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href="/wiki/File:Schematic_of_particle_illustrating_diffusiophoresis.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d7/Schematic_of_particle_illustrating_diffusiophoresis.png/220px-Schematic_of_particle_illustrating_diffusiophoresis.png" decoding="async" width="220" height="202" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d7/Schematic_of_particle_illustrating_diffusiophoresis.png/330px-Schematic_of_particle_illustrating_diffusiophoresis.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d7/Schematic_of_particle_illustrating_diffusiophoresis.png/440px-Schematic_of_particle_illustrating_diffusiophoresis.png 2x" data-file-width="740" data-file-height="680" /></a><figcaption>Schematic illustrating diffusiophoretic motion of a colloidal particle (blue) in a concentration gradient of a solute (red). Note that there is also a concentration gradient of the solvent (green). The particle is moving a diffusiophoretic velocity <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bf {v}}_{\text{dp}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>dp</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bf {v}}_{\text{dp}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea34d83cc311c6829609a4989b2aba7acb774f80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.471ex; height:2.343ex;" alt="{\displaystyle {\bf {v}}_{\text{dp}}}"></span>, in a fluid that is stationary far from the particle. The fluid velocity decays from <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bf {v}}_{\text{dp}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>dp</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bf {v}}_{\text{dp}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea34d83cc311c6829609a4989b2aba7acb774f80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.471ex; height:2.343ex;" alt="{\displaystyle {\bf {v}}_{\text{dp}}}"></span> for fluid in contact with the surface of the particle, to near zero, within the interface at the particle's surface.</figcaption></figure> <p><b>Diffusiophoresis</b> is the spontaneous motion of <a href="/wiki/Colloid" title="Colloid">colloidal particles</a> or <a href="/wiki/Molecules" class="mw-redirect" title="Molecules">molecules</a> in a <a href="/wiki/Fluid" title="Fluid">fluid</a>, induced by a <a href="/wiki/Concentration_gradient" class="mw-redirect" title="Concentration gradient">concentration gradient</a> of a different substance.<sup id="cite_ref-:0_1-0" class="reference"><a href="#cite_note-:0-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:5_2-0" class="reference"><a href="#cite_note-:5-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:6_3-0" class="reference"><a href="#cite_note-:6-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:7_4-0" class="reference"><a href="#cite_note-:7-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> In other words, it is motion of one species, A, in response to a concentration gradient in another species, B. Typically, A is colloidal particles which are in aqueous solution in which B is a dissolved salt such as sodium chloride, and so the particles of A are much larger than the ions of B. But both A and B could be polymer molecules, and B could be a small molecule. For example, concentration gradients in ethanol solutions in water move 1 μm diameter colloidal particles with diffusiophoretic velocities <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bf {v}}_{\text{dp}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>dp</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bf {v}}_{\text{dp}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea34d83cc311c6829609a4989b2aba7acb774f80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.471ex; height:2.343ex;" alt="{\displaystyle {\bf {v}}_{\text{dp}}}"></span> of order <span class="nowrap"><span data-sort-value="6993100000000000000♠"></span>0.1 to 1 μm/s</span>, the movement is towards regions of the solution with lower ethanol concentration (and so higher water concentration).<sup id="cite_ref-:2_6-0" class="reference"><a href="#cite_note-:2-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> Both species A and B will typically be diffusing but diffusiophoresis is distinct from <a href="/wiki/Diffusion" title="Diffusion">simple diffusion</a>: in simple diffusion a species A moves down a gradient in its own concentration. </p><p><b>Diffusioosmosis</b>, also referred to as capillary osmosis, is flow of a solution relative to a fixed wall or pore surface, where the flow is driven by a concentration gradient in the solution. This is distinct from flow relative to a surface driven by a gradient in the hydrostatic pressure in the fluid. In diffusioosmosis the hydrostatic pressure is uniform and the flow is due to a concentration gradient. </p><p>Diffusioosmosis and diffusiophoresis are essentially the same phenomenon. They are both relative motion of a surface and a solution, driven by a concentration gradient in the solution. This motion is called diffusiophoresis when the solution is considered static with particles moving in it due to relative motion of the fluid at the surface of these particles. The term diffusioosmosis is used when the surface is viewed as static, and the solution flows. </p><p>A well studied example of diffusiophoresis is the motion of <a href="/wiki/Colloid" title="Colloid">colloidal</a> particles in an aqueous solution of an <a href="/wiki/Electrolyte" title="Electrolyte">electrolyte</a> solution, where a gradient in the concentration of the electrolyte causes motion of the colloidal particles.<sup id="cite_ref-:2_6-1" class="reference"><a href="#cite_note-:2-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:1_7-0" class="reference"><a href="#cite_note-:1-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> Colloidal particles may be hundred of nanometres or larger in diameter, while the <a href="/wiki/Double_layer_(interfacial)" class="mw-redirect" title="Double layer (interfacial)">interfacial double layer</a> region at the surface of the colloidal particle will be of order the <a href="/wiki/Debye_length" title="Debye length">Debye length</a> wide, and this is typically only nanometres. So here, the interfacial width is much smaller than the size of the particle, and then the gradient in the smaller species drives diffusiophoretic motion of the colloidal particles largely through motion in the <a href="/wiki/Double_layer_(interfacial)" class="mw-redirect" title="Double layer (interfacial)">interfacial double layer</a>.<sup id="cite_ref-:0_1-1" class="reference"><a href="#cite_note-:0-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>Diffusiophoresis was first studied by Derjaguin and coworkers in 1947.