensor
In physics, \mathbf Q-tensor is an orientational order parameter that describes uniaxial and biaxial nematic liquid crystals and vanishes in the isotropic liquid phase. The \mathbf Q tensor is a second-order, traceless, symmetric tensor and is defined byDe Gennes, P. G., & Prost, J. (1993). The physics of liquid crystals (No. 83). Oxford university press.Kleman, M., & Lavrentovich, O. D. (Eds.). (2003). Soft matter physics: an introduction. New York, NY: Springer New York. :\mathbf = S\left(\mathbf n\otimes\mathbf n - \tfrac\mathbf I\right) + R\left(\mathbf m\otimes\mathbf m - \tfrac\mathbf I\right) where S=S(T) and R=R(T) are scalar order parameters, (\mathbf n,\mathbf m) are the two directors of the nematic phase and T is the temperature; in uniaxial liquid crystals, R=0. The components of the tensor are :Q_ = S\left(n_in_j - \tfrac\delta_\right) + R\left(m_im_j - \tfrac\delta_\right) The states with directors \mathbf n and -\mathbf n are physically equivalent and similarly ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Order Parameter
In physics, chemistry, and other related fields like biology, a phase transition (or phase change) is the physical process of transition between one state of a medium and another. Commonly the term is used to refer to changes among the basic states of matter: solid, liquid, and gas, and in rare cases, plasma. A phase of a thermodynamic system and the states of matter have uniform physical properties. During a phase transition of a given medium, certain properties of the medium change as a result of the change of external conditions, such as temperature or pressure. This can be a discontinuous change; for example, a liquid may become gas upon heating to its boiling point, resulting in an abrupt change in volume. The identification of the external conditions at which a transformation occurs defines the phase transition point. Types of phase transition States of matter Phase transitions commonly refer to when a substance transforms between one of the four states of matter ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Biaxial Nematic
A biaxial nematic is a spatially homogeneous liquid crystal with three distinct optical axes. This is to be contrasted to a simple nematic, which has a single preferred axis, around which the system is rotationally symmetric. The symmetry group of a biaxial nematic is D_ i.e. that of a rectangular right parallelepiped, having 3 orthogonal C_2 axes and three orthogonal mirror planes. In a frame co-aligned with optical axes the second rank order parameter tensor, the so-called Q tensor of a biaxial nematic has the form : \mathbf Q= \begin -\frac(S+P) & 0 &0 \\ 0 &-\frac(S-P) & 0 \\ 0 & 0& S\\ \end where S is the standard Liquid_crystal#Order_parameter, nematic scalar order parameter and P is a measure of the biaxiality. The first report of a thermotropic biaxial nematic appeared in 2004 based on a boomerang shaped oxadiazole bent-core mesogen. The biaxial nematic phase for this particular compound only occurs at temperatures around 200 °C and is preceded by as yet unidentified ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Liquid Crystals
Liquid crystal (LC) is a state of matter whose properties are between those of conventional liquids and those of solid crystals. For example, a liquid crystal can flow like a liquid, but its molecules may be oriented in a common direction as in a solid. There are many types of LC phases, which can be distinguished by their optical properties (such as textures). The contrasting textures arise due to molecules within one area of material ("domain") being oriented in the same direction but different areas having different orientations. An LC material may not always be in an LC state of matter (just as water may be ice or water vapour). Liquid crystals can be divided into three main types: thermotropic, lyotropic, and metallotropic. Thermotropic and lyotropic liquid crystals consist mostly of organic molecules, although a few minerals are also known. Thermotropic LCs exhibit a phase transition into the LC phase as temperature changes. Lyotropic LCs exhibit phase transitions a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isotrope
In physics and geometry, isotropy () is uniformity in all orientations. Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix ' or ', hence ''anisotropy''. ''Anisotropy'' is also used to describe situations where properties vary systematically, dependent on direction. Isotropic radiation has the same intensity regardless of the direction of measurement, and an isotropic field exerts the same action regardless of how the test particle is oriented. Mathematics Within mathematics, ''isotropy'' has a few different meanings: ; Isotropic manifolds: A manifold is isotropic if the geometry on the manifold is the same regardless of direction. A similar concept is homogeneity. ; Isotropic quadratic form: A quadratic form ''q'' is said to be isotropic if there is a non-zero vector ''v'' such that ; such a ''v'' is an isotropic vector or null vector. In complex geometry, a line through the origin in the direction of an isotropi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Temperature
Temperature is a physical quantity that quantitatively expresses the attribute of hotness or coldness. Temperature is measurement, measured with a thermometer. It reflects the average kinetic energy of the vibrating and colliding atoms making up a substance. Thermometers are calibrated in various temperature scales that historically have relied on various reference points and thermometric substances for definition. The most common scales are the Celsius scale with the unit symbol °C (formerly called ''centigrade''), the Fahrenheit scale (°F), and the Kelvin scale (K), with the third being used predominantly for scientific purposes. The kelvin is one of the seven base units in the International System of Units (SI). Absolute zero, i.e., zero kelvin or −273.15 °C, is the lowest point in the thermodynamic temperature scale. Experimentally, it can be approached very closely but not actually reached, as recognized in the third law of thermodynamics. It would be impossible ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Invariants Of Tensors
