Casimir interaction of rodlike particles in a two-dimensional critical system

We consider the fluctuation-induced interaction of two thin, rodlike particles, or “needles,” immersed in a two-dimensional critical fluid of Ising symmetry right at the critical point. Conformally mapping the plane containing the needles onto a simpler geometry in which the stress tensor is known, we analyze the force and torque between needles of arbitrary length, separation, and orientation. For infinite and semi-infinite needles we utilize the mapping of the plane bounded by the needles onto the half plane, and for two needles of finite length we use the mapping onto an annulus. For semi-infinite and infinite needles the force is expressed in terms of elementary functions, and we also obtain analytical results for the force and torque between needles of finite length with separation much greater than their length. Evaluating formulas in our approach numerically for several needle geometries and surface universality classes, we study the full crossover from small to large values of the separation to length ratio. In these two limits the numerical results agree with results for infinitely long needles and with predictions of the small-particle operator expansion, respectively.

Figure 1

The configuration of the semi-infinite needles I [with end point $z\left(1\right)]$ and II (with end point 0) in the full $z$ plane, shown on the left, is generated by the conformal mapping (3.8) from the upper half $w$ plane with needles I and II on the positive and negative real axis, respectively, shown on the right. The two edges of needles I and II in the $z$ plane and their preimages in the upper half $w$ plane are denoted by i, ii and iv, v, respectively. The neighborhoods of $z=\infty$ denoted by iii and vi correspond to the neighborhoods of $w=\infty$ and $w=0$, respectively, in the upper half $w$ plane. The integration contour ${\mathcal{C}}_{\text{I}}$ enclosing needle I and its preimage $\mathcal{C}$ play a role in the calculation of the force and torque, as discussed in Secs. 2 and 3.

Figure 2

The configuration of two finite needles in the full $z$ plane, shown on the left, is generated by the conformal mapping with derivative (4.2) from the annulus $h<\left|w\right|<1$ shown on the right. The labels 1, 2 and 3, 4 denote the end points $z_1,z_2$ and $z_3,z_4$ of needles I and II in the $z$ plane, respectively, and their preimages $w_1,w_2$ and $w_3,w_4$. The edges of needles I and II in the $z$ plane and their preimages in the $w$ plane are denoted by i, ii and iii, iv, respectively. The point $w=c\equiv C{h}^{1/2}$ in the annulus labeled by v is mapped onto $z=\infty$ by the transformation. The integration contour ${\mathcal{C}}_{\mathrm{I}}$ enclosing needle I and its preimage $\mathcal{C}$ as well as the concentric circle ${\mathcal{C}}_c$ passing through the point $w=c$ play a role in the calculation of the force and torque, as discussed in Secs. 2 and 4 and Appendix pp2-s1a.

Figure 3

Simple needle configurations described in Sec. 4a1, for which the six mapping parameters $C,h,{\phi }_1$,..., ${\phi }_4$ of the conformal mapping are restricted to subspaces of lower dimension. The labels 1, 2 and 3, 4 denote the end points $z_1,z_2$ and $z_3,z_4$ of needles I and II, respectively. Configurations (A)–(D) are oriented as in Sec. 4a1, with the line from the midpoint $z_{\text{II}}$ of needle II to the midpoint $z_{\text{I}}$ of needle I parallel to the $x$ or horizontal axis. Configuration $({\mathrm{D}}^{\prime })$ is the same as (D), apart from a rotation to orient the needles along the $x$ axis. Our results for the force and torque in configurations (A), (B), (C), and $({\mathrm{D}}^{\prime })$, as functions of the needle lengths $D_{\text{I}},D_{\text{II}},D$, the minimum separation $c$, the angle $\alpha$ between the needles, and the distances $W$ and $V$, are shown in Figs. 4, 5, 6, 7, 8.

Figure 4

Component $f_x$ of the force exerted on needle I by needle II for needles of equal length $D_{\text{I}}=D_{\text{II}}=D$ in the symmetric-perpendicular configuration (A) shown in Fig. 3. Here $c=z_{\text{I}}-z_4$ is the distance from the right tip of needle II to the midpoint of needle I. The points indicate the numerical predictions of our exact approach, and the two curves show the asymptotic forms derived in the text for large and small $c/D$. The force component $f_y$ and the torque vanish due to symmetry. For more details, see Sec. 5.