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Applications_of_diffusiophoresis">Applications of diffusiophoresis</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diffusiophoresis_and_diffusioosmosis&action=edit&section=1" title="Edit section: Applications of diffusiophoresis"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Diffusiophoresis, by definition, moves colloidal particles, and so the applications of diffusiophoresis are to situations where we want to move colloidal particles. Colloidal particles are typically between 10 nanometres and a few micrometres in size. <a href="/wiki/Diffusion" title="Diffusion">Simple diffusion</a> of colloids is fast on length scales of a few micrometres, and so diffusiophoresis would not be useful, whereas on length scales larger than millimetres, diffusiophoresis may be slow as its speed decreases with decreasing size of the solute concentration gradient. Thus, typically diffusiophoresis is employed on length scales approximately in the range a micrometre to a millimetre. Applications include moving particles into or out of pores of that size,<sup id="cite_ref-:1_7-1" class="reference"><a href="#cite_note-:1-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> and helping or inhibiting mixing of colloidal particles.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p><p>In addition, solid surfaces that are slowly dissolving will create concentration gradients near them, and these gradients may drive movement of colloidal particles towards or away from the surface. This was studied by Prieve<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> in the context of latex particles being pulled towards, and coating, a dissolving steel surface. </p> <div class="mw-heading mw-heading2"><h2 id="Relation_to_thermophoresis,_multicomponent_diffusion_and_the_Marangoni_effect"><span id="Relation_to_thermophoresis.2C_multicomponent_diffusion_and_the_Marangoni_effect"></span>Relation to thermophoresis, multicomponent diffusion and the Marangoni effect</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diffusiophoresis_and_diffusioosmosis&action=edit&section=2" title="Edit section: Relation to thermophoresis, multicomponent diffusion and the Marangoni effect"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Diffusiophoresis is an analogous phenomenon to <a href="/wiki/Thermophoresis" title="Thermophoresis">thermophoresis</a>, where a species A moves in response to a temperature gradient. Both diffusiophoresis and thermophoresis are governed by <a href="/wiki/Onsager_reciprocal_relations" title="Onsager reciprocal relations">Onsager reciprocal relations</a>. Simply speaking, a gradient in any thermodynamic quantity, such as the concentration of any species, or temperature, will drive motion of all thermodynamic quantities, i.e., motion of all species present, and a temperature flux. Each gradient provides a thermodynamic force that moves the species present, and the <a href="/wiki/Onsager_reciprocal_relations" title="Onsager reciprocal relations">Onsager reciprocal relations</a> govern the relationship between the forces and the motions. </p><p>Diffusiophoresis is a special case of <a href="/wiki/Diffusion#Onsager's_equations_for_multicomponent_diffusion_and_thermodiffusion" title="Diffusion">multicomponent diffusion</a>. Multicomponent diffusion is diffusion in mixtures, and diffusiophoresis is the special case where we are interested in the movement of one species that is usually a colloidal particle, in a gradient of a much smaller species, such as dissolved salt such as sodium chloride in water. or a miscible liquid, such as ethanol in water. Thus diffusiophoresis always occurs in a mixture, typically a three-component mixture of water, salt and a colloidal species, and we are interested in the cross-interaction between the salt and the colloidal particle. </p><p>It is the very large difference in size between the colloidal particle, which may be 1μm across, and the size of the ions or molecules, which are less than 1 nm across, that makes diffusiophoresis closely related to diffusioosomosis at a flat surface. In both cases the forces that drive the motion are largely localised to the interfacial region, which is a few molecules across and so typically of order a nanometer across. Over distances of order a nanometer, there is little difference between the surface of a colloidal particle 1 μm across, and a flat surface. </p><p>Diffusioosmosis is flow of a fluid at a solid surface, or in other words, flow at a solid/fluid interface. The <a href="/wiki/Marangoni_effect" title="Marangoni effect">Marangoni effect</a> is flow at a fluid/fluid interface. So the two phenomena are analogous with the difference being that in diffusioosmosis one of the phases is a solid. Both diffusioosmosis and the <a href="/wiki/Marangoni_effect" title="Marangoni effect">Marangoni effect</a> are driven by gradients in the interfacial free energy, i.e., in both cases the induced velocities are zero if the interfacial free energy is uniform in space, and in both cases if there are gradients the velocities are directed along the direction of increasing interfacial free energy.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Theory_for_diffusioosmotic_flow_of_a_solution">Theory for diffusioosmotic flow of a solution</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diffusiophoresis_and_diffusioosmosis&action=edit&section=3" title="Edit section: Theory for diffusioosmotic flow of a solution"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div><p> In diffusioosmosis, for a surface at rest the velocity increases from zero at the surface to the diffusioosmotic velocity, over the width of the interface between the surface and the solution. Beyond this distance, the diffusioosmotic velocity does not vary with distance from the surface. The driving force for diffusioosmosis is thermodynamic, i.e., it acts to reduce the free energy if the system, and so the direction of flow is away from surface regions of low surface free energy, and towards regions of high surface free energy. For a solute that adsorbs at surface, diffusioosmotic flow is away from regions of high solute concentration, while for solutes that are repelled by the surface, flow is away from regions of low solute concentration.</p><figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Schematic_illustrating_diffusioosmosis.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/39/Schematic_illustrating_diffusioosmosis.png/220px-Schematic_illustrating_diffusioosmosis.png" decoding="async" width="220" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/39/Schematic_illustrating_diffusioosmosis.png/330px-Schematic_illustrating_diffusioosmosis.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/39/Schematic_illustrating_diffusioosmosis.png/440px-Schematic_illustrating_diffusioosmosis.png 2x" data-file-width="972" data-file-height="444" /></a><figcaption>This schematic illustrates diffusioosmotic flow above a surface in contact with a solution that has a concentration gradient of a solute (red). The flow as a function of height above the surface is shown as black arrows of length proportional to the flow velocity at that height. The flow is left-to-right as this solute is repelled by the surface, and its concentration increases from left-to-right. Therefore, the surface free energy increases right-to-left, which drives flow from right-to-left.