<p>We obtain an expression for the sound velocity <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13538_2025_1794_Article_IEq1.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">\(\varvec{v}_{\varvec{s}}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mi mathvariant="bold-italic">v</mi> </mrow> <mrow> <mi mathvariant="bold-italic">s</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> in the isotropic phase of nematic liquid crystals from the nematodynamics that incorporates the energetic coupling between <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13538_2025_1794_Article_IEq2.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="58" /> </InlineMediaObject> <EquationSource Format="TEX">\(\varvec{Q}_{\varvec{\alpha \beta }}\varvec{(} \textbf{r}\varvec{)}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mrow> <mi mathvariant="bold-italic">Q</mi> </mrow> <mrow> <mi mathvariant="bold-italic">α</mi> <mi mathvariant="bold-italic">β</mi> </mrow> </msub> <mrow> <mo mathvariant="bold" stretchy="false">(</mo> </mrow> <mi mathvariant="bold">r</mi> <mrow> <mo mathvariant="bold" stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> (the order parameter of the nematic-isotropic transition) and inhomogeneities in the local mass density <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13538_2025_1794_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="36" /> </InlineMediaObject> <EquationSource Format="TEX">\(\varvec{\rho }\varvec{(} \textbf{r}\varvec{)}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mi mathvariant="bold-italic">ρ</mi> </mrow> <mrow> <mo mathvariant="bold" stretchy="false">(</mo> </mrow> <mi mathvariant="bold">r</mi> <mrow> <mo mathvariant="bold" stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. It differs from the well-known formula <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13538_2025_1794_Article_IEq4.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="102" /> </InlineMediaObject> <EquationSource Format="TEX">\(\varvec{v}_{\varvec{s}}\varvec{=}\sqrt{\varvec{B}_{\varvec{s}}\varvec{/}\varvec{\rho }_{\varvec{0}}}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mrow> <mi mathvariant="bold-italic">v</mi> </mrow> <mrow> <mi mathvariant="bold-italic">s</mi> </mrow> </msub> <mrow> <mo mathvariant="bold">=</mo> </mrow> <msqrt> <mrow> <msub> <mrow> <mi mathvariant="bold-italic">B</mi> </mrow> <mrow> <mi mathvariant="bold-italic">s</mi> </mrow> </msub> <mrow> <mo mathvariant="bold" stretchy="false">/</mo> </mrow> <msub> <mrow> <mi mathvariant="bold-italic">ρ</mi> </mrow> <mrow> <mn mathvariant="bold">0</mn> </mrow> </msub> </mrow> </msqrt> </mrow> </math></EquationSource> </InlineEquation>, in which <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13538_2025_1794_Article_IEq5.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="25" /> </InlineMediaObject> <EquationSource Format="TEX">\(\varvec{B}_{\varvec{s}}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mi mathvariant="bold-italic">B</mi> </mrow> <mrow> <mi mathvariant="bold-italic">s</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> is the adiabatic bulk modulus and <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13538_2025_1794_Article_IEq6.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">\(\varvec{\rho }_{\varvec{0}}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mi mathvariant="bold-italic">ρ</mi> </mrow> <mrow> <mn mathvariant="bold">0</mn> </mrow> </msub> </math></EquationSource> </InlineEquation> is the average value of <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13538_2025_1794_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="36" /> </InlineMediaObject> <EquationSource Format="TEX">\(\varvec{\rho }\varvec{(} \textbf{r}\varvec{)}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mi mathvariant="bold-italic">ρ</mi> </mrow> <mrow> <mo mathvariant="bold" stretchy="false">(</mo> </mrow> <mi mathvariant="bold">r</mi> <mrow> <mo mathvariant="bold" stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. Using the measured values of <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13538_2025_1794_Article_IEq1.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">\(\varvec{v}_{\varvec{s}}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mi mathvariant="bold-italic">v</mi> </mrow> <mrow> <mi mathvariant="bold-italic">s</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> in the regime of low frequencies, we then calculate <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13538_2025_1794_Article_IEq5.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="25" /> </InlineMediaObject> <EquationSource Format="TEX">\(\varvec{B}_{\varvec{s}}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mi mathvariant="bold-italic">B</mi> </mrow> <mrow> <mi mathvariant="bold-italic">s</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> as a function of temperature for three different liquid crystals. The critical form of <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13538_2025_1794_Article_IEq5.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="25" /> </InlineMediaObject> <EquationSource Format="TEX">\(\varvec{B}_{\varvec{s}}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mi mathvariant="bold-italic">B</mi> </mrow> <mrow> <mi mathvariant="bold-italic">s</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> that emerges is unexpected, and its physical interpretation is discussed in terms of the Anderson-Grüneisen parameter. An explanation for the critical dip of <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13538_2025_1794_Article_IEq1.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">\(\varvec{v}_{\varvec{s}}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mi mathvariant="bold-italic">v</mi> </mrow> <mrow> <mi mathvariant="bold-italic">s</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> near the nematic-isotropic transition is provided.</p>

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Adiabatic Bulk Modulus and Sound Velocity of Nematic Liquid Crystals in the Isotropic Phase

  • Carlindo Vitoriano

摘要

We obtain an expression for the sound velocity \(\varvec{v}_{\varvec{s}}\) v s in the isotropic phase of nematic liquid crystals from the nematodynamics that incorporates the energetic coupling between \(\varvec{Q}_{\varvec{\alpha \beta }}\varvec{(} \textbf{r}\varvec{)}\) Q α β ( r ) (the order parameter of the nematic-isotropic transition) and inhomogeneities in the local mass density \(\varvec{\rho }\varvec{(} \textbf{r}\varvec{)}\) ρ ( r ) . It differs from the well-known formula \(\varvec{v}_{\varvec{s}}\varvec{=}\sqrt{\varvec{B}_{\varvec{s}}\varvec{/}\varvec{\rho }_{\varvec{0}}}\) v s = B s / ρ 0 , in which \(\varvec{B}_{\varvec{s}}\) B s is the adiabatic bulk modulus and \(\varvec{\rho }_{\varvec{0}}\) ρ 0 is the average value of \(\varvec{\rho }\varvec{(} \textbf{r}\varvec{)}\) ρ ( r ) . Using the measured values of \(\varvec{v}_{\varvec{s}}\) v s in the regime of low frequencies, we then calculate \(\varvec{B}_{\varvec{s}}\) B s as a function of temperature for three different liquid crystals. The critical form of \(\varvec{B}_{\varvec{s}}\) B s that emerges is unexpected, and its physical interpretation is discussed in terms of the Anderson-Grüneisen parameter. An explanation for the critical dip of \(\varvec{v}_{\varvec{s}}\) v s near the nematic-isotropic transition is provided.