Figure 5

Component $f_x$ of the force exerted on needle I by needle II for needles with separation $c=z_{\text{I}}-z_{\text{II}}$ in the symmetric-parallel configuration (B) in Fig. 3. The results in the left and right columns are for needles of the same length $D_{\text{I}}=D_{\text{II}}=D$ and for needles with different lengths, $D_{\text{I}}=D$ and $D_{\text{II}}=c$, respectively. The points indicate the numerical predictions of our exact approach, and the curves show the asymptotic forms derived in the text for large and small $D/c$. The force component $f_y$ and the torque vanish due to symmetry. For more details, see Sec. 5.

Figure 6

Force $f_x$ and torque $\mathrm{\Theta }$ exerted on needle I by needle II for needles of equal length $D$ in the mirror-symmetric configuration (C) shown in Fig. 3. Here $c=z_2-z_4$ is the distance between the closest points of the needles, and the angle $\alpha$ between them is $\pi /5$. The points indicate the numerical predictions of our exact approach, and the two curves show the asymptotic forms derived in the text for large and small $D/c$. The force component $f_y$ vanishes due to symmetry. For more details, see Sec. 5.

Figure 7

Components $f_x,f_y$ of the force exerted on needle I by needle II for needles of equal length $D$ in the nonsymmetric parallel configuration $({\mathrm{D}}^{\prime })$ shown in Fig. 3. Here $W=r_{1,y}-r_{4,y}\equiv \mathrm{Im}(z_1-z_4)$ is the vertical separation of the needles, and $V=r_{4,x}-r_{1,x}\equiv \mathrm{Re}(z_4-z_1)$ is the relative horizontal displacement. Results are shown for the fixed ratio $V/W=1.4$. The points indicate numerical predictions of our exact approach, and the two curves show the asymptotic forms derived in the text for large and small $D/W$. For more details, see Sec. 5.

Figure 8

Torque $\mathrm{\Theta }$ on needle I for the same nonsymmetric parallel configuration considered in Fig. 7.

Figure 9

Dependence of the force $f_y$ and torque $\mathrm{\Theta }$ on a needle of length $D$ in the upper half plane on the angle $\mathrm{\Phi }$ between the needle and the boundary. The ratio $\tilde{B}=r_{\text{I},y}/D$, where $r_{\text{I},y}$ is the distance of the needle midpoint from the boundary, has the value $\tilde{B}=10$. The points indicate numerical predictions from the exact approach, and the curves show the analytic results (B56) and (B57) of the operator expansion for a distant needle, i.e., for large $\tilde{B}$. The force component $f_x$ vanishes. For more details see Sec. 5.

Figure 10

Same as Fig. 9, except that the ratio $\tilde{B}$ has the value $\tilde{B}=1$ instead of 10. Since the distant needle condition $\tilde{B}\gg 1$ is not satisfied, the numerical predictions from the exact approach (points) deviate significantly from the predictions (curves) of the operator expansion for a distant needle. For more details, see Sec. 5.

Figure 11

Same as Fig. 9 except that $\tilde{B}=\sqrt{3}/4=0.433$. Both $f_y$ and $\mathrm{\Theta }$ diverge as $\mathrm{\Phi }$ approaches $\pi /3$, the angle at which the needle tip touches the boundary. For $\mathrm{\Phi }$ close to $\pi /3$, the numerical predictions (points) for $f_y$ agree with the asymptotic expression (curves) given in the last paragraph of Sec. 5. Like $f_y,\mathrm{\Theta }$ also appears to diverge as ${\left(\frac{\pi }{3}-\mathrm{\Phi }\right)}^{-1}$. The numerical results for $\mathrm{\Theta }$ are compared with fits of the form ${\mathrm{\Theta }}_{\mathrm{fit}}\left(\mathrm{\Phi }\right)=A{(\frac{\pi }{3}-\mathrm{\Phi })}^{-1}$, with $A$ chosen to reproduce the rightmost point in each graph.