</figcaption></figure> <p>For gradients that are not-too-large, the diffusioosmotic slip velocity, i.e., the relative flow velocity far from the surface will be proportional to the gradient in the concentration gradient<sup id="cite_ref-:0_1-2" class="reference"><a href="#cite_note-:0-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:3_12-0" class="reference"><a href="#cite_note-:3-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bf {v}}_{\text{slip}}=-K\nabla c_{\text{sol}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>slip</mtext> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mi>K</mi> <mi mathvariant="normal">∇</mi> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>sol</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bf {v}}_{\text{slip}}=-K\nabla c_{\text{sol}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/584e24e4f606fa101b2c71c586e9782adbae1917" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.195ex; height:2.843ex;" alt="{\displaystyle {\bf {v}}_{\text{slip}}=-K\nabla c_{\text{sol}}}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"></span> is a diffusioosmotic coefficient, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{\text{sol}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>sol</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{\text{sol}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c512037f6ce06a900c488c7708e1f096fdc6e0cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.166ex; height:2.009ex;" alt="{\displaystyle c_{\text{sol}}}"></span> is the solute concentration. When the solute is ideal and interacts with a surface in the x–y plane at <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b92bfc06485cc90286474b14a516a68d8bfdd7b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.349ex; height:2.176ex;" alt="{\displaystyle z=0}"></span> via a potential <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi _{\text{solute}}(z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>solute</mtext> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi _{\text{solute}}(z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac8edddb87788ef802dd706839c91b95c2ff650b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.726ex; height:2.843ex;" alt="{\displaystyle \phi _{\text{solute}}(z)}"></span>, the coefficient <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"></span> is given by<sup id="cite_ref-:0_1-3" class="reference"><a href="#cite_note-:0-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K={\frac {kT}{\eta }}\int _{0}^{\infty }z\left[\exp(-\phi _{\text{solute}}/kT)-1\right]{\rm {d}}z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>k</mi> <mi>T</mi> </mrow> <mi>η</mi> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>z</mi> <mrow> <mo>[</mo> <mrow> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>solute</mtext> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>k</mi> <mi>T</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mn>1</mn> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K={\frac {kT}{\eta }}\int _{0}^{\infty }z\left[\exp(-\phi _{\text{solute}}/kT)-1\right]{\rm {d}}z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b073094bbd98bf5820fd31013f077de2b8999f64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:40.51ex; height:5.843ex;" alt="{\displaystyle K={\frac {kT}{\eta }}\int _{0}^{\infty }z\left[\exp(-\phi _{\text{solute}}/kT)-1\right]{\rm {d}}z}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> is the <a href="/wiki/Boltzmann_constant" title="Boltzmann constant">Boltzmann constant</a>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> is the absolute temperature, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>η</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4d701857cf5fbec133eebaf94deadf722537f64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.169ex; height:2.176ex;" alt="{\displaystyle \eta }"></span> is the <a href="/wiki/Viscosity" title="Viscosity">viscosity</a> in the interfacial region, assumed to be constant in the interface. This expression assumes that the fluid velocity for fluid in contact with the surface is forced to be zero, by interaction between the fluid and the wall. This is called the <a href="/wiki/No-slip_condition" title="No-slip condition">no-slip condition</a>. </p><p>To understand these expressions better, we can consider a very simple model, where the surface simply excludes an ideal solute from an interface of width <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span>, this is would be the <a href="/wiki/Depletion_force#Asakura–Oosawa_model" title="Depletion force">Asakura–Oosawa</a> model of an ideal polymer against a hard wall.<sup id="cite_ref-:4_13-0" class="reference"><a href="#cite_note-:4-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> Then the integral is simply <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -(1/2)R^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -(1/2)R^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bc1bece7e4deb75d539f736b7043274b41d19a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.923ex; height:3.176ex;" alt="{\displaystyle -(1/2)R^{2}}"></span> and the diffusioosmotic slip velocity </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bf {v}}_{\text{slip}}={\frac {kTR^{2}}{2\eta }}\nabla c_{\text{sol}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>slip</mtext> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>k</mi> <mi>T</mi> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mi>η</mi> </mrow> </mfrac> </mrow> <mi mathvariant="normal">∇</mi> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>sol</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bf {v}}_{\text{slip}}={\frac {kTR^{2}}{2\eta }}\nabla c_{\text{sol}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52daa75447d9e00772c2cc3f870c73e795f84fe3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:18.822ex; height:6.176ex;" alt="{\displaystyle {\bf {v}}_{\text{slip}}={\frac {kTR^{2}}{2\eta }}\nabla c_{\text{sol}}}"></span></dd></dl> <p>Note that the slip velocity is directed towards increasing solute concentrations. </p><p>A particle much larger than <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> moves with a diffusiophoretic velocity <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bf {v}}_{\text{dp}}=-{\bf {v}}_{\text{slip}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>dp</mtext> </mrow> </msub> <mo>=</mo> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>slip</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bf {v}}_{\text{dp}}=-{\bf {v}}_{\text{slip}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4f693abf60f348083fcf056b0b76ed5a2b0bafa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.498ex; height:2.676ex;" alt="{\displaystyle {\bf {v}}_{\text{dp}}=-{\bf {v}}_{\text{slip}}}"></span> relative to the surrounding solution. So diffusiophoresis moves particles towards lower solute concentrations, in this case. </p> <div class="mw-heading mw-heading3"><h3 id="Derivation_of_diffusioosmotic_velocity_from_Stokes_flow">Derivation of diffusioosmotic velocity from Stokes flow</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diffusiophoresis_and_diffusioosmosis&action=edit&section=4" title="Edit section: Derivation of diffusioosmotic velocity from Stokes