In mathematics, in the fields of multilinear algebra and representation theory, the principal invariants of the second rank tensor \mathbf are the coefficients of the characteristic polynomial :\ p(\lambda)=\det (\mathbf-\lambda \mathbf), where \mathbf is the identity operator and \lambda_i \in\mathbb are the roots of the polynomial \ p and the eigenvalues of \mathbf. More broadly, any scalar-valued function f(\mathbf) is an invariant of \mathbf if and only if f(\mathbf\mathbf\mathbf^T)=f(\mathbf) for all orthogonal \mathbf. This means that a formula expressing an invariant in terms of components, A_, will give the same result for all Cartesian bases. For example, even though individual diagonal components of \mathbf will change with a change in basis, the sum of diagonal components will not change. Properties The principal invariants do not change with rotations of the coordinate system (they are objective, or in more modern terminology, satisfy the principle of material frame ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Solid Angle
In geometry, a solid angle (symbol: ) is a measure of the amount of the field of view from some particular point that a given object covers. That is, it is a measure of how large the object appears to an observer looking from that point. The point from which the object is viewed is called the ''apex'' of the solid angle, and the object is said to '' subtend'' its solid angle at that point. In the International System of Units (SI), a solid angle is expressed in a dimensionless unit called a ''steradian'' (symbol: sr), which is equal to one square radian, sr = rad2. One steradian corresponds to one unit of area (of any shape) on the unit sphere surrounding the apex, so an object that blocks all rays from the apex would cover a number of steradians equal to the total surface area of the unit sphere, 4\pi. Solid angles can also be measured in squares of angular measures such as degrees, minutes, and seconds. A small object nearby may subtend the same solid angle as a larger object ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Landau–de Gennes Theory
In physics, Landau–de Gennes theory describes the NI transition, i.e., phase transition between nematic liquid crystals and isotropic liquids, which is based on the classical Landau's theory and was developed by Pierre-Gilles de Gennes in 1969. The phenomonological theory uses the \mathbf tensor as an order parameter in expanding the free energy density. Mathematical description The NI transition is a first-order phase transition, albeit it is very weak. The order parameter is the \mathbf tensor, which is symmetric, traceless, second-order tensor and vanishes in the isotropic liquid phase. We shall consider a uniaxial \mathbf Q tensor, which is defined by :\mathbf Q = S(\mathbf n\otimes\mathbf n - \tfrac\mathbf I) where S=S(T) is the scalar order parameter and \mathbf n is the director. The \mathbf Q tensor is zero in the isotropic liquid phase since the scalar order parameter S is zero, but becomes non-zero in the nematic phase. Near the NI transition, the (Helmholtz ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Soft Matter
Soft matter or soft condensed matter is a type of matter that can be deformed or structurally altered by thermal or mechanical stress which is of similar magnitude to thermal fluctuations. The science of soft matter is a subfield of condensed matter physics. Soft materials include liquids, colloids, polymers, foams, gels, granular materials, liquid crystals, flesh, and a number of biomaterials. These materials share an important common feature in that predominant physical behaviors occur at an energy scale comparable with room temperature thermal energy (of order of kT), and that entropy is considered the dominant factor. At these temperatures, quantum aspects are generally unimportant. When soft materials interact favorably with surfaces, they become squashed without an external compressive force. Pierre-Gilles de Gennes, who has been called the "founding father of soft matter," received the Nobel Prize in Physics in 1991 for discovering that methods developed f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Phase Transitions
In physics, chemistry, and other related fields like biology, a phase transition (or phase change) is the physical process of transition between one state of a medium and another. Commonly the term is used to refer to changes among the basic State of matter, states of matter: solid, liquid, and gas, and in rare cases, plasma (physics), plasma. A phase of a thermodynamic system and the states of matter have uniform physical property, physical properties. During a phase transition of a given medium, certain properties of the medium change as a result of the change of external conditions, such as temperature or pressure. This can be a discontinuous change; for example, a liquid may become gas upon heating to its boiling point, resulting in an abrupt change in volume. The identification of the external conditions at which a transformation occurs defines the phase transition point. Types of phase transition States of matter Phase transitions commonly refer to when a substance tran ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