flow"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In this simple model, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bf {v}}_{\text{slip}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>slip</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bf {v}}_{\text{slip}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f192327c604ef68de514a42d8cf9d10e8d9c9eb0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.12ex; height:2.343ex;" alt="{\displaystyle {\bf {v}}_{\text{slip}}}"></span> can also be derived directly from the expression for fluid flow<sup id="cite_ref-:3_12-1" class="reference"><a href="#cite_note-:3-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:0_1-4" class="reference"><a href="#cite_note-:0-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:4_13-1" class="reference"><a href="#cite_note-:4-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> in the <a href="/wiki/Stokes_flow" title="Stokes flow">Stokes limit for an incompressible fluid</a>, which is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta \nabla ^{2}{\bf {u}}=\nabla p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>η</mi> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mrow> <mo>=</mo> <mi mathvariant="normal">∇</mi> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta \nabla ^{2}{\bf {u}}=\nabla p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aafa9c617c5ebc384f5f4077c57bb943ad035b59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.849ex; height:3.176ex;" alt="{\displaystyle \eta \nabla ^{2}{\bf {u}}=\nabla p}"></span></dd></dl> <p>for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bf {u}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bf {u}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0a76d8342af5d8a8d4a5196d76afde88d4593ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:1.676ex;" alt="{\displaystyle {\bf {u}}}"></span> the fluid flow velocity and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> the pressure. We consider an infinite surface in the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle xy}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle xy}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c72eb345e496513fb8b2fa4aa8c4d89b855f9a01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.485ex; height:2.009ex;" alt="{\displaystyle xy}"></span> plane at <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b92bfc06485cc90286474b14a516a68d8bfdd7b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.349ex; height:2.176ex;" alt="{\displaystyle z=0}"></span>, and enforce stick boundary conditions there, i.e., <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bf {u}}(x,y,z=0)=(0,0,0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo>=</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bf {u}}(x,y,z=0)=(0,0,0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87687ee6e111fee757b418c16d2ca7b7883ad6d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.66ex; height:2.843ex;" alt="{\displaystyle {\bf {u}}(x,y,z=0)=(0,0,0)}"></span>. We take the concentration gradient to be along the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> axis, i.e., <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla c_{\text{sol}}=(\partial c_{\text{sol}}/\partial x,0,0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>sol</mtext> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">∂</mi> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>sol</mtext> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla c_{\text{sol}}=(\partial c_{\text{sol}}/\partial x,0,0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3d72e85e61a265fc81c38db071137fdc3a31e3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.697ex; height:2.843ex;" alt="{\displaystyle \nabla c_{\text{sol}}=(\partial c_{\text{sol}}/\partial x,0,0)}"></span>. Then the only non-zero component of the flow velocity <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bf {u}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bf {u}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0a76d8342af5d8a8d4a5196d76afde88d4593ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:1.676ex;" alt="{\displaystyle {\bf {u}}}"></span> is along x, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u_{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u_{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/045896a773b67c720bdff8d660cfcc906f5f6b20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.502ex; height:2.009ex;" alt="{\displaystyle u_{x}}"></span>, and it depends only on height <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"></span>. So the only non-zero component of the Stokes' equation is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta {\frac {\partial ^{2}u_{x}(z)}{\partial z^{2}}}={\frac {\partial p}{\partial x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>p</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta {\frac {\partial ^{2}u_{x}(z)}{\partial z^{2}}}={\frac {\partial p}{\partial x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83b959f140cf0a4994f0fc4f66d7bf76e86757f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:16.385ex; height:6.176ex;" alt="{\displaystyle \eta {\frac {\partial ^{2}u_{x}(z)}{\partial z^{2}}}={\frac {\partial p}{\partial x}}}"></span></dd></dl> <p>In diffusioosmosis, in the bulk of the fluid (i.e., outside the interface) the hydrostatic pressure is assumed to be uniform (as we expect any gradients to relax away by fluid flow) and so in bulk<sup id="cite_ref-:4_13-2" class="reference"><a href="#cite_note-:4-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:3_12-2" class="reference"><a href="#cite_note-:3-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{In bulk:}}~~~~~~p=\Pi +p_{\text{solv}}={\text{constant}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>In bulk:</mtext> </mrow> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mi>p</mi> <mo>=</mo> <mi mathvariant="normal">Π</mi> <mo>+</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>solv</mtext> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>constant</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{In bulk:}}~~~~~~p=\Pi +p_{\text{solv}}={\text{constant}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27e02e24f7ae3118dd97deb8726426280a1c82c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:36.118ex; height:2.509ex;" alt="{\displaystyle {\text{In bulk:}}~~~~~~p=\Pi +p_{\text{solv}}={\text{constant}}}"></span></dd></dl> <p>for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{\text{solv}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>solv</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{\text{solv}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1faed76992f2cc9d07be23f70dccbb101f638406" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:4.286ex; height:2.009ex;" alt="{\displaystyle p_{\text{solv}}}"></span> the solvent's contribution to the hydrostatic pressure, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eed3e3db6cc2028a183af948212ed2551d25c954" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle \Pi }"></span> the contribution of the solute, called the <a href="/wiki/Osmotic_pressure" title="Osmotic pressure">osmotic pressure</a>. Thus in the bulk the gradients obey </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{In bulk:}}~~~~~~0={\frac {\partial \Pi }{\partial x}}+{\frac {\partial p_{\text{solv}}}{\partial x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>In bulk:</mtext> </mrow> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mn>0</mn> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi mathvariant="normal">Π</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>solv</mtext> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{In bulk:}}~~~~~~0={\frac {\partial \Pi }{\partial x}}+{\frac {\partial p_{\text{solv}}}{\partial x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd7c5df772d3cf51951a25a330277e423eb39d26" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:28.653ex; height:5.509ex;" alt="{\displaystyle {\text{In bulk:}}~~~~~~0={\frac {\partial \Pi }{\partial x}}+{\frac {\partial p_{\text{solv}}}{\partial x}}}"></span></dd></dl> <p>As we have assumed the solute is ideal, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pi =kTc_{\text{sol}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Π</mi> <mo>=</mo> <mi>k</mi> <mi>T</mi> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>sol</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pi =kTc_{\text{sol}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1233f0eeaa60269608415e721a659e190320a329" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.855ex; height:2.509ex;" alt="{\displaystyle \Pi =kTc_{\text{sol}}}"></span>, and so </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{In bulk:}}~~~~~~{\frac {\partial p_{\text{solv}}}{\partial x}}=-kT{\frac {\partial c_{\text{sol}}}{\partial x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>In bulk:</mtext> </mrow> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>solv</mtext> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mi>k</mi> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>sol</mtext> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{In bulk:}}~~~~~~{\frac {\partial p_{\text{solv}}}{\partial x}}=-kT{\frac {\partial c_{\text{sol}}}{\partial x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/822c833360b5796a84d8972f3d3cd819be71a878" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:30.729ex; height:5.509ex;" alt="{\displaystyle {\text{In bulk:}}~~~~~~{\frac {\partial p_{\text{solv}}}{\partial x}}=-kT{\frac {\partial c_{\text{sol}}}{\partial x}}}"></span></dd></dl> <p>Our solute is excluded from a region of width <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> (the interfacial region) from the surface, and so in interface <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pi =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Π</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pi =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d04a04e9975c5a5408e7cf9a8b85741d2141ccf3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.004ex; height:2.176ex;" alt="{\displaystyle \Pi =0}"></span>, and so there <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=p_{\text{solv}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>solv</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=p_{\text{solv}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/671a5bbbe7fcd94221b05987ef12fa00ca9d3dd1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:8.554ex; height:2.009ex;" alt="{\displaystyle p=p_{\text{solv}}}"></span>. Assuming continuity of the solvent contribution into the interface we have a gradient of the hydrostatic pressure in the interface </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{In interface:}}~~~~~~{\frac {\partial p}{\partial x}}={\frac {\partial p_{\text{solv}}}{\partial x}}=-kT{\frac {\partial c_{\text{sol}}}{\partial x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>In interface:</mtext> </mrow> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>p</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>solv</mtext> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mi>k</mi> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>sol</mtext> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{In interface:}}~~~~~~{\frac {\partial p}{\partial x}}={\frac {\partial p_{\text{solv}}}{\partial x}}=-kT{\frac {\partial c_{\text{sol}}}{\partial x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be8644257187472d29955d526fb179464b798cce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:41.579ex; height:5.509ex;" alt="{\displaystyle {\text{In interface:}}~~~~~~{\frac {\partial p}{\partial x}}={\frac {\partial p_{\text{solv}}}{\partial x}}=-kT{\frac {\partial c_{\text{sol}}}{\partial x}}}"></span></dd></dl> <p>i.e., in the interface there is a gradient of the hydrostatic pressure equal to the negative of the bulk gradient in the osmotic pressure. It is this gradient in the interface in the hydrostatic pressure <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> that creates the diffusioosmotic flow. Now that we have <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial p/\partial x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∂</mi> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial p/\partial x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68c47a2ede1c9ab6631b09002c7c2a3aa20cb17f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.298ex; height:2.843ex;" alt="{\displaystyle \partial p/\partial x}"></span>, we can substitute into the Stokes equation, and integrate twice, then </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{In interface:}}~~~~~~u_{x}(z)=-{\frac {kTz^{2}}{2\eta }}{\frac {\partial c_{\text{sol}}}{\partial x}}+Az+B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>In interface:</mtext> </mrow> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>k</mi> <mi>T</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mi>η</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>sol</mtext> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>A</mi> <mi>z</mi> <mo>+</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{In interface:}}~~~~~~u_{x}(z)=-{\frac {kTz^{2}}{2\eta }}{\frac {\partial c_{\text{sol}}}{\partial x}}+Az+B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85628ce578e7a3d4d54e34ddd6df1808567a7af8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:47.302ex; height:6.176ex;" alt="{\displaystyle {\text{In interface:}}~~~~~~u_{x}(z)=-{\frac {kTz^{2}}{2\eta }}{\frac {\partial c_{\text{sol}}}{\partial x}}+Az+B}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{In bulk:}}~~~~~~u_{x}(z)=Cz+D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>In bulk:</mtext> </mrow> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>C</mi> <mi>z</mi> <mo>+</mo> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{In bulk:}}~~~~~~u_{x}(z)=Cz+D}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79ef26dbb83ebcd6556dfe5998d1bdf57040b090" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.42ex; height:2.843ex;" alt="{\displaystyle {\text{In bulk:}}~~~~~~u_{x}(z)=Cz+D}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}"></span> are integration constants. Far from the surface the flow velocity must be a constant, so <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f750a7094a396d89a81974cdf35783db2bb287b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.027ex; height:2.176ex;" alt="{\displaystyle C=0}"></span>. We have imposed zero flow velocity at <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b92bfc06485cc90286474b14a516a68d8bfdd7b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.349ex; height:2.176ex;" alt="{\displaystyle z=0}"></span>, so <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af1d79a64f775646f76d79a452ca8e1082fb1f17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.025ex; height:2.176ex;" alt="{\displaystyle B=0}"></span>. Then imposing continuity where the interface meets the bulk, i.e., forcing <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u_{x}(z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u_{x}(z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8633cadefad89df58dfdfadeccf4c0a6a3ad9a4c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.4ex; height:2.843ex;" alt="{\displaystyle u_{x}(z)}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial u_{x}(z)/\partial z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∂</mi> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial u_{x}(z)/\partial z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/742435c0ecb738e0f0cbc280fd78049bdd0a6cf1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.286ex; height:2.843ex;" alt="{\displaystyle \partial u_{x}(z)/\partial z}"></span> to be continuous at <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89b56c8b7ee8f6e8ac2347dddb8229ca2a2cdda7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.951ex; height:2.176ex;" alt="{\displaystyle z=R}"></span> we determine <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}"></span>, and so get </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{In interface:}}~~~~~~u_{x}(z)={\frac {kTR^{2}}{2\eta }}\left({\frac {\partial c_{\text{sol}}}{\partial x}}\right)\left({\frac {2z}{R}}-{\frac {z^{2}}{R^{2}}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>In interface:</mtext> </mrow> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>k</mi> <mi>T</mi> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mi>η</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>sol</mtext> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>z</mi> </mrow> <mi>R</mi> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{In interface:}}~~~~~~u_{x}(z)={\frac {kTR^{2}}{2\eta }}\left({\frac {\partial c_{\text{sol}}}{\partial x}}\right)\left({\frac {2z}{R}}-{\frac {z^{2}}{R^{2}}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52246f06cbdb3bada012b172a375a27121027d6b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:53.089ex; height:6.343ex;" alt="{\displaystyle {\text{In interface:}}~~~~~~u_{x}(z)={\frac {kTR^{2}}{2\eta }}\left({\frac {\partial c_{\text{sol}}}{\partial x}}\right)\left({\frac {2z}{R}}-{\frac {z^{2}}{R^{2}}}\right)}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{In bulk:}}~~~~~~u_{x}(z)={\frac {kTR^{2}}{2\eta }}\left({\frac {\partial c_{\text{sol}}}{\partial x}}\right)=v_{\text{slip}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>In bulk:</mtext> </mrow> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>k</mi> <mi>T</mi> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mi>η</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>sol</mtext> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>slip</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{In bulk:}}~~~~~~u_{x}(z)={\frac {kTR^{2}}{2\eta }}\left({\frac {\partial c_{\text{sol}}}{\partial x}}\right)=v_{\text{slip}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e320609ecc4a06982b14b45a53c3cd1c5c1cd63b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:42.367ex; height:6.343ex;" alt="{\displaystyle {\text{In bulk:}}~~~~~~u_{x}(z)={\frac {kTR^{2}}{2\eta }}\left({\frac {\partial c_{\text{sol}}}{\partial x}}\right)=v_{\text{slip}}}"></span></dd></dl> <p>Which gives, as it should, the same expression for the slip velocity, as above. This result is for a specific and very simple model, but it does illustrate general features of diffusioosmoisis: 1) the hydrostatic pressure is, by definition (<a href="/wiki/Stokes_flow" title="Stokes flow">flow induced by pressure gradients in the bulk</a> is a common but separate physical phenomenon) uniform in the bulk, but there is a gradient in the pressure in the interface, 2) this pressure gradient in the interface causes the velocity to vary in the direction perpendicular to the surface, and this results in a slip velocity, i.e., for the bulk of the fluid to move relative to the surface, 3) away from the interface the velocity is constant, this type of flow is sometimes called <a href="/wiki/Plug_flow" title="Plug flow">plug flow</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Diffusiophoresis_in_salt_solutions">Diffusiophoresis in salt solutions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diffusiophoresis_and_diffusioosmosis&action=edit&section=5" title="Edit section: Diffusiophoresis in salt solutions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In many applications of diffusiophoresis, the motion is driven by gradients in the concentration of a salt (electrolyte) concentration,<sup id="cite_ref-:5_2-1" class="reference"><a href="#cite_note-:5-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:6_3-1" class="reference"><a href="#cite_note-:6-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> such as sodium chloride in water. Colloidal particles in water are typically charged, and there is an electrostatic potential, called a <a href="/wiki/Zeta_potential" title="Zeta potential">zeta potential</a> at their surface. This charged surface of the colloidal particle interacts with a gradient in salt concentration, and this gives rise to diffusiophoretic velocity <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bf {U}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">U</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bf {U}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3bc4f1a94129fef73ca785453e03bffeed0cf23c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.057ex; height:2.176ex;" alt="{\displaystyle {\bf {U}}}"></span> given by<sup id="cite_ref-:6_3-2" class="reference"><a href="#cite_note-:6-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:1_7-2" class="reference"><a href="#cite_note-:1-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bf {U}}={\frac {\epsilon }{\eta }}\left[{\frac {kT}{e}}\beta \zeta +{\frac {\zeta ^{2}}{8}}\right]\nabla \ln c_{\text{salt}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">U</mi> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϵ</mi> <mi>η</mi> </mfrac> </mrow> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>k</mi> <mi>T</mi> </mrow> <mi>e</mi> </mfrac> </mrow> <mi>β</mi> <mi>ζ</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>ζ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>8</mn> </mfrac> </mrow> </mrow> <mo>]</mo> </mrow> <mi mathvariant="normal">∇</mi> <mi>ln</mi> <mo>⁡</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>salt</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bf {U}}={\frac {\epsilon }{\eta }}\left[{\frac {kT}{e}}\beta \zeta +{\frac {\zeta ^{2}}{8}}\right]\nabla \ln c_{\text{salt}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6858d9178e3f508641491ebc6ff4c22a3c103cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:30.805ex; height:6.343ex;" alt="{\displaystyle {\bf {U}}={\frac {\epsilon }{\eta }}\left[{\frac {kT}{e}}\beta \zeta +{\frac {\zeta ^{2}}{8}}\right]\nabla \ln c_{\text{salt}}}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \epsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϵ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \epsilon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3837cad72483d97bcdde49c85d3b7b859fb3fd2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.944ex; height:1.676ex;" alt="{\displaystyle \epsilon }"></span> is the <a href="/wiki/Permittivity" title="Permittivity">permittivity</a> of water, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>η</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4d701857cf5fbec133eebaf94deadf722537f64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.169ex; height:2.176ex;" alt="{\displaystyle \eta }"></span> is the viscosity of water, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \zeta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ζ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5c3916703cae7938143d38865f78f27faadd4ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.095ex; height:2.509ex;" alt="{\displaystyle \zeta }"></span> is the <a href="/wiki/Zeta_potential" title="Zeta potential">zeta potential</a> of the colloidal particle in the salt solution, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta =(D_{+}-D_{-})/(D_{+}+D_{-})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> <mo>−</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> <mo>+</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta =(D_{+}-D_{-})/(D_{+}+D_{-})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3661cdb9be1a8a162fbfa8a30619282ef64903b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.633ex; height:2.843ex;" alt="{\displaystyle \beta =(D_{+}-D_{-})/(D_{+}+D_{-})}"></span> is the reduced difference between the diffusion constant of the positively charged ion, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{+}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/220945dd2a1b6cace4c6ddc572fbd12fd02839a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.435ex; height:2.509ex;" alt="{\displaystyle D_{+}}"></span>, and the diffusion constant of the negatively charged ion, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{-}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{-}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f77f71ee0e53c0b074ed1a43c0024de3388b7ca7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.435ex; height:2.509ex;" alt="{\displaystyle D_{-}}"></span>, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{\text{salt}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>salt</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{\text{salt}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3efb3bfff51d00af04872a0cfcfebef61094df99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.806ex; height:2.009ex;" alt="{\displaystyle c_{\text{salt}}}"></span> is the salt concentration. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla \ln c_{\text{salt}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mi>ln</mi> <mo>⁡</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>salt</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla \ln c_{\text{salt}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0c983f514e9ec872b611ce68903e0d1066c3d30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.455ex; height:2.509ex;" alt="{\displaystyle \nabla \ln c_{\text{salt}}}"></span> is the gradient, i.e., rate of change with position, of the logarithm of the salt concentration, which is equivalent to the rate of change of the salt concentration, divided by the salt concentration – it is effectively one over the distance over which the concentration decreases by a factor of e. The above equation is approximate, and only valid for 1:1 electrolytes such as sodium chloride. </p><p>Note that there are two contributions to diffusiophoresis of a charged particle in a salt gradient, which give rise to the two terms in the above equation for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bf {U}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">U</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bf {U}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3bc4f1a94129fef73ca785453e03bffeed0cf23c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.057ex; height:2.176ex;" alt="{\displaystyle {\bf {U}}}"></span>. The first is due to the fact that whenever there is a salt concentration gradient, then unless the diffusion constants of the positive and negative ions are exactly equal to each other, there is an electric field, i.e., the gradient acts a little like a capacitor. This electric filed generated by the salt gradient drives <a href="/wiki/Electrophoresis" title="Electrophoresis">electrophoresis</a> of the charged particle, just as an externally applied electric field does. This gives rise to the first term in the equation above, i.e., diffusiophoresis at a velocity <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\epsilon /\eta )(kT/e)\beta \zeta \nabla \ln c_{\text{salt}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>η</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>e</mi> <mo stretchy="false">)</mo> <mi>β</mi> <mi>ζ</mi> <mi mathvariant="normal">∇</mi> <mi>ln</mi> <mo>⁡</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>salt</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\epsilon /\eta )(kT/e)\beta \zeta \nabla \ln c_{\text{salt}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d98106379babc65aa8e584b73268f69bd85422fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.871ex; height:2.843ex;" alt="{\displaystyle (\epsilon /\eta )(kT/e)\beta \zeta \nabla \ln c_{\text{salt}}}"></span>. </p><p>The second part is due to the surface free energy of the surface of a charged particle, decreasing with increasing salt concentration, this is a similar mechanism to that found in diffusiophoresis in gradients of neutrial substances. This gives rise to the second part of the diffusiophoretic velocity <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\epsilon \zeta ^{2}/8\eta )\nabla \ln c_{\text{salt}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>ϵ</mi> <msup> <mi>ζ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>8</mn> <mi>η</mi> <mo stretchy="false">)</mo> <mi mathvariant="normal">∇</mi> <mi>ln</mi> <mo>⁡</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>salt</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\epsilon \zeta ^{2}/8\eta )\nabla \ln c_{\text{salt}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbb9824c98bd50dd8817f7a7d4ec8445e6ac5a9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.876ex; height:3.176ex;" alt="{\displaystyle (\epsilon \zeta ^{2}/8\eta )\nabla \ln c_{\text{salt}}}"></span>. Note that this simple theory predicts that this contribution to the diffusiophoretic motion is always up a salt concentration gradient, it always moves particles towards higher salt concentration. By contrast, the sign of the electric-field contribution to diffusiophoresis depends on the sign of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta \zeta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mi>ζ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta \zeta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a4897936ddf9b6f93a6c33d6de60d28c0d9e337" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.427ex; height:2.509ex;" alt="{\displaystyle \beta \zeta }"></span>. So for example, for a negatively charged particle, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \zeta <0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ζ</mi> <mo><</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta <0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/beb6ce3d70794d9e209c2440cd8aa721af3e3b4f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.356ex; height:2.509ex;" alt="{\displaystyle \zeta <0}"></span>, and if the positively charged ions diffuse faster than the negatively charged ones, then this term will push particles down a salt gradient, but if it is the negatively charged ions that diffuse faster, then this term pushes the particles up the salt gradient. </p> <div class="mw-heading mw-heading2"><h2 id="Practical_applications">Practical applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diffusiophoresis_and_diffusioosmosis&action=edit&section=6" title="Edit section: Practical applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A group from Princeton University<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> reported the application of diffusiophoresis to water purification. Contaminated water is treated with CO<sub>2</sub> to create carbonic acid, and to split the water into a waste stream and a potable water stream.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> This allows for easy ionic separation of suspended particles. This has huge energy cost and time savings opportunity to make drinking water safe compared to traditional water filtration methods for dirty water sources. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diffusiophoresis_and_diffusioosmosis&action=edit&section=7" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Electrokinetic_phenomena" title="Electrokinetic phenomena">Electrokinetic phenomena</a></li> <li><a href="/wiki/Electrophoresis" title="Electrophoresis">Electrophoresis</a></li> <li><a href="/wiki/Marangoni_effect" title="Marangoni effect">Marangoni effect</a> – the analog of diffusioosmosis at a fluid/fluid interface</li> <li><a href="/wiki/Thermophoresis" title="Thermophoresis">Thermophoresis</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diffusiophoresis_and_diffusioosmosis&action=edit&section=8" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-:0-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-:0_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:0_1-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-:0_1-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-:0_1-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-:0_1-4"><sup><i><b>e</b></i></sup></a></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFAnderson1989" class="citation journal cs1">Anderson, J L (1989-01-01). 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(2017-05-02). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5418569">"Membraneless water filtration using CO<sub>2</sub>"</a>. <i>Nature Communications</i>. <b>8</b>: 15181. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2017NatCo...815181S">2017NatCo...815181S</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2Fncomms15181">10.1038/ncomms15181</a>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5418569">5418569</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/28462929">28462929</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nature+Communications&rft.atitle=Membraneless+water+filtration+using+CO%3Csub%3E2%3C%2Fsub%3E&rft.volume=8&rft.pages=15181&rft.date=2017-05-02&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC5418569%23id-name%3DPMC&rft_id=info%3Apmid%2F28462929&rft_id=info%3Adoi%2F10.1038%2Fncomms15181&rft_id=info%3Abibcode%2F2017NatCo...815181S&rft.aulast=Shin&rft.aufirst=Sangwoo&rft.au=Shardt%2C+Orest&rft.au=Warren%2C+Patrick+B.&rft.au=Stone%2C+Howard+A.&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC5418569&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADiffusiophoresis+and+diffusioosmosis" class="Z3988"></span></span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation news cs1"><a rel="nofollow" class="external text" href="https://www.economist.com/news/science-and-technology/21722159-lets-hear-it-diffusiophoresis-way-make-water-potable-using-carbon-dioxide">"A way to make water potable using carbon dioxide"</a>. <i>The Economist</i>. 2017-05-18<span class="reference-accessdate">. Retrieved <span class="nowrap">2018-04-29</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Economist&rft.atitle=A+way+to+make+water+potable+using+carbon+dioxide&rft.date=2017-05-18&rft_id=https%3A%2F%2Fwww.economist.com%2Fnews%2Fscience-and-technology%2F21722159-lets-hear-it-diffusiophoresis-way-make-water-potable-using-carbon-dioxide&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADiffusiophoresis+and+diffusioosmosis" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diffusiophoresis_and_diffusioosmosis&action=edit&section=9" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin" style=""> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAndersonPrieve2006" class="citation journal cs1">Anderson, John L.; Prieve, Dennis C. (2006). "Diffusiophoresis: Migration of Colloidal Particles in Gradients of Solute Concentration". <i>Separation & Purification Reviews</i>. <b>13</b> (1): 67–103. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F03602548408068407">10.1080/03602548408068407</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Separation+%26+Purification+Reviews&rft.atitle=Diffusiophoresis%3A+Migration+of+Colloidal+Particles+in+Gradients+of+Solute+Concentration&rft.volume=13&rft.issue=1&rft.pages=67-103&rft.date=2006&rft_id=info%3Adoi%2F10.1080%2F03602548408068407&rft.aulast=Anderson&rft.aufirst=John+L.&rft.au=Prieve%2C+Dennis+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADiffusiophoresis+and+diffusioosmosis" class="Z3988"></span></li></ul> </div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Diffusiophoresis_and_diffusioosmosis&oldid=1249353526">https://en.wikipedia.org/w/index.php?title=Diffusiophoresis_and_diffusioosmosis&oldid=1249353526</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Colloidal_chemistry" title="Category:Colloidal chemistry">Colloidal chemistry</a></li><li><a href="/wiki/Category:Fluid_mechanics" title="Category:Fluid mechanics">Fluid mechanics</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 4 October 2024, at 13:52<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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